Existence results for nonlinear fourth-order elliptic boundary value problems
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- 24-11-2025
- Research
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Abstract
1 Introduction
In this paper, we discuss the existence of the nonlinear fourth-order elliptic boundary value problem (BVP) where Ω is a smooth bounded domain in \(\mathbb{R}^{N}\) with \(C^{2+\mu}\)-boundary ∂Ω (\(0<\mu <1\)), \(N\ge 2\), Δ is the Laplace operator, \(f: \overline{\Omega}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) is a continuous function. A classical solution u of BVP (1.1) means that \(u\in C^{4}(\overline{\Omega })\) and it satisfies Equation (1.1). To discuss the classical solution of BVP (1.1) requires the Hölder continuous continuity of f on bounded domains, that is, the following condition:
For the general continuous f, we also discuss the existence of \(L^{p}\)-solutions for some \(p>1\). A \(L^{p}\)-solution u means that \(u\in W^{4,p}(\Omega )\cap C^{2}(\overline{\Omega })\) and satisfies the equation in the sense of \(W^{4,p}(\Omega )\).
$$ \left \{ \textstyle\begin{array}{l} {\Delta}^{2} u = f(x,\,u,\,\Delta u),\qquad x\in \Omega , \\ u=\Delta u=0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$
(1.1)
(H0)
For every \(R>0\), there exists \(\alpha \in (0,\,1)\), such that \(f(x,\,\xi ,\,\eta )\) is α-order Hölder continuous on \(\overline{\Omega }\times [-R,\,R]\times [-R,\,R]\).
The fourth order elliptic equation boundary value such as BVP (1.1) arises in the study of traveling waves in suspension bridges and the study of the static deflection of an elastic plate in a fluid, see [1‐4]. The special case of BVP (1.1) that the nonlinearity f does not contain Δu, that is, has been discussed by many researchers, see [5‐12] and references therein. Dalmass [5] obtained uniqueness and positivity results of radial solutions when \(f(x,\,u)=|u|^{p}\) and Ω is a Ball. In [6‐10], some existence results are obtained by applying mountain pass lemma and critical point theory. Recently, the authors of [11, 12] using the fixed point theorem on a cone obtained the existence results of positive solutions.
$$ \left \{ \textstyle\begin{array}{l} \Delta ^{2}u=f(x,\,u),\quad \; x\in \Omega , \\ u=\triangle u=0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$
(1.2)
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There are some researchers discussed the case when the right side of the equation is added a linear term of Δu, see [13‐17]. The authors of [13‐17] mainly applied variational methods and critical theory to discuss the existence of nontrivial solutions of BVP (1.3).
$$ \left \{ \textstyle\begin{array}{l} \Delta ^{2}u=c\,\Delta u+f(x,\,u),\quad \; x\in \Omega , \\ u=\triangle u=0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$
(1.3)
For the general case that the nonlinearity f contains Δu, the existence of a solution has also been discussed by Pao [18] and Wang [19]. In [18], Pao introduced the concept of a pair of coupled upper and lower solutions, and he showed that if BVP (1.1) has a pair of coupled upper and lower solutions ũ and û, then BVP (1.1) has a solution between ũ and û under f satisfies proper regularity condition. See [18, Theorem 2.1]. However, it is not easy to find a pair of coupled upper and lower solutions in the applications. In [19], Wang used a pair of uncoupled upper and lower solutions ũ and û with \(\tilde{u}\ge \hat{u}\) and \(\Delta \tilde{u}\le \Delta \hat{u}\), to replace the pair of Pao’s coupled upper and lower solutions, under the nonlinearity \(f(x,\,u,\,v)\) satisfies an order condition obtained the existence of a solution between ũ and û and established a monotonic iterative program for seeking the solutions. See [19, Theorem 2.1]. Recently, for the case of that f is nonnegative, the authors of [20, 21] obtained existence of positive solutions by applying the theory of fixed point index in cons, which extended the results in [11, 12].
The purpose of this paper is further to obtain the existence results of BVP (1.1). In §2, we build an existence result and an existence and uniqueness result under that \(f(x,\,\xi ,\,\eta )\) satisfies some growth conditions on ζ and η. See Theorem 2.1 and Theorem 2.2. In §3, we establish a theorem of lower and upper solutions providing a pair of upper and lower solutions ũ and û with \(\tilde{u}\ge \hat{u}\) and \(\Delta \tilde{u}\le \Delta \hat{u}\) to obtain a solution u satisfying \(\hat{u}\le u\le \tilde{u}\) and \(\Delta \hat{u}\le \Delta u\le \Delta \tilde{u}\). See Theorem 3.1. In this theorem, we delete the wang’s order condition in [19]. In §4, we use the theorem of lower and upper solutions to obtain the existence result of a positive solution, and present an example to illustrates the applicability of our results.
