1 Introduction and preliminaries
In this paper, we consider a p-biharmonic equation with Hardy potential and negative exponents:
where \(0\in \varOmega \subset \mathbb{R}^{N}\) is a bounded smooth domain with \(1< p<\frac{N}{2}\), \(\Delta ^{2}_{p}u=\Delta (\vert \Delta u\vert ^{p-2} \Delta u)\) is the p-biharmonic operator. \(\lambda >0\) is a parameter, \(0<\mu <\mu _{N,p}=(\frac{(p-1)N(N-2p)}{p^{2}})^{p}\), \(0< q<1\) and \(p-1<\gamma <p^{*}-1\), where \(p^{*}=\frac{Np}{N-2p}\) is called the critical Sobolev exponent. \(f(x)\geq 0\), \(f(x)\not \equiv 0\), \(g(x)\) satisfies the requirement that the set \(\{x\in \varOmega : g(x)>0 \}\) has positive measures, \(\operatorname{supp}f \cap \{x\in \varOmega : g(x)>0 \} \neq \emptyset \) and \(f, g\in C(\overline{\varOmega })\). Biharmonic equations describe the sport of a rigid body and the deformations of an elastic beam. For example, this type of equation provides a model for considering traveling wave in suspension bridges [5, 16, 27, 30, 36]. Various methods and tools have been adopted to deal with singular problems, such that fixed point theorems [14], topological methods [37], Fourier and Laurent transformation [18, 19], monotone iterative methods [21], global bifurcation theory [12], and degree theory [22, 31].
$$ \textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}= f(x)u^{-q}+\lambda g(x)u ^{\gamma } & \text{in } \varOmega \setminus \{0\}, \\ u(x)>0 &\text{in } \varOmega \setminus \{0\}, \\ u=\Delta u=0 &\text{on } \partial \varOmega , \end{cases} $$
(1.1)
In recent years, there was much attention focused on the existence, multiplicity and qualitative properties of solutions for p-biharmonic equations under Dirichlet boundary conditions or Navier boundary conditions with Hardy terms [4, 15, 17, 32, 34]. Xie and Wang [32] studied the following p-biharmonic equation with Dirichlet boundary conditions:
where \(\frac{\partial }{\partial n}\) is the outer normal derivative. By using the variational method, the existence of infinitely many solutions with positive energy levels for (1.2) was established. Huang and Liu [15] considered the following p-biharmonic equation with Navier boundary conditions:
where \(1< p<\frac{N}{2}\). By using invariant sets of gradient flows, the authors proved that (1.3) possesses a sign-changing solution. Furthermore, Yang, Zhang and Liu [34] showed that (1.3) has a positive solution, a negative solution and a sequence of sign-changing solutions when f satisfies appropriate conditions. Bhakta [4] established the qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with Hardy terms.
$$ \textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}= f(x,u) & \text{in } \varOmega , \\ u=\frac{\partial u}{\partial n}=0 & \text{on } \partial \varOmega , \end{cases} $$
(1.2)
$$ \textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}= f(x,u) & \text{in } \varOmega , \\ u=\Delta u=0 & \text{on } \partial \varOmega , \end{cases} $$
(1.3)
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On the other hand, nonlinear biharmonic equations with negative exponents have been studied expensively [1, 6, 8, 13, 20]. Guerra [13] gave a complete description of entire radially symmetric solutions for the following biharmonic equation:
where \(q>1\). Moreover, Cowan et al. [8] dealt with the regularity of the extremal solution of the following fourth order boundary value problems:
Very recently, Ansari, Vaezpour and Hesaaraki [1] considered fourth order elliptic problem with the combinations of Hardy term and negative exponents,
where \(\varOmega \subset \mathbb{R}^{N}\) (\(N\geq 1\)) is a bounded \(C^{4}\)-domain, λ and μ are positive parameters and \(0<\alpha <1\), \(0<\gamma <1\) are constants. Here M, h and k are given continuous functions satisfying suitable hypotheses. By using the Galerkin method and the sharp angle lemma, the authors proved that problem (1.4) has a positive solution for \(0<\mu < (\frac{N(N-4)}{4} ) ^{2}\).
$$ \Delta ^{2} u=-u^{-q}, \qquad u>0 \quad \text{in } \mathbb{R}^{3}, $$
$$ \textstyle\begin{cases} \Delta ^{2}u=\frac{\lambda }{(1-u)^{2}} &\text{in } \varOmega , \\ 0< u< 1 &\text{in } \varOmega , \\ u=\frac{\partial u}{\partial n}=0 &\text{on } \partial \varOmega . \end{cases} $$
$$ \textstyle\begin{cases} \Delta ^{2}u-\lambda M( \Vert \nabla u \Vert ^{2})\Delta u -\frac{\mu }{ \vert x \vert ^{4}}u= \frac{h(x)}{u^{\gamma }}+k(x) u^{\alpha } & \text{in }\varOmega , \\ u=\Delta u=0 & \text{on } \partial \varOmega , \end{cases} $$
(1.4)
We say that \(u\in W:=W^{2,p}(\varOmega )\cap W_{0}^{1,p}(\varOmega )\) is a weak solution of (1.1), if for every \(\varphi \in W\), there holds
$$ \int _{\varOmega } \vert \Delta u \vert ^{p-2}\Delta u \Delta \varphi \,dx- \int _{\varOmega }\frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p-2}u \varphi \,dx= \int _{\varOmega }f(x)u^{-q}\varphi \,dx+\lambda \int _{\varOmega } g(x)u^{\gamma }\varphi \,dx. $$
(1.5)
The following Rellich inequality will be used in this paper:
and it is not achieved [9, 24]. For any \(u\in W\), and \(0<\mu <\mu _{N,p}\). The energy functional corresponding to (1.1) is defined by
For \(\mu \in [0,\mu _{N,p})\), W is equipped with the following norm:
Negative exponent term \(u^{-q}\) implies that \(I_{\lambda ,\mu }\) is not differential on W, therefore, critical point theory cannot be applied to the problem (1.1) directly. We consider the following manifold:
and make the following splitting for \(\mathcal{M}\):
$$ \int _{\varOmega } \vert \Delta u \vert ^{p}\,dx\geq \mu _{N,p} \int _{\varOmega }\frac{ \vert u \vert ^{p}}{ \vert x \vert ^{2p}}\,dx, \quad \forall u\in W, $$
$$ \begin{aligned}[b] I_{\lambda ,\mu }(u)={}&\frac{1}{p} \int _{\varOmega } \biggl( \vert \Delta u \vert ^{p}- \frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p} \biggr)\,dx- \frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \\ &{}- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx. \end{aligned} $$
(1.6)
$$ \Vert u \Vert ^{p}_{\mu }= \int _{\varOmega } \biggl( \vert \Delta u \vert ^{p}- \frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p} \biggr)\,dx. $$
$$ \mathcal{M}= \biggl\{ u\in W: \Vert u \Vert ^{p}_{\mu } = \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx+\lambda \int _{\varOmega } g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} , $$
$$\begin{aligned}& \mathcal{M}^{+}= \biggl\{ u\in \mathcal{M}: (p+q-1 ) \Vert u \Vert ^{p}_{ \mu } >\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} , \end{aligned}$$
(1.7)
$$\begin{aligned}& \mathcal{M}^{0}= \biggl\{ u\in \mathcal{M}: (p+q-1 ) \Vert u \Vert ^{p}_{ \mu } =\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} , \end{aligned}$$
(1.8)
$$\begin{aligned}& \mathcal{M}^{-}= \biggl\{ u\in \mathcal{M}: (p+q-1 ) \Vert u \Vert ^{p}_{ \mu } < \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr\} . \end{aligned}$$
(1.9)
In this paper, we will study the dependence of problem (1.1) on q, γ, f, g and Ω and evaluate the extremal value of λ related to multiplicity of positive solutions for problem (1.1). Our idea comes from [7, 28, 29]. Our results improve and complement previous ones obtained in [23, 25]. Denote \(\Vert u\Vert _{t}^{t}= \int _{\varOmega }\vert u\vert ^{t}\,dx\) and \(D^{2, p}(\mathbb{R}^{N})\) be the closure of \(C_{0}^{\infty }(\mathbb{R}^{N})\) with respect to the norm \((\int _{\mathbb{R}^{N}}\vert \Delta u\vert ^{p} \,dx )^{\frac{1}{p}}\).
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\(\lambda _{1}\) denotes the smallest eigenvalue for
and \(\varphi _{1}\) denotes the corresponding eigenfunction with \(\varphi _{1}>0\) in Ω [3, 10, 26, 33, 35]. The following minimization problem will be useful in the following discussions:
and \(S_{\mu }\) is achieved by a family of functions [4, 11]. Thus, for every \(u\in W\setminus \{0\}\), \(\Vert u\Vert _{p^{*}}\leq \frac{ \Vert u\Vert _{\mu }}{\sqrt[p]{S_{\mu }}}\). Therefore, combining with the Hölder inequality, we deduce that
and
$$ \Delta _{p}^{2}u-\frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p-2}u=\lambda _{1} \vert u \vert ^{p-2}u, \quad x\in \varOmega \setminus \{0\}, u\in W, $$
(1.10)
$$ S_{\mu }=\inf \biggl\{ \int _{\mathbb{R}^{N}} \biggl( \vert \Delta u \vert ^{p}- \frac{\mu }{ \vert x \vert ^{2p}} \vert u \vert ^{p} \biggr)\,dx, u\in D^{2,p} \bigl(\mathbb{R}^{N} \bigr), \int _{\mathbb{R}^{N}} \vert u \vert ^{p^{*}}\,dx=1 \biggr\} >0, $$
(1.11)
$$\begin{aligned}& \begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{\gamma +1}\,dx&\leq \biggl[ \int _{\varOmega } \vert u \vert ^{(\gamma +1)\frac{p^{*}}{\gamma +1}}\,dx \biggr]^{\frac{ \gamma +1}{p^{*}}} \biggl( \int _{\varOmega }1\,dx \biggr)^{\frac{p^{*}-\gamma -1}{p^{*}}} \\ &= \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \Vert u \Vert ^{\gamma +1} _{p^{*}} \\ &\leq \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \biggl( \frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{\gamma +1}, \end{aligned} \end{aligned}$$
(1.12)
$$\begin{aligned}& \begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{1-q}\,dx&\leq \biggl[ \int _{\varOmega } \vert u \vert ^{(1-q)\frac{p^{*}}{1-q}}\,dx \biggr]^{\frac{1-q}{p ^{*}}} \biggl( \int _{\varOmega }1\,dx \biggr)^{\frac{p^{*}-1+q}{p^{*}}} \\ &= \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}} \Vert u \Vert ^{1-q} _{p^{*}} \\ &\leq \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}} \biggl( \frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{1-q}, \end{aligned} \end{aligned}$$
(1.13)
$$ \begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{1-q}\,dx&\leq \biggl[ \int _{\varOmega } \vert u \vert ^{(1-q)\frac{\gamma +1}{1-q}}\,dx \biggr]^{\frac{1-q}{ \gamma +1}} \biggl( \int _{\varOmega }1\,dx \biggr)^{\frac{\gamma +q}{\gamma +1}} \\ &= \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}} \Vert u \Vert ^{1-q} _{\gamma +1}. \end{aligned} $$
(1.14)
Our main results are stated in the following theorems.
Theorem 1.1
Assume that
\(\lambda \in (0,\varLambda )\), where
Then problem (1.1) admits at least two solutions
\(u_{0}\in \mathcal{M}^{+}\), \(U_{0}\in \mathcal{M}^{-}\), with
\(\Vert U_{0}\Vert _{\mu }> \Vert u_{0}\Vert _{\mu }\).
$$\begin{aligned} \begin{aligned}[b] \varLambda \geq{}& T_{\mu }= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl( \frac{S_{\mu }}{ \vert \varOmega \vert ^{ \frac{p}{N}}} \biggr)^{\frac{q+\gamma }{p+q-1}} \\ >{}&0. \end{aligned} \end{aligned}$$
(1.15)
Corollary 1.2
Let
\(U_{\lambda , \mu ,\varepsilon } \in \mathcal{M}^{-}\)
be the solution of problem (1.1) with
\(\gamma = \varepsilon +p-1\), where
\(\lambda \in (0,T_{\mu })\). Then
with
$$\begin{aligned}& \Vert U_{\lambda , \mu ,\varepsilon } \Vert _{\mu }>C_{\mu ,\varepsilon } \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\varepsilon }} \end{aligned}$$
$$ C_{\mu ,\varepsilon }= \vert \varOmega \vert ^{\frac{1}{p}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl(1+ \frac{p+q-1}{\varepsilon } \biggr)^{ \frac{1}{p+q-1}} \biggl(\frac{ \vert \varOmega \vert ^{\frac{2}{N}}}{\sqrt[p]{S _{\mu }}} \biggr)^{\frac{1-q}{p+q-1}} \rightarrow \infty , \quad \textit{as }\varepsilon \rightarrow 0. $$
(1.16)
Theorem 1.3
There exists
\(\lambda ^{*} =\lambda ^{*} (N, \varOmega , \mu , q, \gamma )>0\)
such that problem (1.1) with
\(f=g=1\)
admits at least a positive solution for every
\(0<\lambda <\lambda ^{*}\)
and has no solution for every
\(\lambda >\lambda ^{*}\).
2 Some lemmas
Lemma 2.1
Assume that
\(\lambda \in (0,T_{\mu })\), where
\(T_{\mu }\)
is defined in (1.15). Then
\(\mathcal{M}^{\pm }\neq \emptyset \)
and
\(\mathcal{M}^{0}=\{0\}\).
Proof
(i) We can choose \(u^{*}\in \mathcal{M}\setminus \{0 \}\) such that \(\int _{\varOmega }f(x)\vert u^{*}\vert ^{1-q}\,dx>0\) and \(\int _{\varOmega }g(x) \vert u^{*}\vert ^{\gamma +1}\,dx>0\) from the conditions imposed on f and g. Denote
Note that \(\varphi '_{\mu }(t)=(p-1-\gamma )t^{p-2-\gamma }\Vert u^{*}\Vert _{\mu }^{p}+(q+\gamma ) t^{-1-q-\gamma } \int _{\varOmega }f(x)\vert u^{*}\vert ^{1-q}\,dx\). Let \(\varphi '_{\mu }(t)=0\), we have
It is easy to check that \(\varphi _{\mu }(t)\rightarrow -\infty \) as \(t\rightarrow 0^{+}\) and \(\varphi _{\mu }(t)\rightarrow -\lambda \int _{\varOmega } g(x)\vert u^{*}\vert ^{\gamma +1}\,dx<0\) as \(t\rightarrow \infty \). Furthermore, \(\varphi _{\mu }(t)\) attains its maximum at \(t_{\max }\). By (1.12) and (1.13), we obtain
When \(A(\mu ,\lambda )=0\), we get
where we use the following two equalities:
and
In turn, this is also true. Hence \(A(\mu ,\lambda )=0\) if and only if \(\lambda =T_{\mu }\). Thus for \(\lambda \in (0,T_{\mu })\), we have \(A(\mu ,\lambda )>0\). Moreover, by (2.2), we derive that \(\varphi _{ \mu }(t_{\max })>0\). Consequently, there exist two numbers \(t_{\mu } ^{-}\) and \(t_{\mu }^{+}\) such that \(0< t_{\mu }^{-}< t_{\max }< t_{ \mu }^{+}\), and
It follows that \(t_{\mu }^{-}u^{*}\in \mathcal{M}^{+}\), and \(t_{\mu }^{+}u^{*}\in \mathcal{M}^{-}\). In fact, if \(\varphi _{\mu }(t)=0\), then
namely
Hence \(tu\in \mathcal{M}\). Furthermore, if \(\varphi '_{\mu }(t)>0\), then
That is
i.e.,
Note that \(tu\in \mathcal{M}\), we have
Thus \(tu\in \mathcal{M}^{+}\). By a similar argument, if \(\varphi _{ \mu }(t)=0\) and \(\varphi '_{\mu }(t)<0\), then \(tu\in \mathcal{M}^{-}\). Therefore, both \(\mathcal{M}^{+}\) and \(\mathcal{M}^{-}\) are non-empty sets for every \(\lambda \in (0,T_{\mu })\).
$$\begin{aligned} \varphi _{\mu }(t) :=&\frac{1}{t^{\gamma }} \frac{d}{dt}I_{\lambda , \mu } \bigl(tu^{*} \bigr) \\ =&t^{p-1-\gamma } \bigl\Vert u^{*} \bigr\Vert _{\mu }^{p}-t^{-q- \gamma } \int _{\varOmega }f(x) \bigl\vert u^{*} \bigr\vert ^{1-q}\,dx- \lambda \int _{\varOmega }g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx, \quad t>0. \end{aligned}$$
$$ t:=t_{\max }= \biggl[\frac{(\gamma -p+1) \Vert u^{*} \Vert _{\mu }^{p}}{(q+ \gamma )\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx} \biggr]^{\frac{1}{1-q-p}}. $$
(2.1)
$$\begin{aligned}& \varphi _{\mu }(t_{\max }) \\& \quad = \biggl[\frac{(\gamma -p+1) \Vert u^{*} \Vert _{\mu }^{p}}{(q+\gamma ) \int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx} \biggr]^{\frac{p-\gamma -1}{1-q-p}} \bigl\Vert u^{*} \bigr\Vert _{\mu }^{p} \\& \quad\quad{} - \biggl[\frac{(\gamma -p+1) \Vert u^{*} \Vert _{\mu }^{p}}{(q+ \gamma )\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx} \biggr]^{ \frac{-q-\gamma }{1-q-p}} \int _{\varOmega }f(x) \bigl\vert u^{*} \bigr\vert ^{1-q}\,dx \\& \quad \quad {} -\lambda \int _{\varOmega } g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx \\& \quad = \biggl(\frac{\gamma -p+1}{q+\gamma } \biggr)^{ \frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u^{*} \Vert _{\mu }^{p})^{\frac{- \gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}} \\& \quad\quad{} - \biggl( \frac{\gamma -p+1}{q+\gamma } \biggr) ^{\frac{-q- \gamma }{1-q-p}}\frac{( \Vert u^{*} \Vert _{\mu }^{p})^{ \frac{-\gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}} \\& \quad \quad {} -\lambda \int _{\varOmega } g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx \\& \quad = \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u^{*} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u^{*} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}}-\lambda \int _{\varOmega }g(x) \bigl\vert u^{*} \bigr\vert ^{\gamma +1}\,dx \\& \quad \geq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u^{*} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{ [ \Vert f \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}(\frac{ \Vert u^{*} \Vert _{\mu }}{\sqrt[p]{S_{ \mu }}})^{1-q} ]^{\frac{p-\gamma -1}{1-q-p}}} \\& \quad \quad {} -\lambda \Vert g \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \biggl( \frac{ \Vert u^{*} \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{\gamma +1} \\& \quad = \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \frac{(\sqrt[p]{S_{ \mu }})^{\frac{(1-q)(p-\gamma -1)}{1-q-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p ^{*}}\frac{p-\gamma -1}{1-q-p}}} \bigl\Vert u^{*} \bigr\Vert _{\mu } ^{\gamma +1} \\& \quad \quad {} -\lambda \Vert g \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p ^{*}}} \biggl(\frac{ \Vert u^{*} \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr) ^{\gamma +1} \\& \quad = \biggl[ \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \frac{(\sqrt[p]{S_{ \mu }})^{\frac{(1-q)(p-\gamma -1)}{1-q-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p ^{*}}\frac{p-\gamma -1}{1-q-p}}} \\& \quad \quad {}-\lambda \Vert g \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}}}{({\sqrt[p]{S_{\mu }}})^{ \gamma +1}} \biggr] \bigl\Vert u^{*} \bigr\Vert _{\mu }^{\gamma +1} \\& \quad :=A(\mu ,\lambda ) \bigl\Vert u^{*} \bigr\Vert _{\mu }^{\gamma +1} \\& \quad >0. \end{aligned}$$
(2.2)
$$ \begin{aligned} \lambda &= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl( \frac{\gamma -p+1}{q+ \gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \frac{(\sqrt[p]{S_{\mu }})^{\frac{(1-q)(p-\gamma -1)}{1-q-p}+\gamma +1}}{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}\frac{p- \gamma -1}{1-q-p} +\frac{p^{*}-\gamma -1}{p^{*}}}} \\ &= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl( \frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl[\frac{S_{\mu }}{ \vert \varOmega \vert ^{\frac{2p}{N}}} \biggr] ^{\frac{q+\gamma }{p+q-1}} =T_{\mu }, \end{aligned} $$
$$ \frac{(1-q)(p-\gamma -1)}{1-q-p}+\gamma +1= \frac{p(q+\gamma )}{q+p-1}, $$
$$ \frac{(p^{*}-1+q)(p-\gamma -1)}{p^{*}(1-q-p)}+\frac{p^{*}-\gamma -1}{p ^{*}} =\frac{2p(q+\gamma )}{N(q+p-1)}. $$
$$ \varphi _{\mu } \bigl(t^{-}_{\mu } \bigr)=0=\varphi _{\mu } \bigl(t^{+}_{\mu } \bigr), \quad\quad \varphi '_{\mu } \bigl(t^{-}_{\mu } \bigr)>0> \varphi '_{\mu } \bigl(t^{+}_{\mu } \bigr). $$
$$ \varphi _{\mu }(t)=t^{p-1-\gamma } \Vert u \Vert _{\mu }^{p}-t^{-q-\gamma } \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx-\lambda \int _{\varOmega } g(x) \vert u \vert ^{\gamma +1}\,dx=0, $$
$$ \Vert tu \Vert _{\mu }^{p}= \int _{\varOmega }f(x) \vert tu \vert ^{1-q}\,dx +\lambda \int _{\varOmega }g(x) \vert tu \vert ^{\gamma +1}\,dx. $$
$$ (p-1-\gamma )t^{p-2-\gamma } \Vert u \Vert _{\mu }^{p} +(q+ \gamma )t^{-1-q- \gamma } \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx>0. $$
$$ (p-1-\gamma ) \Vert tu \Vert _{\mu }^{p}+(q+\gamma ) \int _{\varOmega }f(x) \vert tu \vert ^{1-q}\,dx>0, $$
$$ (p-1-\gamma ) \Vert tu \Vert _{\mu }^{p}+(q+\gamma ) \biggl[ \Vert tu \Vert _{\mu }^{p} - \lambda \int _{\varOmega }g(x) \vert tu \vert ^{\gamma +1}\,dx \biggr]>0. $$
$$ (p+q-1) \Vert tu \Vert _{\mu }^{p}-\lambda (q+\gamma ) \int _{\varOmega }g(x) \vert tu \vert ^{\gamma +1}\,dx>0. $$
(ii) We claim that \(\mathcal{M}^{0}=\{0\}\). Otherwise, we suppose that there exists \(u_{*}\in \mathcal{M}^{0}\) and \(u_{*}\neq 0\). Since \(u_{*}\in \mathcal{M}^{0}\), we have
moreover
For \(\lambda \in (0,T_{\mu })\) and \(u_{*}\neq 0\), combining with (2.2), we deduce that
which is a contradiction, Thus \(u_{*}=0\). That is, \(\mathcal{M}^{0}= \{0\}\). □
$$\begin{aligned}& (p+q-1 ) \Vert u_{*} \Vert ^{p}_{\mu }=\lambda ( \gamma +q ) \int _{\varOmega }g(x) \vert u_{*} \vert ^{\gamma +1} \,dx, \end{aligned}$$
$$ \begin{aligned} 0&= \Vert u_{*} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{*}^{1-q}\,dx -\lambda \int _{\varOmega }g(x)u_{*}^{\gamma +1}\,dx \\ &= \Vert u_{*} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{*}^{1-q}\,dx - \frac{p+q-1}{\gamma +q} \Vert u_{*} \Vert _{\mu }^{p} \\ &=\frac{\gamma -p+1}{\gamma +q} \Vert u_{*} \Vert _{\mu } ^{p}- \int _{\varOmega } f(x)u_{*}^{1-q}\,dx. \end{aligned} $$
$$ \begin{aligned} 0&< A(\mu ,\lambda ) \Vert u_{*} \Vert _{\mu }^{\gamma +1} \\ &\leq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u_{*} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\frac{\gamma -p+1}{q+\gamma } \Vert u_{*} \Vert _{\mu }^{p})^{\frac{p-\gamma -1}{1-q-p}}} - \biggl(\frac{q+p-1}{q+ \gamma } \biggr) \Vert u_{*} \Vert _{\mu }^{p}=0, \end{aligned} $$
The gap structure in \(\mathcal{M}\) is embodied in the following lemma.