Recently, the case when the nonlinearity f is also dependent on the gradient ∇u, has be studied, see [22, 23]. In [22], the authors considered the case that Ω is a annulus and \(f (x,\,\xi ,\,\zeta ,\,\eta )\) is radially symmetric on x, and obtained the existence and uniqueness results of radially symmetry solutions by discussing the corresponding boundary value problem of fourth-order ordinary differential equation and using the Leray-Schauder fixed-point theorem. In [23], the authors considered the case that the biharmonic operator \(\Delta ^{2} u\) on the right is replaced by the sign-changing Kirchhoff type \(p(x)\)-biharmonic operator \(\Delta ^{2}_{k,p}u\), and obtained the existence of solutions in weak sense by applied topological theory together with Galerkin method and fixed-point arguments. The problem involving \(p(x)\)-Laplacian-like operator and nonlinear gradient term has also been discussed in [24] by the topological degree theory of demi-continuous operators of generalized type. For these problems, due to the appearance of nonlinear gradient terms, they have no variational structure, and the variational method and critical point theory cannot be directly applied to them. It should be pointed out that our method of discussing BVP (1.1) can be applied to these problems, but we have to overcome some technical difficulties caused by gradient terms. For the ideas, See [25].
$$ \left \{ \textstyle\begin{array}{l} {\Delta}^{2} u = f(x,\,u,\,\nabla u,\,\Delta u),\qquad x\in \Omega , \\ u=\Delta u=0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$
(1.4)
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2 Existence and uniqueness
To discuss BVP (1.1), we review some concepts and conclusions on the linear elliptic boundary value problem (LBVP) where \(h\in L^{p}(\Omega )(1< p<\infty )\). Let \(W^{n,p}(\Omega )\) be the usual Sobolev space on domain Ω, and \(H^{n}(\Omega ):=W^{n,2}(\Omega )\). Let \(C^{\mu}(\overline{\Omega })\) be the μ-order Hölder continuous function space on Ω̅ with exponent \(\mu \in (0,\,1)\), and \(C^{n+\mu}(\overline{\Omega }):=\{u\in C^{n}(\overline{\Omega })\;|\;D^{ \alpha }u\in C^{\mu}(\overline{\Omega }),\;|\alpha |=n\,\}\). By the \(L^{p}\)-theory of the linear elliptic equations [26], for every \(h\in L^{p}(\Omega )\), LBVP (2.1) has a unique strong solution \(u:=T h\in W^{2,p}(\Omega )\cap W^{1,p}_{0}(\Omega )\), and the solution operator \(T: L^{p}(\Omega )\to W^{2,p}(\Omega )\) is a linear bounded operator. When \(h\in C^{\mu}(\overline{\Omega })\), by the Schauder theory of the linear elliptic equations [26], \(u=T h\in C^{2+\mu}(\overline{\Omega })\) is a classical solution of LBVP (2.1). When \(p>\tfrac{N}{2}\), by the compactness of the Sobolev embedding \(W^{2,p}(\Omega )\hookrightarrow C(\overline{\Omega })\), \(T: L^{p}(\Omega )\to C(\overline{\Omega })\) is a linear compact operator. Especially, the restriction of T on \(C(\overline{\Omega })\), \(T: C(\overline{\Omega })\to C(\overline{\Omega })\) is compact. When \(h\ge 0\), by the maximum principle of elliptic operators [27], \(u=Th\ge 0\). Hence, \(T: C(\overline{\Omega })\to C(\overline{\Omega })\) is a positive compact linear operator.
$$ \left \{ \textstyle\begin{array}{l} -\Delta u = h(x)\,,\qquad x\in \Omega \,,\qquad \\ u(x)=0,\qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$
(2.1)
It is well-known that the Laplace operator −Δ with the boundary condition \(u|_{\partial \Omega}=0\) has a minimum positive real eigenvalue \(\lambda _{1}\), which given by Moreover, \(\lambda _{1}\) has a positive unit eigenfunction, that is, there exists a function \(\phi _{1}\in C^{2}(\overline{\Omega })\cap C^{+}(\overline{\Omega })\) with \(\|\phi _{1}\|_{C(\overline{\Omega })}=1\) satisfies the equation
$$ \lambda _{1}=\inf \left \{{\frac{\|\nabla u\|_{2}}{\|u\|_{2}}\;\bigg| \;u\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega ),\,\|u\|_{2}\ne 0}\right \}. $$
(2.2)
$$ \left \{ \textstyle\begin{array}{l} -\Delta \phi _{1} = \lambda _{1}\,\phi _{1},\qquad x\in \Omega , \\ \phi _{1}(x)=0,\qquad x\in \partial \Omega . \end{array}\displaystyle \right . $$
(2.3)
For \(h\in L^{2}(\Omega )\), the solution \(u=T h\) of LBVP (2.1) satisfies In fact, since \(u=T h\in H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\) is a \(L^{2}\)-solution of LBVP (2.1), By (2.2) and the Schwartz inequality, we have Hence (2.4) holds.
$$ \|T h\|_{2}\le \frac{1}{\lambda _{1}}\,\|h\|_{2},\qquad h\in L^{2}( \Omega ). $$
(2.4)
$$ \lambda _{1}{\|u\|_{2}}^{2}\le{\|\nabla u\|_{2}}^{2}=(-\Delta u,\,u) \le \|-\Delta u\|_{2}\|u\|_{2}= \|h\|_{2}\|u\|_{2}. $$
Theorem 2.1
Let \(f:\overline{\Omega}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) be continuous. If there exist constant \(a,\,b\ge 0\) satisfying \(\frac{a}{\lambda _{1}^{\;2}}+\frac{b}{\lambda _{1}}<1\) and \(c>0\) such that then BVP (1.1) has a solution \(u\in H^{4}(\Omega )\cap C^{2}(\overline{\Omega })\). Moreover, when f satisfies (H0), the solution \(u\in C^{4}(\overline{\Omega })\) is a classical solution.