Lemma 2.2
Assume that
\(\lambda \in (0,T_{\mu })\), then
where
$$\begin{aligned}& \Vert U \Vert _{\mu }>M_{\mu }(\lambda )>M_{\mu ,0}> \Vert u \Vert _{\mu }, \\& \Vert U \Vert _{\gamma +1}>N_{\mu }(\lambda )>N_{\mu ,0}> \Vert u \Vert _{\gamma +1}, \quad \forall u\in \mathcal{M}^{+}, U\in \mathcal{M}^{-}, \end{aligned}$$
$$\begin{aligned}& M_{\mu ,0}= \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \biggr] ^{\frac{1}{p+q-1}}, \\& M_{\mu }(\lambda )= \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }})^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}}, \\& N_{\mu ,0}= \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}+\frac{(p^{*}-1-\gamma )p}{p^{*}( \gamma +1)}}}{S_{\mu }} \biggr]^{\frac{1}{p+q-1}}, \\& N_{\mu }(\lambda )= \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}}. \end{aligned}$$
Proof
If \(u\in \mathcal{M}^{+}\subset \mathcal{M}\), then
We obtain from (1.13) that
which leads to
By (1.12) and (1.14), we have
which implies that
If \(U\in \mathcal{M}^{-}\subset \mathcal{M}\), combining with (1.12), we derive that
which leads to
Furthermore
which means that
Therefore
where we have used the following facts:
and
Similarly
where we have applied the following equalities:
and
Consequently, \(M_{\mu }(\lambda )=M_{\mu ,0}\) if and only if \(\lambda =T_{\mu }\) and \(N_{\mu }(\lambda )=N_{\mu ,0}\) if and only if \(\lambda =T_{\mu }\) respectively. This completes the proof of Lemma 2.2. □
$$ \begin{aligned} 0&< (p+q-1 ) \Vert u \Vert ^{p}_{\mu }- \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &= (p+q-1 ) \Vert u \Vert ^{p}_{\mu }- (\gamma +q ) \biggl[ \Vert u \Vert ^{p}_{\mu } - \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \biggr] \\ &= (p-\gamma -1 ) \Vert u \Vert ^{p}_{\mu }+ (\gamma +q ) \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx. \end{aligned} $$
$$ \begin{aligned} (\gamma -p+1 ) \Vert u \Vert ^{p}_{\mu }&< (\gamma +q ) \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \\ &\leq (\gamma +q ) \Vert f \Vert _{\infty } \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}} \biggl( \frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S _{\mu }}} \biggr)^{1-q}, \end{aligned} $$
$$\begin{aligned}& \Vert u \Vert _{\mu }< \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \biggr] ^{\frac{1}{p+q-1}}=M_{\mu ,0}. \end{aligned}$$
$$\begin{aligned} & (\gamma -p+1 ) \Vert u \Vert ^{p}_{\gamma +1} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \\ &\quad \leq (\gamma -p+1 )\frac{S_{\mu }}{ \vert \varOmega \vert ^{p\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1)}}} \biggl[ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}} \biggl(\frac{ \Vert u \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr) ^{\gamma +1} \biggr] ^{\frac{p}{\gamma +1}} \\ &\quad = (\gamma -p+1 ) \Vert u \Vert ^{p}_{\mu } \\ &\quad < (\gamma +q ) \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx \\ &\quad \leq (\gamma +q ) \Vert f \Vert _{\infty } \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}} \Vert u \Vert ^{1-q} _{\gamma +1}, \end{aligned}$$
$$\begin{aligned}& \Vert u \Vert _{\gamma +1}< \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}+\frac{(p^{*}-1-\gamma )p}{p^{*}(\gamma +1)}}}{S_{\mu }} \biggr]^{\frac{1}{p+q-1}}=N_{\mu ,0}. \end{aligned}$$
$$ \begin{aligned} (p+q-1 ) \Vert U \Vert ^{p}_{\mu }&< \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U \vert ^{\gamma +1}\,dx \\ &\leq \lambda (\gamma +q ) \Vert g \Vert _{ \infty } \vert \varOmega \vert ^{\frac{p^{*}-\gamma -1}{p^{*}}} \biggl( \frac{ \Vert U \Vert _{\mu }}{\sqrt[p]{S_{\mu }}} \biggr)^{\gamma +1}, \end{aligned} $$
$$\begin{aligned}& \Vert U \Vert _{\mu }> \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }})^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}}=M_{\mu }( \lambda ). \end{aligned}$$
$$\begin{aligned}& (p+q-1 ) \Vert U \Vert ^{p}_{\gamma +1} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \\& \quad \leq (p+q-1 )\frac{S_{\mu }}{ \vert \varOmega \vert ^{p\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1)}}} \biggl[ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}} \biggl(\frac{ \Vert U \Vert _{\mu }}{\sqrt[p]{S_{ \mu }}} \biggr) \biggr]^{\frac{p}{\gamma +1}} \\& \quad = (p+q-1 ) \Vert U \Vert ^{p}_{\mu } \\& \quad < \lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U \vert ^{\gamma +1}\,dx \\& \quad \leq \lambda (\gamma +q ) \Vert g \Vert _{ \infty } \Vert U \Vert ^{\gamma +1} _{\gamma +1}, \end{aligned}$$
$$\begin{aligned}& \Vert U \Vert _{\gamma +1}> \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}}=N_{\mu }(\lambda ). \end{aligned}$$
$$\begin{aligned}& \lambda =T_{\mu }= \biggl( \frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl(\frac{S_{\mu }}{ \vert \varOmega \vert ^{ \frac{2p}{N}}} \biggr)^{\frac{q+\gamma }{p+q-1}} \\& \begin{aligned} \Leftrightarrow \quad M_{\mu }(\lambda )&= \biggl[\frac{p+q-1}{\lambda ( \gamma +q)} \frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }})^{ \gamma +1}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{ \gamma +1-p}} \\ & =\lambda ^{-\frac{1}{\gamma +1-p}} \biggl[\frac{p+q-1}{\gamma +q}\frac{1}{ \Vert g \Vert _{\infty }} \frac{(\sqrt[p]{S_{\mu }}) ^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}} \\ & = \biggl(\frac{q+\gamma }{q+p-1} \biggr) ^{ \frac{1}{\gamma +1-p}} \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr) ^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}} \bigl( \Vert g \Vert _{\infty } \bigr)^{\frac{1}{\gamma +1-p}} \\ & \quad{}\times \frac{ \vert \varOmega \vert ^{\frac{2p}{N}\frac{q+\gamma }{(q+p-1)(\gamma +1-p)}}}{(S_{\mu })^{\frac{q+ \gamma }{(p+q-1) (\gamma +1-p)}}} \biggl[\frac{p+q-1}{\gamma +q}\frac{1}{ \Vert g \Vert _{\infty }}\frac{(\sqrt[p]{S _{\mu }}) ^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p^{*}}}} \biggr] ^{\frac{1}{\gamma +1-p}} \\ & = \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr)^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}}\frac{ \vert \varOmega \vert ^{ \frac{2p}{N} \frac{q+\gamma }{{(\gamma -p+1)(p+q-1)}}-\frac{p^{*}-1- \gamma }{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{p \frac{q+\gamma }{(\gamma -p+1)(p+q-1)} -\frac{\gamma +1}{\gamma +1-p}}} \\ & = \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \biggr] ^{ \frac{1}{p+q-1}}=M_{\mu ,0}, \end{aligned} \end{aligned}$$
$$ \begin{aligned} &\frac{2p}{N}\frac{q+\gamma }{{(\gamma -p+1)(p+q-1)}}- \frac{p^{*}-1- \gamma }{p^{*} (\gamma -p+1)} \\ &\quad =\frac{2p(p^{*}-p)}{2pp^{*}}\frac{q+\gamma }{ {(\gamma -p+1)(p+q-1)}} -\frac{p^{*}-1-\gamma }{p^{*}(\gamma -p+1)} \\ &\quad = \frac{(\gamma -p+1)(p^{*}+q-1)}{p^{*}(\gamma -p+1)(p+q-1)}, \end{aligned} $$
$$ p\frac{q+\gamma }{(\gamma -p+1)(p+q-1)}- \frac{\gamma +1}{\gamma +1-p} =\frac{pq-q\gamma +\gamma -p-q+1}{( \gamma -p+1)(p+q-1)}= \frac{1-q}{p+q-1}. $$
$$\begin{aligned}& \lambda =T_{\mu }= \biggl( \frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert f \Vert _{\infty }} \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) \biggl[\frac{S_{\mu }}{ \vert \varOmega \vert ^{ \frac{2p}{N}}} \biggr]^{\frac{q+\gamma }{p+q-1}}. \\& \Leftrightarrow \quad N_{\mu }(\lambda )= \biggl[\frac{p+q-1}{\lambda ( \gamma +q)} \frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} =\lambda ^{-\frac{1}{\gamma +1-p}} \biggl[\frac{p+q-1}{\lambda ( \gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} = \biggl(\frac{q+\gamma }{q+p-1} \biggr)^{ \frac{1}{\gamma +1-p}} \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr) ^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}} \bigl( \Vert g \Vert _{\infty } \bigr)^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} \quad{}\times \frac{ \vert \varOmega \vert ^{\frac{2p}{N}\frac{q+\gamma }{(q+p-1)(\gamma +1-p)}}}{(S_{\mu })^{\frac{q+ \gamma }{(p+q-1)(\gamma +1-p)}}} \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert W \Vert _{ \infty }} \frac{S_{\mu }}{ \vert \varOmega \vert ^{p(\frac{p^{*}-1-\gamma }{p^{*}( \gamma +1)})}} \biggr] ^{\frac{1}{\gamma +1-p}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} = \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr)^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr)^{\frac{1}{p+q-1}}\frac{ \vert \varOmega \vert ^{ \frac{2p}{N} \frac{q+\gamma }{(\gamma -p+1)(p+q-1)}-p\frac{p^{*}-1- \gamma }{p^{*}(\gamma +1) (\gamma +1-p)}}}{(S_{\mu })^{\frac{q+\gamma }{(\gamma -p+1)(p+q-1)} -\frac{1}{\gamma +1-p}}} \\& \hphantom{\Leftrightarrow \quad N_{\mu }(\lambda )} = \biggl[\frac{\gamma +q}{\gamma -p+1} \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{\gamma +q}{\gamma +1}+\frac{(p^{*}-1-\gamma )p}{p^{*}(\gamma +1)}}}{S_{\mu }} \biggr]^{\frac{1}{p+q-1}}=N_{\mu ,0}, \end{aligned}$$
$$ \begin{aligned} &\frac{2p}{N}\frac{q+\gamma }{(\gamma -p+1)(p+q-1)}-p \frac{p^{*}-1- \gamma }{p^{*}(\gamma +1)(\gamma +1-p)} \\ &\quad =\frac{2p(p^{*}-p)}{2pp^{*}}\frac{q+\gamma }{ {(\gamma -p+1)(p+q-1)}} -\frac{p^{*}-1-\gamma }{p^{*}(\gamma -p+1)} \\ &\quad =\frac{\gamma +q}{\gamma +1}+p\frac{p^{*}-1- \gamma }{p^{*}(\gamma +1)}, \end{aligned} $$
$$ \frac{q+\gamma }{(\gamma -p+1)(p+q-1)}-\frac{1}{\gamma +1-p} =\frac{q+ \gamma -(p+q-1)}{(\gamma -p+1)(p+q-1)}= \frac{1}{p+q-1}. $$
Lemma 2.3
Assume that
\(\lambda \in (0,T_{\mu })\). Then
\(\mathcal{M}^{-}\)
is a closed set in
W-topology.