$$ |f(x,\,\xi ,\,\eta )|\le a |\xi |+b |\eta |+c, \qquad (x,\,\xi ,\, \eta )\in \overline{\Omega}\times \mathbb{R}\times \mathbb{R}, $$
(2.5)
Proof
For BVP (1.1), set \(v=-\Delta u\), then it becomes second order elliptic boundary value problem We firstly show that BVP (2.6) has a \(L^{2}\)-solution \(v_{0}\in H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\). Since the solution operator of LBVP (2.1) \(T: L^{2}(\Omega )\to H^{2}(\Omega )\) is a linear bounded operator, by the compactness of the Sobolev embedding \(H^{2}(\Omega )\hookrightarrow L^{2}(\Omega )\), \(T: L^{2}(\Omega )\to L^{2}(\Omega )\) is a linear compact operator. Define a mapping F on \(L^{2}(\Omega )\) by By (2.5), \(F: L^{2}(\Omega )\to L^{2}(\Omega )\) is continuous. By (2.5) and (2.4), we have Hence, the composite mapping of F and T is completely continuous. By the definition of T, the \(L^{2}\)-solution of BVP (2.6) is equivalent to the fixed point of A. We use the Schauder fixed point theorem [28] to show that A has a fixed point. Set Then \(\bar{B}(R_{0},\,L^{2})\) is a bounded closed convex set in \(L^{2}(\Omega )\). For every \(v\in \bar{B}(R_{0},\,L^{2})\), by (2.4) and (2.8), we have Hence, \(A v\in \bar{B}(R_{0},\,L^{2})\). This means that \(A(\bar{B}(R_{0},\,L^{2}))\subset \bar{B}(R_{0},\,L^{2})\). By the Schauder fixed point theorem, A has a fixed point \(v_{0}\in \bar{B}(R_{0},\,L^{2})\). Since \(v_{0}=A v_{0}=T(F(v_{0}))\). By the definition of T, \(v_{0}\) is a \(L^{2}\)-solution of LBVP (2.1) for \(h=F(v_{0})\in L^{2}(\Omega )\). By the definition of F, \(v_{0}\in H^{2}(\Omega )\cap H_{0}^{1}(\Omega )\) is \(L^{2}\)-solution of BVP (2.6). Hence, \(u_{0}=Tv_{0}\in H^{4}(\Omega )\) is a \(L^{2}\)-solution of BVP (1.1).
$$ \left \{ \textstyle\begin{array}{l} -\Delta v = f(x,\,Tv,\, -v),\qquad x\in \Omega , \\ v(x)=0,\qquad x\in \partial \Omega . \end{array}\displaystyle \right . $$
(2.6)
$$ F(u)(x)=f(x,\,Tv(x),\,-v(x)),\qquad v\in L^{2}(\Omega ),\;\;x\in \Omega . $$
(2.7)
$$\begin{aligned} \|F(v)\|_{2} &\le a\|Tv\|_{2}+b\|v\|_{2}+c|\Omega |^{1/2} \\ &\le \Big(\tfrac{a}{\lambda _{1}}+b\Big)\|v\|_{2}+c|\Omega |^{1/2}, \quad v\in L^{2}(\Omega ). \end{aligned}$$
(2.8)
$$ A=T\circ F: L^{2}(\Omega )\to L^{2}(\Omega ) $$
(2.9)
$$ R_{0}= \frac{c|\Omega |^{1/2}}{\lambda _{1}\big(1-\big(\frac{a}{\lambda _{1}^{\;2}}+\frac{b}{\lambda _{1}}\big)\big)}, \qquad \bar{B}(R_{0},\,L^{2})=\{v\in L^{2}(\Omega )\;|\;\|v\|_{2}\le R_{0} \}. $$
$$\begin{aligned} \|Av\|_{2}=\|T(F(v))\|_{2} &\le \tfrac{1}{\lambda _{1}}\|F(v)\|_{2} \\ &\le \Big(\tfrac{a}{\lambda _{1}^{\;2}}+\tfrac{b}{\lambda _{1}}\Big) \|v\|_{2}+\tfrac{c|\Omega |^{1/2}}{\lambda _{1}} \\ &\le \Big(\tfrac{a}{\lambda _{1}^{\;2}}+\tfrac{b}{\lambda _{1}}\Big)R_{0}+ \tfrac{c|\Omega |^{1/2}}{\lambda _{1}}=R_{0}. \end{aligned}$$
Next, we show that \(u_{0}\in C^{2}(\overline{\Omega })\). For this we need to prove that \(v_{0}\in C^{1}(\overline{\Omega })\). Set \(h_{0}=F(v_{0})\). Then \(v_{0}=Th_{0}\) is the solution of LBVP (2.1) for \(h=h_{0}\). By (2.7) and (2.5)
$$ |h_{0}(x)|\le a\,|Tv_{0}(x)|+b\,|v_{0}(x)|+c,\qquad x\in \Omega . $$
(2.10)
If \(p_{0}:=2< N\), choose \(p_{1}=\tfrac{Np_{0}}{N-p_{0}}(>p_{0})\), then by the Sobolev embedding theorem, \(H^{2}(\Omega )\hookrightarrow L^{p_{1}}(\Omega )\). Since \(v_{0}\in H^{2}(\Omega )\), it follows that \(v_{0}\in L^{p_{1}}(\Omega )\). By the definition of T, \(Tv_{0}\) is a \(L^{p_{1}}\)-solution of LBVP (2.1) for \(h=v_{0}\). Hence, \(Tv_{0}\in W^{2,p_{1}}(\Omega )\subset L^{p_{1}}(\Omega )\). Hence, by (2.10), \(h_{0}\in L^{p_{1}}(\Omega )\). So by existence and uniqueness of \(L^{p}\)-solution of LBVP (2.1), \(v_{0}=T h_{0}\in W^{2,p_{1}}(\Omega )\).