Proof
We choose a sequence \(\{U_{n}\}\) such that \(\{U_{n}\} \subset \mathcal{M}^{-}\) and \(U_{n}\rightarrow U_{0}\) with \(U_{0} \in W\). Then
and
Hence \(U_{0} \in \mathcal{M}^{-} \cup \mathcal{M}^{0}\). By Lemma 2.2, we have
that is, \(U_{0}\neq 0\). Combining with Lemma 2.1, we obtain \(U_{0}\notin \mathcal{M}^{0}\). Thus \(U_{0}\in \mathcal{M}^{-}\). Therefore \(\mathcal{M}^{-}\) is a closed set in W-topology for every \(\lambda \in (0,T_{\mu })\). □
$$ \begin{aligned} \Vert U_{0} \Vert _{\mu }^{p}&= \lim_{n\rightarrow \infty } \Vert U_{n} \Vert _{ \mu }^{p} \\ &=\lim_{n\rightarrow \infty } \biggl[ \int _{\varOmega }f(x) \vert U_{n} \vert ^{1-q}\,dx+ \lambda \int _{\varOmega } g(x) \vert U_{n} \vert ^{\gamma +1}\,dx \biggr] \\ &= \int _{\varOmega }f(x) \vert U_{0} \vert ^{1-q}\,dx+ \lambda \int _{\varOmega } g(x) \vert U_{0} \vert ^{\gamma +1}\,dx, \end{aligned} $$
$$ \begin{aligned} & (p+q-1 ) \Vert U_{0} \Vert ^{p}_{\mu }-\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U_{0} \vert ^{\gamma +1}\,dx \\ &\quad =\lim_{n\rightarrow \infty } \biggl[ (p+q-1 ) \Vert U _{n} \Vert ^{p}_{\mu }-\lambda (\gamma +q ) \int _{\varOmega }g(x) \vert U_{n} \vert ^{\gamma +1}\,dx \biggr]\leq 0. \end{aligned} $$
$$ \Vert U_{0} \Vert _{\mu }=\lim_{n\rightarrow \infty } \Vert U_{n} \Vert _{\mu } \geq M_{\mu ,0}>0, $$
Lemma 2.4
For
\(u\in \mathcal{M}^{\pm }\), there exist a number
\(\varepsilon >0\)
and a continuous function
\(\widetilde{g}(h)>0\)
with
\(h\in W\)
and
\(\Vert h\Vert <\varepsilon \)
such that
$$ \widetilde{g}(0)=1, \quad\quad \widetilde{g}(h) (u+h)\in \mathcal{M}^{\pm }, \quad \forall h\in W, \Vert h \Vert < \varepsilon . $$
Proof
We only prove the case that \(\mathcal{M}^{+}\). Define a function \(\widetilde{F}: W\times \mathbb{R}^{+}\rightarrow \mathbb{R}\) by:
Note that \(u\in \mathcal{M}^{+}\), we obtain
and
At \((0,1)\), using the implicit function theorem, we know that there exists \(\overline{\varepsilon }>0\) such that for \(h\in W\) and \(\Vert h\Vert <\overline{\varepsilon }\), the equation \(\widetilde{F}(h,s)=0\) has a unique continuous solution \(s=\widetilde{g}(h)>0\). Hence \(\widetilde{g}(0)=1\) and
i.e.,
and
together with (2.3), these imply that we can choose \(\varepsilon >0\) small enough (\(\varepsilon <\overline{\varepsilon }\)) such that for every \(h\in W\) and \(\Vert h\Vert <\varepsilon \)
that is,
This completes the proof of Lemma 2.3. □
$$\begin{aligned}& \widetilde{F}(h,s)=s^{p-1+q} \Vert u+h \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert u+h \vert ^{1-q}\,dx-\lambda s^{\gamma +q} \int _{\varOmega }g(x) \vert u+h \vert ^{\gamma +1}\,dx. \end{aligned}$$
$$\begin{aligned}& \widetilde{F}(0,1)= \Vert u \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx -\lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx=0, \end{aligned}$$
$$ \widetilde{F}_{s}(0,1)=(p-1+q) \Vert u \Vert _{\mu }^{p}-(q+\gamma )\lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx>0. $$
(2.3)
$$ \begin{aligned} 0&=\widetilde{g}(h)^{p-1+q} \Vert u+h \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert u+h \vert ^{1-q}\,dx-\lambda \widetilde{g}(h)^{\gamma +q} \int _{\varOmega } g(x) \vert u+h \vert ^{\gamma +1}\,dx \\ &=\frac{ \Vert \widetilde{g}(h)(u+h) \Vert _{\mu }^{p}- \int _{\varOmega }f(x) \vert \widetilde{g}(h)(u+h) \vert ^{1-q}\,dx -\lambda \int _{\varOmega }g(x) \vert \widetilde{g}(h)(u+h) \vert ^{\gamma +1}\,dx}{ \widetilde{g}(h)^{1-q}}, \end{aligned} $$
$$\begin{aligned}& \widetilde{g}(h) (u+h)\in \mathcal{M}, \quad \forall h\in W, \Vert h \Vert < \overline{ \varepsilon }, \end{aligned}$$
$$ \begin{aligned} \widetilde{F}_{s} \bigl(h, \widetilde{g}(h) \bigr)&=(p-1+q)\widetilde{g}(h)^{p+q-2} \Vert u+h \Vert _{\mu }^{p}-(q+ \gamma )\lambda \widetilde{g}(h)^{\gamma +q-1} \int _{\varOmega } g(x) \vert u+h \vert ^{\gamma +1}\,dx \\ &=\frac{(p-1+q) \Vert \widetilde{g}(h)(u+h) \Vert _{\mu }^{p}-(q+\gamma ) \lambda \int _{\varOmega } g(x) \vert \widetilde{g}(h)(u+h) \vert ^{\gamma +1}\,dx}{ \widetilde{g}^{2-q}(h)}, \end{aligned} $$
$$ (p-1+q) \bigl\Vert \widetilde{g}(h) (u+h) \bigr\Vert _{\mu }^{p}-(q+ \gamma )\lambda \int _{\varOmega } g(x) \bigl\vert \widetilde{g}(h) (u+h) \bigr\vert ^{\gamma +1}\,dx>0, $$
$$ \widetilde{g}(h) (u+h)\in \mathcal{M}^{+}, \quad \forall h\in W, \Vert h \Vert < \varepsilon . $$
3 Proof of Theorem 1.1
For every \(u\in \mathcal{M}\), by (1.13), we have
Let
We have
Since \(\mathcal{K}''(\Vert u\Vert _{\mu })>0\) for all \(\Vert u\Vert _{\mu }>0\) with \(\mathcal{K}(\Vert u\Vert _{\mu })\rightarrow 0\) as \(\Vert u\Vert _{\mu }\rightarrow 0\) and \(\mathcal{K}(\Vert u\Vert _{\mu })\rightarrow \infty \) as \(\Vert u\Vert _{ \mu }\rightarrow \infty \). Therefore \(\mathcal{K}(u)\) attains its minimum at \((\Vert u\Vert _{\mu })_{\min }\), and
By (3.1), we deduce that
namely, \(I_{\lambda ,\mu }(u)\) is coercive on \(\mathcal{M}\). Combining with (3.1), we have
Thus \(I_{\lambda ,\mu }(u)\) is bounded below on \(\mathcal{M}\). According to Lemma 2.3, if \(\lambda \in (0,T_{\mu })\), then \(\mathcal{M}^{+} \cup \mathcal{M}^{0}\) and \(\mathcal{M}^{-}\) are two closed sets in \(\mathcal{M}\). Therefore, we apply the Ekeland variational principle [2] to derive a minimizing sequence \(\{u_{n}\}\subset \mathcal{M}^{+}\cup \mathcal{M}^{0}\) satisfying:
$$\begin{aligned} I_{\lambda ,\mu }(u)&=\frac{1}{p} \Vert u \Vert _{\mu }^{p}-\frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx - \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &=\frac{1}{p} \Vert u \Vert _{\mu }^{p}- \frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx- \frac{1}{\gamma +1} \biggl[ \Vert u \Vert _{\mu } ^{p}- \int _{\varOmega }f(x)u^{1-q}\,dx \biggr] \\ &= \biggl(\frac{1}{p}- \frac{1}{\gamma +1} \biggr) \Vert u \Vert _{\mu }^{p}- \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) \int _{\varOmega }f(x)u^{1-q}\,dx \\ &\geq \biggl(\frac{1}{p}-\frac{1}{\gamma +1} \biggr) \Vert u \Vert _{\mu } ^{p}- \biggl(\frac{1}{1-q} -\frac{1}{\gamma +1} \biggr) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \Vert u \Vert _{\mu }^{1-q} \\ &:=\mathcal{K}\bigl( \Vert u \Vert _{\mu }\bigr). \end{aligned}$$
(3.1)
$$ \mathcal{K}'\bigl( \Vert u \Vert _{\mu }\bigr)= \biggl(1- \frac{p}{\gamma +1} \biggr) \Vert u \Vert _{\mu }^{p-1}- \biggl(1- \frac{1-q}{\gamma +1} \biggr) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}} \Vert u \Vert _{\mu }^{-q}=0. $$
$$\Vert u \Vert _{\mu }:=\bigl( \Vert u \Vert _{\mu } \bigr)_{\min }= \biggl[\frac{(1-\frac{1-q}{ \gamma +1}) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S _{\mu }}) ^{1-q}}}{1-\frac{p}{\gamma +1}} \biggr]^{\frac{1}{p+q-1}}. $$
$$ \begin{aligned} \mathcal{K} \bigl(\bigl( \Vert u \Vert _{\mu }\bigr)_{\min } \bigr)&= \biggl(\frac{1}{p}- \frac{1}{ \gamma +1} \biggr) \biggl[ \frac{(1-\frac{1-q}{\gamma +1}) \Vert f \Vert _{ \infty }\frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{ \mu }})^{1-q}}}{1-\frac{p}{\gamma +1}} \biggr]^{\frac{p}{p+q-1}} \\ &\quad {} - \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) \Vert f \Vert _{\infty }\frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S _{\mu }})^{1-q}} \biggl[\frac{(1-\frac{1-q}{\gamma +1}) \Vert f \Vert _{\infty } \frac{ \vert \varOmega \vert ^{\frac{p^{*}-1+q}{p^{*}}}}{(\sqrt[p]{S_{\mu }})^{1-q}}}{1-\frac{p}{ \gamma +1}} \biggr] ^{\frac{1-q}{p+q-1}}. \end{aligned} $$
$$\begin{aligned}& \lim_{ \Vert u \Vert _{\mu }\rightarrow \infty }I_{\lambda ,\mu }(u) \geq \lim_{ \Vert u \Vert _{\mu }\rightarrow \infty } \mathcal{K}\bigl( \Vert u \Vert _{\mu }\bigr)=\infty , \end{aligned}$$
$$ I_{\lambda ,\mu }(u)\geq \mathcal{K}(u)\geq \mathcal{K} \bigl( \bigl( \Vert u \Vert _{ \mu }\bigr)_{\min } \bigr). $$
(3.2)
$$\begin{aligned}& (\mathrm{i}) \quad I_{\lambda ,\mu }(u_{n})< \inf _{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u)+ \frac{1}{n}; \\& (\mathrm{ii}) \quad I_{\lambda ,\mu }(u)\geq I_{\lambda ,\mu }(u_{n})- \frac{1}{n} \Vert u-u _{n} \Vert , \quad \forall u\in \mathcal{M}^{+}\cup \mathcal{M}^{0}. \end{aligned}$$
Assume that \(u_{n}\geq 0\) on \(\varOmega \setminus \{0\}\). Note that \(I_{\lambda ,\mu }(u)\) is bounded below on \(\mathcal{M}\). By (3.2), we get
for n large enough and a positive constant \(C_{1}\). Hence \(\{u_{n}\}\) is bounded in \(\mathcal{M}\). Let us, for a subsequence, suppose that
For every \(u\in \mathcal{M}^{+}\), we deduce from \(p>1\) that
which implies that \(\inf_{\mathcal{M}^{+}}I_{\lambda ,\mu }(u)<0\). For \(\lambda \in (0,T_{\mu })\), it follows from Lemma 2.1 that \(\mathcal{M}^{0}=\{0 \}\). Thus \(u_{n}\in \mathcal{M}^{+}\) for n large enough and \(\inf_{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u) =\inf_{\mathcal{M}^{+}}I_{\lambda ,\mu }(u)<0\). Therefore
i.e., \(u_{0}\geq 0\) and \(u_{0}\neq 0\).