If \(p_{1}< N\), choose \(p_{2}=\frac{Np_{1}}{N-p_{1}}(>p_{1})\), then by the Sobolev embedding theorem, \(W^{2, p_{1}}(\Omega )\hookrightarrow L^{p_{2}}(\Omega )\). Hence \(v_{0}\in L^{p_{2}}(\Omega )\). Using the same argument as above, we can obtain that \(u_{0}=T h_{0}\in W^{2,p_{2}}(\Omega )\).
Continuing such procedure for \(p_{1},\, p_{2},\,p_{3},\,\ldots\, \), since the step length \(p_{k}-p_{k-1}(k=1,\,2,\,\ldots )\) is increasing, we can choose \(p>N\), such that \(h_{0}\in L^{p}(\Omega )\). Hence \(v_{0}=Th_{0}\in W^{2,p}(\Omega )\). By the Sobolev embedding theorem, \(W^{2,p}(\Omega )\hookrightarrow C^{1}(\overline{\Omega })\). Hence \(v_{0}\in C^{1}(\overline{\Omega })\). Since \(u_{0}=Tv_{0}\), by the Schauder theory of the linear elliptic equations, \(u_{0}\in C^{2+\mu}(\overline{\Omega })\) is a classical solution of LBVP (1.1) for \(h=v_{0}\).
Assume that f satisfies (H0). Then from (H0) and (2.7) we easily see the fact: there exists \(\alpha \in (0,\,\mu )\), such that \(h_{0}=F(v_{0})\in C^{\alpha}(\overline{\Omega })\). Hence, by the regularity of the solutions of LBVP (2.1), \(v_{0}=Th_{0}\in C^{2+\alpha}(\overline{\Omega })\), and thus \(u_{0}=Tv_{0}\in C^{4+\alpha}(\overline{\Omega })\) is a classical solution of BVP (1.1). □
Theorem 2.2
Let \(f:\overline{\Omega}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) be continuous. If there exist constant \(a,\,b\ge 0\) satisfying \(\frac{a}{\lambda _{1}^{\;2}}+\frac{b}{\lambda _{1}}<1\), such that then BVP (1.1) has a unique solution in \(H^{4}(\Omega )\cap C^{2}(\overline{\Omega })\).
$$\begin{aligned} |f(x,\,\xi _{2},\,\eta _{2})-f(x,\,\xi _{1},\,\eta _{2})| &\le a | \xi _{2}-\xi _{1}|+b |\eta _{2}-\eta _{1}|, \\ &\quad \textit{for}\quad x\in \overline{\Omega },\;\; (\xi _{1},\,\eta _{1}), \;(\xi _{2},\,\eta _{2})\in \times \mathbb{R}\times \mathbb{R}, \end{aligned}$$
(2.11)
Proof
Obviously, from (2.11) it follows that f satisfies (2.5). By Theorem 2.1, BVP (1.1) has a solution in \(H^{4}(\Omega )\cap C^{2}(\overline{\Omega })\). Let \(u_{1},\,u_{2}\in H^{4}(\Omega )\cap C^{2}(\overline{\Omega })\) be two solutions of BVP (1.1). Set \(v_{1}=-\Delta u_{1}\), \(v_{2}=-\Delta u_{2}\). Then \(v_{1}\), \(v_{2}\) are the solution of BVP (2.6). By the definition of T and F, By (2.7) and (2.11), we have By this and (2.4) we obtain that Hence, by (2.12) and (2.4), we have Since \(\tfrac{a}{\lambda _{1}^{\;2}}+\tfrac{b}{\lambda _{1}}<1\), from this inequality it follows that \(\|v_{2}-v_{1}\|=0\). Thus, \(v_{1}=v_{2}\), so we have \(u_{1}=Tv_{1}=Tv_{2}=u_{2}\). Hence, BVP (1.1) has a unique solution in \(H^{4}(\Omega )\cap C^{2}(\overline{\Omega })\). □
$$ v_{1}=T(F(v_{1})),\qquad v_{2}=T(F(v_{2})). $$
(2.12)
$$\begin{aligned} |F(v_{2})(x)-F(v_{1})(x)|&=|f(x,\,Tv_{2}(x),\,-v_{2}(x))-f(x,\,Tv_{1}(x), \,-v_{1}(x))| \\ &\le a\,|Tv_{2}(x)-Tv_{1}(x)|+b\,|v_{2}(x)-v_{1}(x)|, \qquad x\in \overline{\Omega }. \end{aligned}$$
$$\begin{aligned} \|F(v_{2})-F(v_{1})\|_{2} &\le a\|T(v_{2}-v_{1})\|_{2}+b\|v_{2}-v_{1} \|_{2} \\ &\le \big(\tfrac{a}{\lambda _{1}}+b\big)\|v_{2}-v_{1}\|_{2}. \end{aligned}$$
$$\begin{aligned} \|v_{2}-v_{1}\|_{2}=\|T(F(v_{2})-T(F(v_{1})\|_{2} &=\|T(F(v_{2})-F(v_{1})) \|_{2} \\ &\le \tfrac{1}{\lambda _{1}}\|F(v_{2})-f(v_{1})\|_{2} \\ &\le \big(\tfrac{a}{\lambda _{1}^{\;2}}+\tfrac{b}{\lambda _{1}}\big) \|v_{2}-v_{1}\|_{2}. \end{aligned}$$
Theorem 2.3
Let \(a,\,b\ge 0\) and \(c>0\) be constants, and \(\frac{a}{\lambda _{1}^{\;2}}+\frac{b}{\lambda _{1}}<1\). Then the fourth order linear elliptic boundary value has a unique classical solution \(\tilde{u}\in C^{4}(\overline{\Omega })\), moreover \(\tilde{u}\ge 0\), \(\Delta \tilde{u}\le 0\).