$$ \mathcal{K} \bigl(\bigl( \Vert u_{n} \Vert _{\mu }\bigr)_{\min } \bigr)\leq I_{\lambda ,\mu }(u_{n})< \inf_{\mathcal{M}^{+}\cup \mathcal{M}^{0}}I_{\lambda ,\mu }(u)+ \frac{1}{n}\leq C_{1}, $$
(3.3)
$$ \textstyle\begin{cases} u_{n}\rightharpoonup u_{0} &\text{in }W, \\ u_{n}(x)\rightarrow u_{0}(x) & \text{a.e. in } \varOmega , \\ u_{n}\rightarrow u_{0} &\text{in } L^{1-q}(\varOmega ) \text{ and } L^{\gamma +1}(\varOmega ). \end{cases} $$
$$ \begin{aligned} I_{\lambda ,\mu }(u)&=\frac{1}{p} \Vert u \Vert _{\mu }^{p}-\frac{1}{1-q} \int _{\varOmega }f(x) \vert u \vert ^{1-q}\,dx - \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &=\frac{1}{p} \Vert u \Vert _{\mu }^{p}- \frac{1}{1-q} \biggl[ \Vert u \Vert _{\mu }^{p} - \lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \biggr]- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ & = \biggl(\frac{1}{p}-\frac{1}{1-q} \biggr) \Vert u \Vert _{\mu }^{p}+ \biggl(\frac{1}{1-q}- \frac{1}{ \gamma +1} \biggr)\lambda \int _{\varOmega }g(x) \vert u \vert ^{\gamma +1}\,dx \\ &< \biggl(\frac{1}{p}-\frac{1}{1-q} \biggr) \Vert u \Vert _{\mu }^{p} + \biggl(\frac{1}{1-q}-\frac{1}{ \gamma +1} \biggr)\frac{p+q-1}{\gamma +q} \Vert u \Vert _{\mu }^{p} \\ &=\frac{p+q-1}{\gamma +q} \biggl(\frac{1}{\gamma +1}-\frac{1}{p} \biggr) \Vert u \Vert _{\mu }^{p}< 0, \end{aligned} $$
$$\begin{aligned}& I_{\lambda ,\mu }(u_{0})\leq \liminf_{n\rightarrow \infty }I _{\lambda ,\mu }(u_{n}) =\inf_{\mathcal{M}^{+}\cup \mathcal{M} ^{0}}I_{\lambda ,\mu }< 0, \end{aligned}$$
In the following, we prove that, when \(\lambda \in (0,T_{\mu })\),
For \(\{u_{n}\}\subset \mathcal{M}^{+}\), we have
We suppose that
It follows from \(u_{n}\in \mathcal{M}\), the weak lower semi-continuity of the norm and (3.5) that
Hence, for every \(\lambda \in (0,T_{\mu })\) and \(u_{0}\neq 0\), combining with (2.2), we obtain
which is a contradiction. In view of (3.4), we get
for n large enough and some positive constant \(C_{2}\). Since \(u_{n}\in \mathcal{M}\), we have
Set \(\phi \in \mathcal{M}\) with \(\phi \geq 0\). Using Lemma 2.4, there exists \(\widetilde{g}_{n}(t)\) such that \(\widetilde{g}_{n}(0)=1\) and \(\widetilde{g}_{n}(t)(u_{n}+t\phi )\in \mathcal{M}^{+}\). Thus
and
Therefore
Dividing by \(t>0\) and letting \(t\rightarrow 0\), we have
where \(\widetilde{g}'_{n}(0)\) denotes the right derivative of \(\widetilde{g}_{n}(t)\) at zero. If it does not exist, \(\widetilde{g}'_{n}(0)\) should be replaced by \(\lim_{k\rightarrow \infty }\frac{\widetilde{g}_{n}(t_{k})- \widetilde{g}_{n}(0)}{t_{k}}\) for some sequence \(\{t_{k}\}_{k=1}^{ \infty }\) with \(\lim_{k\rightarrow \infty }t_{k} =0\) and \(t_{k}>0\).
$$ (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx >\lambda ( \gamma -q+1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. $$
(3.4)
$$\begin{aligned}& (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx -\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\& \quad =\lim_{n\rightarrow \infty } \biggl[ (p+q-1 ) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\& \quad =\lim_{n\rightarrow \infty } \biggl\{ (p+q-1 ) \biggl[ \Vert u_{n} \Vert _{\mu }^{p}-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr]- \lambda (\gamma -p+1 ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr\} \\& \quad =\lim_{n\rightarrow \infty } \biggl[ (p+q-1 ) \Vert u _{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega }g(x)u_{n} ^{\gamma +1}\,dx \biggr] \geq 0. \end{aligned}$$
$$ (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx=0. $$
(3.5)
$$ \begin{aligned} 0&=\lim_{n\rightarrow \infty } \biggl[ \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega } f(x)u_{n}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &\geq \Vert u_{0} \Vert _{\mu }^{p}- \int _{\varOmega } f(x)u_{0}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\ &= \textstyle\begin{cases} \Vert u_{0} \Vert _{\mu }^{p}-\lambda \frac{\gamma +q}{p+q-1} \int _{\varOmega } g(x)u_{0}^{\gamma +1}\,dx, \\ \Vert u_{0} \Vert _{\mu }^{p}-\lambda \frac{\gamma +q}{ \gamma -p+1} \int _{\varOmega } f(x)u_{0}^{1-q}\,dx. \end{cases}\displaystyle \end{aligned} $$
$$ \begin{aligned} 0&< A(\mu ,\lambda ) \Vert u_{0} \Vert _{\mu }^{\gamma +1} \\ &\leq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u_{0} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\int _{\varOmega }f(x) \vert u_{0} \vert ^{1-q}\,dx)^{\frac{p- \gamma -1}{1-q-p}}}-\lambda \int _{\varOmega }g(x) \vert u_{0} \vert ^{\gamma +1}\,dx \\ &\leq \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl(\frac{\gamma -p+1}{q+ \gamma } \biggr) ^{\frac{p-\gamma -1}{1-q-p}}\frac{( \Vert u_{0} \Vert _{ \mu }^{p})^{\frac{-\gamma -q}{1-q-p}}}{(\frac{\gamma -p+1}{q+\gamma } \Vert u_{0} \Vert _{\mu }^{p})^{\frac{p-\gamma -1}{1-q-p}}} -\frac{p+q-1}{ \gamma +q} \Vert u_{0} \Vert _{\mu }^{p}=0, \end{aligned} $$
$$ (p+q-1 ) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx\geq C_{2} $$
(3.6)
$$ (p+q-1 ) \Vert u_{n} \Vert _{\mu }^{p}- \lambda (\gamma +q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx\geq C_{2}>0. $$
(3.7)
$$\begin{aligned}& \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx=0 \end{aligned}$$
$$\begin{aligned}& \widetilde{g}_{n}^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p}-\widetilde{g} _{n}^{1-q}(t) \int _{\varOmega }f(x) (u_{n}+t\phi )^{1-q}\,dx- \lambda \widetilde{g}_{n} ^{\gamma +1}(t) \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1} \,dx=0. \end{aligned}$$
$$ \begin{aligned} 0&= \bigl[\widetilde{g}_{n}^{p}(t)-1 \bigr] \Vert u_{n}+t\phi \Vert _{\mu }^{p}+ \bigl( \Vert u_{n}+t\phi \Vert _{\mu }^{p}- \Vert u_{n} \Vert _{\mu }^{p} \bigr) \\ &\quad {}- \bigl[ \widetilde{g}_{n} ^{1-q}(t)-1 \bigr] \int _{\varOmega }f(x) (u_{n}+t\phi )^{1-q}\,dx \\ &\quad {} - \int _{\varOmega }f(x) \bigl[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} \bigr]\,dx- \lambda \bigl[ \widetilde{g}_{n}^{\gamma +1}(t)-1 \bigr] \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1}\,dx \\ &\quad {} -\lambda \int _{\varOmega }g(x) \bigl[(u_{n}+t\phi )^{\gamma +1}-u_{n}^{\gamma +1} \bigr]\,dx \\ & \leq \bigl[\widetilde{g}_{n}^{p}(t)-1 \bigr] \Vert u_{n}+t\phi \Vert _{\mu }^{p}+ \bigl( \Vert u_{n}+t\phi \Vert _{\mu }^{p}- \Vert u_{n} \Vert _{\mu } ^{p} \bigr) \\ &\quad {}- \bigl[ \widetilde{g}_{n} ^{1-q}(t)-1 \bigr] \int _{\varOmega }f(x) (u_{n}+t\phi )^{1-q}\,dx \\ &\quad {} - \lambda \bigl[\widetilde{g}_{n}^{\gamma +1}(t)-1 \bigr] \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1}\,dx- \lambda \int _{\varOmega }g(x) \bigl[(u_{n}+t\phi )^{\gamma +1}-u_{n}^{\gamma +1} \bigr]\,dx. \end{aligned} $$
$$ \begin{aligned}[b] 0& \leq p \widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p}+p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad {}- (1-q ) \widetilde{g}'_{n}(0) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx \\ &\quad {} -\lambda (\gamma +1 )\widetilde{g}'_{n}(0) \int _{\varOmega } g(x)u_{n}^{\gamma +1}\,dx-\lambda ( \gamma +1 ) \int _{\varOmega } g(x)u_{n}^{\gamma }\phi \,dx \\ & =\widetilde{g}'_{n}(0) \biggl[p \Vert u_{n} \Vert _{\mu }^{p}- (1-q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &\quad {}+p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad {} -\lambda (\gamma +1 ) \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ & =\widetilde{g}'_{n}(0) \biggl[ (p+q-1 ) \Vert u_{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &\quad {} +p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx-\lambda (\gamma +1 ) \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx, \end{aligned} $$
(3.8)
Combining with (3.7) and (3.8), we have \(\widetilde{g}'_{n}(0)\neq - \infty \). Now we prove that \(\widetilde{g}'_{n}(0)\neq +\infty \). Otherwise, we suppose that \(\widetilde{g}'_{n}(0)=+\infty \). Note that \(\widetilde{g}_{n}(t)>\widetilde{g}_{n}(0)=1\) for n large enough, and
Using condition (ii) with \(u=\widetilde{g}_{n}(t)(u_{n}+t\phi ) \in \mathcal{M}^{+}\), we deduce that
Dividing by \(t>0\) and letting \(t\rightarrow 0\), we obtain
that is,
which is not true since \(\widetilde{g}'_{n}(0)=+\infty \) and
It follows from (3.7), (3.8) and (3.10) that
for n sufficiently large and a suitable positive constant \(C_{4}\).
$$ \begin{aligned}[b] \bigl\vert \widetilde{g}_{n}(t)-1 \bigr\vert \cdot \Vert u_{n} \Vert +t\widetilde{g}_{n}(t) \Vert \phi \Vert &\geq \bigl\Vert \bigl[ \widetilde{g}_{n}(t)-1 \bigr]u_{n}+t\widetilde{g} _{n}(t)\phi \bigr\Vert \\ &= \bigl\Vert \widetilde{g}_{n}(t) (u_{n}+t\phi )-u_{n} \bigr\Vert . \end{aligned} $$
(3.9)
$$\begin{aligned}& \bigl[\widetilde{g}_{n}(t)-1 \bigr] \cdot \frac{ \Vert u_{n} \Vert }{n}+t \widetilde{g}_{n}(t) \frac{ \Vert \phi \Vert }{n} \\ & \quad \geq \frac{1}{n} \bigl\Vert \widetilde{g}_{n}(t) ( u _{n}+t\phi )-u_{n} \bigr\Vert \\ & \quad \geq I_{\lambda ,\mu }(u_{n})- I_{\lambda ,\mu } \bigl( \widetilde{g}_{n} (t) (u_{n}+t\phi ) \bigr) \\ & \quad =\frac{1}{p} \Vert u_{n} \Vert _{\mu }^{p} -\frac{1}{1-q} \int _{\varOmega }f(x) \vert u_{n} \vert ^{1-q}\,dx- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx- \frac{1}{p}\widetilde{g}_{n} ^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p} \\ & \quad \quad {} +\frac{1}{1-q} \int _{\varOmega }f(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{1-q}\,dx+\frac{ \lambda }{\gamma +1} \int _{\varOmega }g(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{ \gamma +1}\,dx \\ & \quad =\frac{1}{p} \Vert u_{n} \Vert _{\mu }^{p}- \frac{1}{1-q} \biggl[ \Vert u_{n} \Vert _{\mu }^{p}- \lambda \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx \biggr]-\frac{\lambda }{ \gamma +1} \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx \\ & \quad \quad {} -\frac{1}{p}\widetilde{g}_{n}^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p}+ \frac{1}{1-q} \biggl[\widetilde{g}_{n}^{p}(t) \Vert u_{n}+t \phi \Vert _{ \mu }^{p}-\lambda \int _{\varOmega }g(x) \vert u_{n}+t\phi \vert ^{\gamma +1}\,dx \biggr] \\ & \quad \quad {} +\frac{\lambda }{\gamma +1} \widetilde{g}_{n}^{\gamma +1}(t) \int _{\varOmega }g(x) \vert u_{n}+t\phi \vert ^{\gamma +1}\,dx \\ & \quad = \biggl(\frac{1}{p}-\frac{1}{1-q} \biggr) \Vert u_{n} \Vert _{\mu } ^{p}+ \biggl( \frac{1}{1-q}- \frac{1}{\gamma +1} \biggr)\lambda \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx \\ & \quad \quad {} + \biggl(\frac{1}{1-q}-\frac{1}{p} \biggr) \widetilde{g}_{n} ^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p} \\ & \quad\quad{} - \biggl(\frac{1}{1-q}- \frac{1}{ \gamma +1} \biggr)\lambda \widetilde{g}_{n} ^{\gamma +1}(t) \int _{\varOmega }g(x) \vert u_{n}+t\phi \vert ^{\gamma +1}\,dx \\ & \quad = \biggl(\frac{1}{1-q}-\frac{1}{p} \biggr) \bigl( \Vert u_{n}+t \phi \Vert _{\mu }^{p}- \Vert u_{n} \Vert _{\mu }^{p} \bigr)+ \biggl( \frac{1}{1-q} -\frac{1}{p} \biggr) \bigl[\widetilde{g}_{n}^{p}(t)-1 \bigr] \Vert u_{n}+t \phi \Vert _{\mu }^{p} \\ & \quad \quad {} - \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) \lambda \widetilde{g}_{n}^{\gamma +1}(t) \int _{\varOmega }g(x) \bigl[ (u_{n}+t\phi )^{\gamma +1}-u _{n}^{\gamma +1} \bigr]\,dx \\ & \quad \quad {} - \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr)\lambda \bigl[ \widetilde{g}_{n}^{\gamma +1}(t)-1 \bigr] \int _{\varOmega }g(x) u_{n}^{\gamma +1}\,dx. \end{aligned}$$
$$\begin{aligned}& \widetilde{g}'_{n}(0)\cdot \frac{ \Vert u_{n} \Vert }{n}+ \frac{ \Vert \phi \Vert }{n} \\ & \quad \geq \biggl(\frac{1}{1-q}-\frac{1}{p} \biggr)\cdot p \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ & \quad \quad {}+ \biggl(\frac{1}{1-q}- \frac{1}{p} \biggr)\cdot p \widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p} \\ & \quad \quad {}-\lambda \biggl(\frac{1}{1-q}-\frac{1}{\gamma +1} \biggr) (\gamma +1 ) \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ & \quad \quad {}-\lambda \biggl(\frac{1}{1-q} -\frac{1}{\gamma +1} \biggr) (\gamma +1 ) \widetilde{g}'_{n}(0) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \\ & \quad = \frac{p-1+q}{1-q} \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx + \frac{p-1+q}{1-q}\widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p} \\ & \quad \quad {}-\lambda \frac{\gamma +q}{1-q} \int _{\varOmega } g(x)u_{n}^{\gamma }\phi \,dx- \lambda \frac{\gamma +q}{1-q}\widetilde{g}'_{n}(0) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \\ & \quad = \frac{\widetilde{g}'_{n}(0)}{1-q} \biggl[ (p-1+q ) \Vert u_{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega } g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ & \quad \quad {}+\frac{p-1+q}{1-q} \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx-\lambda \frac{ \gamma +q}{1-q} \int _{\varOmega } g(x)u_{n}^{\gamma }\phi \,dx, \end{aligned}$$
$$ \begin{aligned}[b] \frac{ \Vert \phi \Vert }{n}&\geq \frac{\widetilde{g}'_{n}(0)}{1-q} \biggl[(p-1+q) \Vert u_{n} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega } g(x)u_{n}^{\gamma +1}\,dx- \frac{(1-q) \Vert u_{n} \Vert }{n} \biggr] \\ &\quad {} + \frac{p-1+q}{1-q} \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad {}-\lambda \frac{ \gamma +q}{1-q} \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx, \end{aligned} $$
(3.10)
$$\begin{aligned}& (p-1+q ) \Vert u_{n} \Vert _{\mu }^{p}- \lambda ( \gamma +q ) \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx- \frac{(1-q) \Vert u_{n} \Vert }{n}\geq C _{2}-\frac{(1-q)C_{3}}{n}>0. \end{aligned}$$
$$\begin{aligned}& \bigl\vert \widetilde{g}_{n}^{\prime }(0) \bigr\vert \leq C_{4} \end{aligned}$$
In the following, we prove that \(u_{0}\in \mathcal{M}^{+}\) is a solution of problem (1.1). By (3.9) and condition (ii) again, we have
Dividing by \(t>0\) and letting \(t\rightarrow 0^{+}\), we derive that
Noting \(f(x) [(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} ]\geq 0\), for every \(x \in \varOmega \) and \(t>0\), together with the Fatou lemma, we find that
is integrable, and
Applying the Fatou lemma again, we have
Since \(\int _{\varOmega }u_{0}^{-q}\varphi _{1}\,dx<\infty \), we have \(u_{0}>0\) a.e. in Ω. For every \(\phi \in \mathcal{M}\) and \(\phi \geq 0\), we have
Set \(\phi =u_{0}\) in (3.11), we derive that
Furthermore
Hence
Therefore \(u_{n}\rightarrow u_{0}\) in \(\mathcal{M}\) and \(u_{0}\in \mathcal{M}\). By (3.4), we have
i.e., \(u_{0}\in \mathcal{M}^{+}\).
$$\begin{aligned}& \frac{1}{n} \bigl[ \bigl\vert \widetilde{g}_{n}(t)-1 \bigr\vert \cdot \Vert u_{n} \Vert +t \widetilde{g}_{n}(t) \Vert \phi \Vert \bigr] \\& \quad \geq \frac{1}{n} \bigl\Vert \widetilde{g}_{n}(t) (u_{n}+t\phi )-u_{n} \bigr\Vert \\& \quad \geq I_{\lambda ,\mu }(u_{n})-I_{\lambda ,\mu } \bigl( \widetilde{g}_{n}(t) (u_{n}+t\phi ) \bigr) \\& \quad = \frac{1}{p} \Vert u_{n} \Vert _{\mu }^{p} -\frac{1}{1-q} \int _{\varOmega }f(x) \vert u_{n} \vert ^{1-q}\,dx- \frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \vert u_{n} \vert ^{\gamma +1}\,dx- \frac{1}{p}\widetilde{g}_{n} ^{p}(t) \Vert u_{n}+t\phi \Vert _{\mu }^{p} \\& \quad \quad {} +\frac{1}{1-q} \int _{\varOmega }f(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{1-q}\,dx +\frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \bigl\vert \widetilde{g}_{n} (u_{n}+t\phi ) \bigr\vert ^{ \gamma +1}\,dx \\& \quad =-\frac{\widetilde{g}_{n}^{p}(t)-1}{p} \Vert u_{n} \Vert _{\mu }^{p}- \frac{ \widetilde{g}_{n}^{p}(t)}{p} \bigl( \Vert u_{n}+t\phi \Vert _{\mu }^{p}- \Vert u _{n} \Vert _{\mu }^{p} \bigr) \\& \quad \quad {}+\frac{\widetilde{g}_{n}^{1-q}(t)-1}{1-q} \int _{\varOmega } f(x) (u_{n}+t\phi )^{1-q}\,dx \\& \quad \quad {} +\frac{1}{1-q} \int _{\varOmega }f(x) \bigl[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q} \bigr]\,dx +\frac{ \lambda (\widetilde{g}_{n}^{\gamma +1}(t)-1)}{\gamma +1} \int _{\varOmega }g(x) (u_{n}+t\phi )^{\gamma +1}\,dx \\& \quad \quad {} +\frac{\lambda }{\gamma +1} \int _{\varOmega }g(x) \bigl[(u_{n}+t\phi )^{\gamma +1} -u_{n}^{\gamma +1} \bigr]\,dx. \end{aligned}$$
$$\begin{aligned}& \frac{1}{n} \bigl[ \bigl\vert \widetilde{g}'_{n}(0) \bigr\vert \cdot \Vert u_{n} \Vert + \Vert \phi \Vert \bigr] \\ & \quad \geq -\widetilde{g}'_{n}(0) \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx+ \widetilde{g}_{n}^{\prime }(0) \int _{\varOmega }f(x)u_{n}^{1-q}\,dx \\ & \quad \quad {} +\lambda \widetilde{g}'_{n}(0) \int _{\varOmega }g(x)u_{n} ^{\gamma +1}\,dx+\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ & \quad \quad {}+\liminf _{t\rightarrow 0^{+}} \frac{1}{1-q} \int _{\varOmega }\frac{f(x)[(u_{n}+t\phi )^{1-q} -u_{n}^{1-q}]}{t}\,dx \\ & \quad =-\widetilde{g}'_{n}(0) \biggl[ \Vert u_{n} \Vert _{\mu }^{p}- \int _{\varOmega }f(x)u_{n}^{1-q}\,dx-\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ & \quad \quad {} - \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx +\lambda \int _{\varOmega }g(x) u_{n}^{\gamma }\phi \,dx \\ & \quad \quad {} +\liminf_{t\rightarrow 0^{+}}\frac{1}{1-q} \int _{\varOmega }\frac{f(x)[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q}]}{t}\,dx. \end{aligned}$$
$$ \liminf_{t\rightarrow 0^{+}} \biggl[\frac{f(x)[(u_{n}+t\phi )^{1-q}-u _{n}^{1-q}]}{t} \biggr] $$
$$ \begin{aligned} & \int _{\varOmega }f(x)u_{n}^{-q}\phi \,dx \\ &\quad \leq \liminf_{t\rightarrow 0^{+}}\frac{1}{1-q} \int _{\varOmega } \frac{f(x)[(u_{n}+t\phi )^{1-q}-u_{n}^{1-q}]}{t}\,dx \\ &\quad \leq \frac{ \vert \widetilde{g}'_{n}(0) \vert \Vert u_{n} \Vert + \Vert \phi \Vert }{n}+ \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad\quad{} -\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \\ &\quad \leq \frac{C_{3}C_{4}+ \Vert \phi \Vert }{n}+ \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx -\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx. \end{aligned} $$
$$ \begin{aligned} & \int _{\varOmega }f(x)u_{0}^{-q}\phi \,dx \\ &\quad = \int _{\varOmega } \Bigl[\liminf_{n\rightarrow \infty }f(x)u_{n} ^{-q}\phi \Bigr]\,dx \leq \liminf_{n\rightarrow \infty } \int _{\varOmega } f(x)u_{n}^{-q}\phi \,dx \\ &\quad \leq \liminf_{n\rightarrow \infty } \biggl[\frac{C_{3}C_{4}+ \Vert \phi \Vert }{n}+ \int _{\varOmega } \biggl( \vert \Delta u_{n} \vert ^{p-2}\Delta u_{n} \Delta \phi - \mu \frac{ \vert u_{n} \vert ^{p-2}u_{n} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx \\ &\quad\quad{} -\lambda \int _{\varOmega }g(x)u_{n}^{\gamma }\phi \,dx \biggr] \\ &\quad = \int _{\varOmega } \biggl( \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx -\lambda \int _{\varOmega }g(x)u_{0}^{\gamma }\phi \,dx. \end{aligned} $$
$$ \begin{aligned}[b] & \int _{\varOmega } \biggl( \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}} \biggr)\,dx- \int _{\varOmega }f(x) u_{0}^{-q}\phi \,dx \\ &\quad{} -\lambda \int _{\varOmega }g(x)u_{0}^{\gamma }\phi \,dx\geq 0. \end{aligned} $$
(3.11)
$$\begin{aligned}& \Vert u_{0} \Vert _{\mu }^{p}= \int _{\varOmega } \biggl( \vert \Delta u_{0} \vert ^{p}-\mu \frac{ \vert u_{0} \vert ^{p}}{ \vert x \vert ^{2p}} \biggr)\,dx\geq \int _{\varOmega }f(x) u_{0}^{1-q}\,dx+\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. \end{aligned}$$
$$ \begin{aligned}[b] \Vert u_{0} \Vert _{\mu }^{p}&\leq \liminf_{n\rightarrow \infty } \Vert u _{n} \Vert _{\mu }^{p}\leq \limsup _{n\rightarrow \infty } \Vert u_{n} \Vert _{\mu }^{p} \\ & =\limsup_{n\rightarrow \infty } \biggl[ \int _{\varOmega }f(x)u_{n}^{1-q}\,dx +\lambda \int _{\varOmega }g(x)u_{n}^{\gamma +1}\,dx \biggr] \\ &= \int _{\varOmega }f(x)u_{0}^{1-q}\,dx +\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. \end{aligned} $$
(3.12)
$$ \Vert u_{0} \Vert _{\mu }^{p}= \int _{\varOmega }f(x)u_{0}^{1-q}\,dx +\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx. $$
(3.13)
$$ \begin{aligned} & (p+q-1 ) \Vert u_{0} \Vert _{\mu }^{p}-\lambda (\gamma +q ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\ &\quad = (p+q-1 ) \biggl[ \int _{\varOmega }f(x)u_{0}^{1-q}\,dx+\lambda \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \biggr]- \lambda (\gamma +q ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx \\ &\quad = (p+q-1 ) \int _{\varOmega }f(x)u_{0}^{1-q}\,dx-\lambda ( \gamma -1 ) \int _{\varOmega }g(x)u_{0}^{\gamma +1}\,dx>0, \end{aligned} $$
Next, we only need to show that \(u_{0}\) is a positive weak solution of problem (1.1). Define
Substituting Φ into (3.11), combining with (3.12), we deduce that
where \(\varOmega _{1}=\{x\vert u_{0}(x)+\varepsilon \phi (x)>0, x\in \varOmega \}\) and \(\varOmega _{2}=\{x\vert u_{0}(x)+\varepsilon \phi (x)\leq 0, x\in \varOmega \}\). Since the measure of \(\varOmega _{2}\) tends to zero as \(\varepsilon \rightarrow 0\), we have \(\int _{\varOmega _{2}} \vert \Delta u_{0}\vert ^{p-2}\Delta u_{0} \Delta \phi \,dx \rightarrow 0\) as \(\varepsilon \rightarrow 0\). By the same arguments, we have \(\lambda \varepsilon ^{\gamma }\Vert g\Vert _{\infty } \int _{\varOmega _{2}}\vert \phi \vert ^{\gamma +1}\,dx \longrightarrow 0\) and \(\lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma }\phi \,dx\longrightarrow 0\) as \(\varepsilon \rightarrow 0\). Dividing by ε and taking the limit for \(\varepsilon \rightarrow 0\), we deduce that
Therefore \(u_{0}\) is a positive weak solution of problem (1.1).