$$ \left \{ \textstyle\begin{array}{l} \Delta ^{2} u = a u-b\Delta u+c,\qquad x\in \Omega \,,\qquad \\ u=\Delta u=0, \qquad x\in \partial \Omega \end{array}\displaystyle \right . $$
(2.13)
Proof
Consider fourth order the elliptic boundary value It is easy to verify that the corresponding nonlinearity satisfies the condition (2.11). By Theorem 2.2, BVP (2.14) has a unique solution \(\tilde{u}\in H^{4}(\Omega )\cap C^{2}(\overline{\Omega })\). Set then \(\tilde{u}=T\tilde{v}\), \(\tilde{v}=T\tilde{h}\). By (2.16), \(h\ge 0\). By the positivity of T, \(\tilde{v}=T\tilde{h}\ge 0\), \(\tilde{u}=T\tilde{v}\ge 0\). Hence, \(\tilde{u}\ge 0\), \(\Delta \tilde{u}=-\tilde{v}\le 0\). This implies that ũ is also a solution of BVP (2.13). Similarly to BVP (2.14), it is easy to prove that ũ is the unique solution of BVP (2.13).
$$ \left \{ \textstyle\begin{array}{l} \Delta ^{2} u = a\,|u|+b\,|\Delta u|+c,\qquad x\in \Omega \,,\qquad \\ u=\Delta u=0, \qquad x\in \partial \Omega \end{array}\displaystyle \right . $$
(2.14)
$$ f(x,\,\xi ,\,\eta ):=a\,|\xi |+b\,|\eta |+c, \qquad (x,\,\xi ,\,\eta ) \in{\overline{\Omega }}\times \mathbb{R}\times \mathbb{R} $$
(2.15)
$$ \tilde{v}=-\Delta \tilde{u},\qquad \tilde{h}=F(\tilde{v})=a|(T \tilde{v})(x)|+b|\tilde{v}(x)|+c, $$
(2.16)
3 Theorem of upper and lower solutions
In this section, we present an existence result of BVP (1.1) by upper and lower solutions. If a function \(\tilde{u}\in C^{4}(\overline{\Omega })\) satisfies we call it an upper solution of BVP (1.1), and if a function \(\hat{u}\in C^{4}(\overline{\Omega })\) satisfies we call it a lower solution of BVP (1.1).
$$ \left \{ \textstyle\begin{array}{l} {\Delta}^{2} \tilde{u} \ge f(x,\,\tilde{u},\,\Delta \tilde{u}), \qquad x\in \Omega , \\ \tilde{u}\ge 0,\quad \Delta \tilde{u}\le 0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$
(3.1)
$$ \left \{ \textstyle\begin{array}{l} {\Delta}^{2} \hat{u} \le f(x,\,\hat{u},\,\Delta \hat{u}),\qquad x\in \Omega , \\ \hat{u}\le 0,\quad \Delta \hat{u}\ge 0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$
(3.2)
Lemma 3.1
Let ũ be an upper solution of BVP (1.1) and û a lower solution with \(\Delta \tilde{u}\le \Delta \hat{u}\). Then \(\tilde{u}\ge \hat{u}\).
Proof
Consider \(u=\tilde{u}-\hat{u}\). By the definitions of upper and lower solutions, \(u\in C^{2}(\overline{\Omega })\) satisfies: \(-\Delta u\ge 0\), \(u|_{\partial \Omega}\ge 0\), By the maximum principle of elliptic operators, \(u\ge 0\). Hence, \(\tilde{u}\ge \hat{u}\). □
Theorem 3.1
Let \(f: \overline{\Omega }\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) satisfy the condition (H0) and BVP (1.1) have an upper solution ũ and an lower solution û with \(\Delta \tilde{u}\le \Delta \hat{u}\). If f satisfies the following condition
(H1) for any \(x\in \overline{\Omega }\) and \(\xi \in [\hat{u}(x),\,\tilde{u}(x)]\), then BVP (1.1) has at least one classical solution in \(C^{4}(\overline{\Omega })\) satisfies: \(\hat{u}\le u\le \tilde{u}\), \(\Delta \hat{u}\ge \Delta u\ge \Delta \tilde{u}\).
$$\begin{aligned} &f(x,\,\xi ,\Delta \tilde{u}(x)) \le f(x,\,\tilde{u}(x),\,\Delta \tilde{u}(x)), \\ &f(x,\,\xi ,\Delta \hat{u} (x)) \ge f(x,\,\hat{u}(x),\,\Delta \hat{u}(x)), \end{aligned}$$
Proof
By Lemma 3.1, \(\hat{u}\le \tilde{u}\). Define functions \(\sigma _{1},\;\sigma _{2}: \overline{\Omega }\times \mathbb{R}\to \mathbb{R}\) by Clearly, \(\sigma _{1},\;\sigma _{2}: \overline{\Omega }\times \mathbb{R}\to \mathbb{R}\) are Lipschitz continuous on ξ or η and satisfy Choose a constant \(a\in (0,\,\lambda _{1}^{\;2})\), and define a function \(f^{*}: \overline{\Omega }\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) by Since f satisfies (H0), by the Lipschitz continuity of \(\sigma _{1}\) and \(\sigma _{2}\), we can easily see that \(f^{*}\) also satisfies (H0). By the definition (3.3) of \(\sigma _{2}\), we obtain that Hence, by (3.5) and (3.4), we have where Hence, \(f^{*}: \overline{\Omega }\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) satisfies the condition (2.5) of Theorem 2.1. By Theorem 2.1, the fourth order elliptic boundary value problem has a classical solution \(u^{*}\in C^{4}(\overline{\Omega })\). We prove that: \(\Delta \tilde{u}\le \Delta u^{*}\le \Delta \hat{u}\).