$$ \varPhi =(u_{0}+\varepsilon \phi )^{+}, \quad \phi \in W, \varepsilon >0. $$
$$\begin{aligned} 0&\leq \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \varPhi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \varPhi }{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q}\varPhi - \lambda g(x)u_{0} ^{\gamma }\varPhi \biggr]\,dx \\ &= \int _{\varOmega _{1}} \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \varPhi -\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \varPhi }{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} \varPhi -\lambda g(x)u_{0} ^{\gamma }\varPhi \biggr]\,dx \\ &\quad {} + \int _{\varOmega _{2}} \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \varPhi -\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \varPhi }{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} \varPhi -\lambda g(x)u_{0} ^{\gamma }\varPhi \biggr]\,dx \\ &= \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta (u_{0}+ \varepsilon \phi )-\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} (u_{0}+\varepsilon \phi )}{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} (u_{0}+ \varepsilon \phi ) \\ &\quad {} -\lambda g(x)u_{0}^{\gamma }(u_{0}+ \varepsilon \phi ) \biggr]\,dx \\ &\quad {} - \int _{\varOmega _{2}} \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta (u _{0}+\varepsilon \phi )-\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} (u_{0}+\varepsilon \phi )}{ \vert x \vert ^{2p}} -f(x)u_{0}^{-q} (u_{0}+ \varepsilon \phi ) \\ &\quad {} -\lambda g(x)u_{0}^{\gamma }(u_{0}+ \varepsilon \phi ) \biggr]\,dx \\ &= \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p}-\mu \frac{ \vert u_{0} \vert ^{p}}{ \vert x \vert ^{2p}}-f(x)u_{0}^{1-q}- \lambda g(x)u_{0}^{ \gamma +1} \biggr]\,dx \\ &\quad {} +\varepsilon \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma }\phi \biggr]\,dx \\ &\quad {} - \int _{\varOmega _{2}} \biggl[ \vert \Delta u_{0} \vert ^{p}+\varepsilon \vert \Delta u_{0} \vert ^{p-2} \Delta u_{0} \Delta \phi -\mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} (u_{0}+ \varepsilon \phi )}{ \vert x \vert ^{2p}} \biggr]\,dx \\ &\quad {} - \int _{\varOmega _{2}} \bigl[-f(x) u_{0}^{-q}(u_{0}+ \varepsilon \phi )- \lambda g(x)u_{0}^{\gamma +1}-\varepsilon \lambda g(x)u_{0}^{\gamma } \phi \bigr]\,dx \\ &\leq \varepsilon \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma } \phi \biggr]\,dx \\ &\quad {} -\varepsilon \int _{\varOmega _{2}} \vert \Delta u_{0} \vert ^{p-2} \Delta u_{0} \Delta \phi \,dx+ \lambda \Vert g \Vert _{\infty } \int _{\varOmega _{2}} \vert \varepsilon \phi \vert ^{\gamma +1} \,dx+ \varepsilon \lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma }\phi \,dx \\ &=\varepsilon \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma }\phi \biggr]\,dx \\ &\quad {} -\varepsilon \int _{\varOmega _{2}} \vert \Delta u_{0} \vert ^{p-2} \Delta u_{0} \Delta \phi \,dx+ \varepsilon \lambda \varepsilon ^{\gamma } \Vert g \Vert _{\infty } \int _{\varOmega _{2}} \vert \phi \vert ^{\gamma +1}\,dx+ \varepsilon \lambda \int _{\varOmega _{2}}g(x)u_{0}^{\gamma } \phi \,dx, \end{aligned}$$
$$\begin{aligned}& \int _{\varOmega } \biggl[ \vert \Delta u_{0} \vert ^{p-2}\Delta u_{0} \Delta \phi - \mu \frac{ \vert u_{0} \vert ^{p-2}u_{0} \phi }{ \vert x \vert ^{2p}}-f(x)u_{0}^{-q} \phi - \lambda g(x)u_{0}^{\gamma }\phi \biggr]\,dx\geq 0. \end{aligned}$$
We adopt the Ekeland variational principle again to derive a minimizing sequence \(U_{n}\subset \mathcal{M}^{-}\) for the minimization problem \(\inf_{\mathcal{M}^{-}} I_{\lambda ,\mu }\) such that for \(U_{n}\in \mathcal{M}\), \(U_{n} \rightharpoonup U_{0}\) weakly in \(\mathcal{M}\) and pointwise a.e. in Ω. By similar arguments to those in (3.4) and (3.6), for \(\lambda \in (0,T_{\mu })\), we have
which leads to
for n large enough and a positive constant \(C_{5}\). Therefore \(U_{0}>0\) is the positive weak solution of problem (1.1). Furthermore \(U_{0}\in \mathcal{M}\). By (3.14), we obtain
i.e., \(U_{0}\in \mathcal{M}^{-}\). According to Lemma 2.2, we know that problem (1.1) has at least two positive weak solutions \(u_{0}\in \mathcal{M}^{+}\) and \(U_{0}\in \mathcal{M}^{-}\) with \(\Vert U_{0}\Vert _{ \mu }>\Vert u_{0}\Vert _{\mu }\) for every \(\lambda \in (0,T_{\mu })\). This completes the proof of Theorem 1.1.
$$ (p+q-1 ) \int _{\varOmega }f(x) \vert U_{0} \vert ^{1-q}\,dx- \lambda (\gamma -p+1 ) \int _{\varOmega }g(x) \vert U_{0} \vert ^{\gamma +1}\,dx< 0, $$
(3.14)
$$\begin{aligned}& (p+q-1 ) \int _{\varOmega }f(x) \vert U_{n} \vert ^{1-q}\,dx- \lambda (\gamma -p+1 ) \int _{\varOmega }g(x) \vert U_{n} \vert ^{\gamma +1}\,dx \leq -C_{5}, \end{aligned}$$
$$ \begin{aligned} & (p+q-1 ) \Vert U_{0} \Vert _{\mu }^{p}- (q+\gamma ) \lambda \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx \\ & \quad = (p+q-1 ) \biggl[ \int _{\varOmega }f(x)U_{0}^{1-q}\,dx+\lambda \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx \biggr]- \lambda (\gamma +q ) \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx \\ & \quad = (p+q-1 ) \int _{\varOmega }f(x)U_{0}^{1-q}\,dx-\lambda ( \gamma -p+1 ) \int _{\varOmega }g(x)U_{0}^{\gamma +1}\,dx< 0, \end{aligned} $$
4 Proof of Corollary 1.2
For every \(U\in \mathcal{M}^{-}\), by Lemma 2.2, we deduce that
Combining with the definition of \(T_{\mu }\), we have
where we adopted the following facts:
$$ \begin{aligned} \Vert U \Vert _{\mu }&>M_{\mu }( \lambda ) \\ & = \biggl[\frac{p+q-1}{\lambda (\gamma +q)}\frac{1}{ \Vert g \Vert _{\infty }}\frac{(\sqrt[p]{S_{\mu }})^{\gamma +1}}{ \vert \varOmega \vert ^{\frac{p ^{*}-1-\gamma }{p^{*}}}} \biggr]^{\frac{1}{\gamma +1-p}} \\ &= \biggl(\frac{1}{\lambda } \biggr)^{\frac{1}{\gamma +1-p}} \biggl(\frac{p+q-1}{ \gamma +q} \biggr)^{\frac{1}{\gamma +1-p}} \biggl(\frac{1}{ \Vert g \Vert _{ \infty }} \biggr)^{\frac{1}{\gamma +1-p}} \frac{(\sqrt[p]{S_{\mu }}) ^{\frac{\gamma +1}{\gamma +1-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1-p)}}} \\ &= (T_{\mu } )^{-\frac{1}{\gamma +1-p}} \biggl(\frac{p+q-1}{ \gamma +q} \biggr)^{\frac{1}{\gamma +1-p}} \biggl(\frac{1}{ \Vert g \Vert _{ \infty }} \biggr)^{\frac{1}{\gamma +1-p}} \frac{(\sqrt[p]{S_{\mu }}) ^{\frac{\gamma +1}{\gamma +1-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1-p)}}} \biggl(\frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{ \gamma +1-p}}. \end{aligned} $$
$$\begin{aligned} \Vert U \Vert _{\mu }&> \biggl(\frac{q+\gamma }{q+p-1} \biggr)^{\frac{1}{ \gamma +1-p}} \biggl(\frac{q+\gamma }{\gamma -p+1} \biggr)^{ \frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \bigl( \Vert g \Vert _{\infty } \bigr)^{\gamma -p+1}\frac{ \vert \varOmega \vert ^{ \frac{2p}{N}\frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}}}{S_{\mu } ^{\frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}}} \\ & \quad{}\times \biggl(\frac{p+q-1}{\gamma +q} \biggr)^{ \frac{1}{\gamma +1-p}} \biggl(\frac{1}{ \Vert g \Vert _{\infty }} \biggr) ^{\frac{1}{ \gamma +1-p}}\frac{(\sqrt[p]{S_{\mu }})^{ \frac{\gamma +1}{\gamma +1-p}}}{ \vert \varOmega \vert ^{\frac{p^{*}-1-\gamma }{p ^{*}(\gamma +1-p)}}} \biggl(\frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{ \gamma +1-p}} \\ & = \biggl( \frac{q+\gamma }{\gamma -p+1} \biggr)^{ \frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl(\frac{ \vert \varOmega \vert ^{\frac{2p}{N}\frac{q+\gamma }{p+q-1} \frac{1}{ \gamma +1-p}-\frac{p^{*}-1-\gamma }{p^{*}(\gamma +1-p)}}}{(\sqrt[p]{S _{\mu }})^{p\cdot \frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}- \frac{ \gamma +1}{\gamma +1-p}}} \biggr) \biggl( \frac{T_{\mu }}{\lambda } \biggr) ^{\frac{1}{\gamma +1-p}} \\ &= \vert \varOmega \vert ^{\frac{1}{p}} \biggl( \frac{q+\gamma }{ \gamma -p+1} \biggr)^{\frac{1}{p+q-1}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl( \frac{ \vert \varOmega \vert ^{\frac{2}{N}}}{\sqrt[p]{S _{\mu }}} \biggr) ^{\frac{1-q}{p+q-1}} \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\gamma +1-p}} \\ &= \vert \varOmega \vert ^{\frac{1}{p}} \bigl( \Vert f \Vert _{\infty } \bigr) ^{\frac{1}{p+q-1}} \biggl(1+\frac{p+q-1}{\gamma -p+1} \biggr)^{ \frac{1}{p+q-1}} \biggl( \frac{ \vert \varOmega \vert ^{\frac{2}{N}}}{\sqrt[p]{S _{\mu }}} \biggr) ^{\frac{1-q}{p+q-1}} \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\gamma +1-p}}, \end{aligned}$$
$$\begin{aligned}& \begin{aligned} &\frac{2p}{N}\frac{q+\gamma }{p+q-1} \frac{1}{\gamma +1-p}-\frac{p ^{*}-1-\gamma }{p^{*}(\gamma +1-p)} \\ &\quad =\frac{p^{*}-1+q}{p^{*}(p+q-1)}=\frac{\frac{Np}{N-2p}+q-1}{ \frac{Np}{N-2p}(p+q-1)} \\ &\quad =\frac{N(p+q-1)+2p(1-q)}{Np(p+q-1)}=\frac{1}{p}+\frac{2}{N}\cdot \frac{1-q}{p+q-1}, \end{aligned} \\& p\cdot \frac{q+\gamma }{p+q-1}\frac{1}{\gamma +1-p}-\frac{\gamma +1}{\gamma +1-p}= \frac{(1-q)(\gamma +1-p)}{(p+q-1)(\gamma +1-p)}= \frac{1-q}{p+q-1}. \end{aligned}$$
Let \(U_{\lambda , \mu ,\varepsilon }\in \mathcal{M}^{-}\) be the solution of problem (1.1) with \(\gamma =\varepsilon +p-1\), where \(\lambda \in (0,T_{\mu })\). Then
where \(C_{\mu , \varepsilon }\) is given in (1.16). This completes the proof of Corollary 1.2.