$$\begin{aligned} \begin{aligned} &\sigma _{1}(x,\,\xi )=\min \{\max \{\hat{u}(x),\;\xi \},\;\tilde{u}(x) \}, \\ &\sigma _{2}(x,\,\eta )=\min \{\max \{\Delta \tilde{u}(x),\;\eta \}, \;\Delta \hat{u}(x)\}. \end{aligned} \end{aligned}$$
(3.3)
$$\begin{aligned} \begin{aligned} &\quad \hat{u}(x)\le \sigma _{1}(x,\,\xi )\le \tilde{u}(x),\quad (x, \,\xi )\in \overline{\Omega }\times \mathbb{R}; \\ &\Delta \tilde{u}(x)\le \sigma _{2}(x,\,\eta ) \le \Delta \hat{u}(x), \quad (x,\,\eta )\in \overline{\Omega }\times \mathbb{R}. \end{aligned} \end{aligned}$$
(3.4)
$$ f^{*}(x,\,\xi ,\,\eta )= f(x,\,\sigma _{1}(x,\,\xi ),\,\sigma _{2}(x, \,\eta ))+a(\eta -\sigma _{2}(x,\,\eta )),\quad x\in \overline{\Omega },\;\;\xi ,\,\eta \in \mathbb{R}. $$
(3.5)
$$ |\eta -\sigma _{2}(x,\,\eta )|\le |\eta |+\|\Delta \tilde{u}\|_{C}+\| \Delta \hat{u}\|_{C}. $$
$$ |f^{*}(x,\,\xi ,\,\eta )|\le a|\eta |+a(\|\Delta \tilde{u}\|_{C}+\| \Delta \hat{u}\|_{C})+C_{0},\qquad (x,\,\xi ,\,\eta )\in \overline{\Omega }\times \mathbb{R}\times \mathbb{R}, $$
$$ C_{0}=\max \{\,|f(x,\,\xi ,\,\eta )|\;|\;\;x\in \overline{\Omega },\; \hat{u}(x)\le \xi \le \tilde{u}(x),\;\Delta \tilde{u}(x)\le \eta \le \Delta \hat{u}(x)\,\}. $$
$$ \left \{ \textstyle\begin{array}{l} {\Delta}^{2} u = f^{*}(x,\,u,\,\Delta u),\qquad x\in \Omega , \\ u=\Delta u=0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$
(3.6)
Conversely, if \(\Delta \tilde{u}\not \le \Delta u^{*}\), observing the function \(\phi (x)=\Delta u^{*}(x)-\Delta \tilde{u}(x)\), we have \(\min \limits _{x\in \overline{\Omega }}\phi (x)<0\). Since \(\phi |_{\partial \Omega}=\Delta u^{*}|_{\partial \Omega}-\Delta \tilde{u}|_{\partial \Omega}\ge 0\), by the maximum and minimum value theorem of continuous functions, there exists \(x_{0}\in \Omega \) such that \(\phi (x_{0})=\min \limits _{x\in \overline{\Omega }}\phi (x)<0\). By the properties of the minimum point of a \(C^{2}\)-function, \(\nabla \phi (x_{0})=0\) and \(\Delta \phi (x_{0})\ge 0\). Hence, by \(\phi (x_{0})=\Delta u^{*}(x_{0})-\Delta \tilde{u}(x_{0})<0\) and \(\Delta \phi (x_{0})=\Delta ^{2} u^{*}(x_{0})-\Delta ^{2}\tilde{u}(x_{0}) \ge 0\), we obtain that Hence, by the definitions (3.3), \(\sigma (x_{0},\,\Delta u^{*}(x_{0}))=\Delta \tilde{u}(x_{0})\). By the equation (3.6), the definition of \(f^{*}\), Condition (H1) and the definition of upper solution ũ, we have that is, \({\Delta}^{2} u^{*}(x_{0})<\Delta ^{2}\tilde{u}(x_{0})\), which contradict to the second inequality of (3.7). Hence, \(\Delta \tilde{u}\le \Delta u^{*}\).