$$\begin{aligned}& \Vert U_{\lambda , \mu ,\varepsilon } \Vert _{\mu }>C_{\mu ,\varepsilon } \biggl( \frac{T_{\mu }}{\lambda } \biggr)^{\frac{1}{\varepsilon }}, \end{aligned}$$
5 Proof of Theorem 1.3
For simplicity, we consider problem (1.1) with \(f=g=1\),
Let us define
Using Theorem 1.1, we provide uniform estimates for \(\lambda ^{*}(N, \varOmega ,\mu ,q,\gamma )\).
$$ \textstyle\begin{cases} \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}=u^{-q}+\lambda u^{ \gamma } &\text{in } \varOmega \setminus \{0\}, \\ u(x)>0 &\text{in }\varOmega \setminus \{0\}, \\ u=\Delta u=0 &\text{on } \partial \varOmega . \end{cases} $$
(5.1)
$$\begin{aligned}& \lambda ^{*}=\lambda ^{*}(N,\varOmega ,\mu ,q,\gamma )=\sup \bigl\{ \lambda >0: \text{problem (5.1) has a positive solution} \bigr\} . \end{aligned}$$
Lemma 5.1
For
\(1< p<\frac{N}{2}\), \(0<\mu <\mu _{N,p}\), \(0< q<1<\gamma <p^{*}-1\)
and
\(\varOmega \in \mathbb{U}\), where
\(\mathbb{U}=\{\varOmega \in \mathbb{R}^{N}: \varOmega \textit{ is an open and bounded domain}\}\), we have
where
and
$$\begin{aligned}& 0< \lambda ^{-}\leq \lambda ^{*}\leq \lambda ^{+}< \infty , \end{aligned}$$
$$ \lambda ^{-}= \biggl(\frac{q+p-1}{q+\gamma } \biggr) \biggl( \frac{ \gamma -p+1}{q+\gamma } \biggr)^{\frac{p-\gamma -1}{1-q-p}} \biggl[\frac{S _{\mu }}{ \vert \varOmega \vert ^{\frac{2p}{N}}} \biggr]^{\frac{q+\gamma }{p+q-1}} $$
$$ \lambda ^{+}=\lambda _{1}^{\frac{\gamma +q}{q-1+p}} \biggl( \frac{\gamma -p+1}{\gamma +q} \biggr)^{\frac{\gamma -p+1}{q+p-1}}\frac{-1+p+q}{ \gamma +q}+ \frac{1}{2}. $$
Proof
(1) Assume that \(\lambda \in (0,\lambda ^{-})\), then problem (5.1) has at least two solutions. By the definition of \(\lambda ^{*}\), we have \(\lambda ^{*}\geq \lambda ^{-}>0\).
(2) Assume that (5.1) has a positive solution u. Integrating over Ω by multiplying (5.1) by \(\varphi _{1}\), we obtain
We claim that there exists \(\lambda ^{+}>0\) such that
In fact, letting
We have
i.e.,
Then \(G_{\lambda }(t)\) attains minimum at \(t_{\min }\), and
We may choose \(\lambda =\lambda _{1}^{\frac{\gamma +q}{q-1+p}} (\frac{ \gamma -p+1}{\gamma +q} )^{\frac{\gamma -p+1}{q+p-1}}\frac{-1+p+q}{ \gamma +q}+\frac{1}{2}=\lambda ^{+} >0\) such that
Therefore
$$ \lambda _{1} \int _{\varOmega } \vert u \vert ^{p-2}u \varphi _{1}\,dx= \int _{\varOmega } \biggl(\Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}} \biggr) \varphi _{1}\,dx= \int _{\varOmega }u^{-q}\varphi _{1}\,dx+\lambda \int _{\varOmega }u^{\gamma } \varphi _{1}\,dx. $$
(5.2)
$$ t^{-q}+\lambda ^{+}t^{\gamma }>\lambda _{1}t^{p-1}, \quad \forall t>0. $$
(5.3)
$$ F_{\lambda }(t)=t^{-q}+\lambda t^{\gamma }- \lambda _{1}t^{p-1}=t^{ \gamma } \bigl(t^{-q-\gamma }+ \lambda -\lambda _{1}t^{-\gamma +p-1} \bigr) :=t^{ \gamma } \cdot G_{\lambda }(t), \quad t>0. $$
(5.4)
$$ G'_{\lambda }(t)=(-\gamma -q)t^{-\gamma -q-1}+\lambda _{1} (\gamma -p+1)t ^{-\gamma +p-2}=0, $$
$$ t:=t_{\min }= \biggl(\frac{\gamma +q}{\lambda _{1} (\gamma -p+1)} \biggr) ^{\frac{1}{q-1+p}}. $$
$$ G_{\lambda }(t_{\min })=\lambda +\lambda _{1}^{\frac{\gamma +q}{q-1+p}} \biggl(\frac{\gamma -p+1}{\gamma +q} \biggr)^{ \frac{\gamma -p+1}{q+p-1}}\frac{1-p-q}{\gamma +q}. $$
$$ G_{\lambda ^{+}}(t)\geq G_{\lambda ^{+}}(t_{\min })= \frac{1}{2}>0, \quad \text{for }t>0. $$
$$ F_{\lambda ^{+}}(t)=t^{\gamma }\cdot G_{\lambda ^{+}}(t)>0 \quad \text{for }t>0. $$
Using (5.3) with \(t=u\), we have
Combining with (5.2) and (5.5), we obtain \(\lambda \leq \lambda ^{+}\). Since λ is arbitrary, we have \(\lambda ^{*}\leq \lambda ^{+}< \infty \). □
$$ \int _{\varOmega }u^{-q}\varphi _{1}\,dx+\lambda ^{+} \int _{\varOmega } u^{\gamma }\varphi _{1}\,dx\geq \lambda _{1} \int _{\varOmega } \vert u \vert ^{p-2}u \varphi _{1}\,dx. $$
(5.5)
Proof of Theorem 1.3
We only prove the case that \(0<\lambda <\lambda ^{*}\). By the definition of \(\lambda ^{*}\), there exists \(\overline{\lambda }\in (\lambda , \lambda ^{*})\) such that the problem
has a positive solution, denoted by \(u_{\overline{\lambda }}\). It follows that
Hence \(u_{\overline{\lambda }}\) is an upper solution of (5.1). Note that \(\lim_{t\rightarrow 0^{+}}G_{\lambda }(t)=\infty \), we can take \(\varepsilon >0\) small enough with \(\varepsilon \varphi _{1}< u_{\overline{ \lambda }}\) and \(G_{\lambda }(\varepsilon \varphi _{1})\geq 0\). Thus
i.e.,
Combining with (1.10) and (5.6), we obtain
namely, \(\varepsilon \varphi _{1}\) is a lower solution of (5.1). Note that \(\Delta _{p}^{2}-\frac{\mu }{\vert x\vert ^{2p}}\) is monotone, then problem (5.1) has a positive solution \(u_{\lambda }\) with \(\varepsilon \varphi _{1}\leq u_{\lambda }\leq u_{\overline{\lambda }}\). □
$$\begin{aligned}& \Delta ^{2}_{p}u-\mu \frac{ \vert u \vert ^{p-2}u}{ \vert x \vert ^{2p}}=u^{-q}+ \overline{ \lambda } u^{\gamma } \end{aligned}$$
$$\begin{aligned}& \Delta ^{2}_{p}u_{\overline{\lambda }}-\mu \frac{ \vert u_{\overline{ \lambda }} \vert ^{p-2}u_{\overline{\lambda }}}{ \vert x \vert ^{2p}} =u_{\overline{ \lambda }}^{-q}+\overline{\lambda } u_{\overline{\lambda }}^{\gamma } \geq u_{\overline{\lambda }}^{-q}+\lambda u_{\overline{\lambda }}^{ \gamma }. \end{aligned}$$
$$\begin{aligned}& F_{\lambda }(\varepsilon \varphi _{1})= (\varepsilon \varphi _{1})^{ \gamma }G_{\lambda }(\varepsilon \varphi _{1})\geq 0, \quad \text{for all } \lambda >0, \end{aligned}$$
$$ \lambda _{1}(\varepsilon \varphi _{1})^{p-1} \leq (\varepsilon \varphi _{1})^{-q}+\lambda (\varepsilon \varphi _{1})^{\gamma }, \quad \text{for all } \lambda >0. $$
(5.6)
$$ \begin{aligned} \Delta ^{2}_{p}(\varepsilon \varphi _{1})-\mu \frac{ \vert (\varepsilon \varphi _{1}) \vert ^{p-2} (\varepsilon \varphi _{1})}{ \vert x \vert ^{2p}}&=\varepsilon ^{p-1} \biggl(\Delta ^{2}_{p}\varphi _{1}- \mu \frac{ \vert \varphi _{1} \vert ^{p-2} \varphi _{1}}{ \vert x \vert ^{2p}} \biggr) \\ & =\varepsilon ^{p-1}\lambda _{1} \vert \varphi _{1} \vert ^{p-1}= \lambda _{1}(\varepsilon \varphi _{1})^{p-1}\leq (\varepsilon \varphi _{1})^{-q}+\lambda (\varepsilon \varphi _{1})^{\gamma }, \end{aligned} $$
6 Conclusions
In this paper, we study a class of p-biharmonic equations with Hardy potential and negative exponents. We establish the dependence of the above problem on q, γ, f, g and Ω and evaluate the extremal value of λ related to the multiplicity of positive solutions for this problem.
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