$$ \Delta u^{*}(x_{0})< \Delta \tilde{u}(x_{0}),\qquad \Delta ^{2} u^{*}(x_{0}) \ge \Delta ^{2}\tilde{u}(x_{0}). $$
(3.7)
$$\begin{aligned} {\Delta}^{2} u^{*}(x_{0}) &= f^{*}(x_{0},\,u^{*}(x_{0}),\,\Delta u^{*}(x_{0})) \\ & =f(x_{0},\sigma _{1}(x_{0},u^{*}(x_{0})),\sigma _{2}(x_{0},\Delta u^{*}(x_{0}))+a( \Delta u^{*}(x_{0})-\sigma _{2}(x_{0},\Delta u^{*}(x_{0}))) \\ & =f(x_{0},\sigma _{1}(x_{0},u^{*}(x_{0})),\Delta \tilde{u}(x_{0}))+a( \Delta u^{*}(x_{0})-\Delta \tilde{u}(x_{0})) \\ & < f(x_{0},\,\sigma _{1}(x_{0},u^{*}(x_{0})),\,\Delta \tilde{u}(x_{0})) \\ & \le f(x_{0},\,\tilde{u}(x_{0}),\,\Delta \tilde{u}(x_{0})) \\ & \le \Delta ^{2}\tilde{u}(x_{0}), \end{aligned}$$
Similarly, we can show that \(\Delta u^{*}\le \Delta \hat{u}\). Hence, by Lemma 3.1, \(\hat{u}\le u^{*}\le \tilde{u}\). Now, by the definitions (3.3) of \(\sigma _{1}\) and \(\sigma _{2}\), we have Hence by the equation (3.6) and the definition (3.5) of \(f^{*}\), we have Hence, \(u^{*}\) is a classical solution of BVP (1.1) and it satisfies: \(\hat{u}\le u^{*}\le \tilde{u}\), \(\Delta \hat{u}\ge \Delta u^{*}\ge \Delta \tilde{u}\). □
$$ \sigma _{1}(x,\,u^{*}(x))=u^{*}(x),\qquad \sigma _{2}(x,\,\Delta u^{*}(x))= \Delta u^{*}(x),\quad x\in \overline{\Omega }. $$
$$\begin{aligned} {\Delta}^{2} u^{*}(x) &= f^{*}(x,\,u^{*}(x),\,\Delta u^{*}(x)) \\ & =f(x,\,\sigma _{1}(x,u^{*}(x)),\,\sigma _{2}(x,\Delta u^{*}(x))+a( \Delta u^{*}(x)-\sigma _{2}(x,\Delta u^{*}(x)) \\ & =f(x,\,u^{*}(x),\,\Delta u^{*}(x)),\qquad x\in \overline{\Omega }. \end{aligned}$$
Remark 3.1
In Theorem 3.1, clearly, if for every \(x\in \overline{\Omega }\), \(f(x,\,\xi ,\,\Delta \tilde{u}(x))\) and \(f(x,\,\xi ,\,\Delta \hat{u}(x))\) are monotone nondecreasing on ξ in \([\hat{u}(x),\;\tilde{u}(x)]\), then the condition (H1) holds. Especially, if for every \(x\in \overline{\Omega }\) and \(\eta \in [\Delta \tilde{u}(x),\;\Delta \hat{u}(x)]\), \(f(x,\,\xi ,\,\eta )\) is monotone nondecreasing on ξ in \([\hat{u}(x),\;\tilde{u}(x)]\), then the condition (H1).
4 Positive solutions
In this section we use Theorem 3.1 and the theorems of Sect. 2 to present an existence result of positive solution for BVP (1.1).
Theorem 4.1
Assume that \(f: \overline{\Omega }\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\) satisfies (H0) and the following conditions:
Then BVP (1.1) has at least one positive classical solution.
(F1)
for every \(x\in \overline{\Omega }\) and \(\eta \in (-\infty ,\,0]\), \(f(x,\, \xi ,\,\eta )\) is monotone nondecreasing on ξ in \([0,\,+\infty )\);
(F2)
there exist constants \(a_{0},\,b_{0}\ge 0\) with \(\frac{a_{0}}{\lambda _{1}^{\;2}}+\frac{b_{0}}{\lambda _{1}}>1\) and \(\delta >0\), such that
$$ f(x,\,\xi ,\,\eta )\ge a_{0}\xi -b_{0}\eta , \qquad x\in \overline{\Omega },\;\;\xi \ge 0,\;\;\eta \le 0,\;\; |(\xi ,\,\eta )| < \delta , $$
(F3)
there exist constants \(a_{1},\,b_{1}\ge 0\) with \(\frac{a_{1}}{\lambda _{1}^{\;2}}+\frac{b_{1}}{\lambda _{1}}<1\) and \(H>0\), such that
$$ f(x,\,\xi ,\,\eta )\le a_{1}\xi -b_{1}\eta , \qquad x\in \overline{\Omega },\;\;\xi \ge 0,\;\;\eta \le 0,\;\; |(\xi ,\,\eta )| >H. $$
Proof
For the condition (F3), choosing a positive constant by we have By Theorem 2.3, the linear fourth order elliptic boundary value has a unique solution \(\tilde{u}\in C^{4}(\overline{\Omega })\) satisfies: \(\tilde{u}\ge 0\), \(\Delta \tilde{u}\le 0\). By (4.1), ũ is an upper solution of BVP (1.1).
$$ C_{1}=\max \{\,|f(x,\,\xi ,\,\eta )-(a_{1}\xi -b_{1}\eta )\;|\;\; x \in \overline{\Omega },\;\xi \ge 0,\;\eta \le 0,\; |(\xi ,\,\eta )| \le H\}+1, $$
$$ f(x,\,\xi ,\,\eta )\le a_{1}\xi -b_{1}\eta +C_{1},\qquad x\in \overline{\Omega },\;\;\xi \ge 0,\;\;\eta \le 0. $$
(4.1)
$$ \left \{ \textstyle\begin{array}{l} \Delta ^{2} u = a_{1}u-b_{1}\Delta u+C_{0},\qquad x\in \Omega \,, \qquad \\ u=\Delta u=0, \qquad x\in \partial \Omega \end{array}\displaystyle \right . $$
(4.2)
On the other hand, choose \(\varepsilon =\min \big\{\tfrac{\delta}{2\sqrt{1+\lambda _{1}^{\;2}}}, \;\tfrac{C_{0}}{\lambda _{1}^{\;2}}\big\}\), and set \(\hat{u}=\varepsilon \phi _{1}(x)\), where δ is the constant in (F2) and \(\phi _{1}\) is the positive eigenfunction in (2.3). For every \(x\in \overline{\Omega }\), since \(\hat{u}(x)\ge 0\), \(\Delta \hat{u}(x)=-\lambda _{1}\hat{u}(x)\le 0\), and \(|(\hat{u}(x),\,\Delta \hat{u}(x))|\le \varepsilon \sqrt{1+\lambda _{1}^{ \;2}}<\delta \), by the condition (F2), we have Hence, û is a lower solution of BVP (1.1).
$$\begin{aligned} f^{*}(x,\,\hat{u}(x),\,\Delta \hat{u}(x)))&\ge a_{0}\hat{u}(x)-b_{0} \Delta \hat{u}(x) \\ & \ge (a_{0}+b_{0}\lambda _{1})\hat{u}(x) \\ & \ge \lambda _{1}^{\;2}\hat{u}(x)=\Delta ^{2}\hat{u}. \end{aligned}$$
Investigate the function \(v=\Delta \hat{u}-\Delta \tilde{u}\). Since by the maximum principle of elliptic operators, \(v\ge 0\). Hence, \(\Delta \tilde{u}\le \Delta \hat{u}\).
$$\begin{aligned} -\Delta v(x) &=-\Delta ^{2}\tilde{u}(x)-\Delta ^{2}\hat{u}(x) \\ & = a_{1}\tilde{u}(x)-b_{1}\Delta \tilde{u}(x)+C_{0}-\varepsilon \lambda _{1}^{\;2}\phi _{1}(x) \\ & \ge a_{1}\tilde{u}(x)-b_{1}\Delta \tilde{u}(x)+(C_{0}-\varepsilon \lambda _{1}^{\;2}) \\ & \ge a_{1}\tilde{u}(x)-b_{1}\Delta \tilde{u}(x)\ge 0,\quad x\in \overline{\Omega }, \end{aligned}$$
For these ũ and û, by the condition (F1), we easily sse that \(f(x,\,\xi ,\,\eta )\) satisfies the condition (H1). Hence, by Theorem 3.1, BVP (1.1) has a classical solution \(u^{*}\in C^{4}(\overline{\Omega })\) satisfies: \(\hat{u}\le u^{*}\le \tilde{u}\), \(\Delta \hat{u}\ge \Delta u^{*}\ge \Delta \tilde{u}\). This means that for every \(x\in \Omega \), \(u^{*}(x)\ge \hat{u}(x)=\varepsilon \phi _{1}(x)>0\). Hence, \(u^{*}\) is a positive solution of BVP (1.1). □
Example 4.1
Consider the following nonlinear fourth-order elliptic boundary value problem where a, b, c, d are positive constants. Clearly, BVP (4.3) has the trivial solution 0, we use Theorem 4.1 to show that it has at least one positive solution when \(\tfrac{a}{\lambda _{1}^{\;2}}+\tfrac{b}{\lambda _{1}}<1\).
$$ \left \{ \textstyle\begin{array}{l} \Delta ^{2} u =au-b\Delta u+c \sqrt{|u|}+d(\Delta u)^{3},\qquad x\in \Omega , \\ u=\Delta u=0, \qquad x\in \partial \Omega , \end{array}\displaystyle \right . $$
(4.3)
Corresponding to BVP (1.1), the nonlinearity is Obviously, f satisfies the conditions (H0) and (F1). Assume that \(\tfrac{a}{\lambda _{1}^{\;2}}+\tfrac{b}{\lambda _{1}}<1\), we verify that f satisfies (F2) and (F3).
$$ f(t,\,\xi ,\,\eta )=a\,\xi -b\eta + c\sqrt{|\xi |}+d\eta ^{3}. $$
(4.4)
Choose \(\delta =\min \big\{1,\,\tfrac{c^{2}}{\lambda _{1}^{\;4}},\,\sqrt{ \tfrac{b}{2d^{2}}}\big\}\). Then when \(\xi \ge 0\), \(\eta \le 0\), and \(|(\xi ,\,\eta )|<\delta \), by (4.4) we have Hence, f satisfies (F2) for \(a_{0}=a+\lambda _{1}^{\;2}\), \(b_{0}=\tfrac{b}{2}\).
$$\begin{aligned} f(x,\,\xi ,\,\eta ) &\ge a\xi -b\eta + c\sqrt{|\xi |}+d|(\xi ,\,\eta )|^{2} \eta \\ & \ge a\xi -b\eta + c\sqrt{|\xi |}+\tfrac{b}{2}\eta \\ & \ge a\xi -\tfrac{b}{2}\eta + c \sqrt{\xi}\cdot \tfrac{\sqrt{\xi}}{\sqrt{|(\xi ,\,\eta )|}} \\ & \ge (a+\lambda _{1}^{\;2})\xi -\tfrac{b}{2}\eta . \end{aligned}$$
Since \(\tfrac{a}{\lambda _{1}^{\;2}}+\tfrac{b}{\lambda _{1}}<1\), we can choose \(\varepsilon >0\) such that \(\tfrac{a+2\varepsilon}{\lambda _{1}^{\;2}}+ \tfrac{b+\varepsilon}{\lambda _{1}}<1\). Choose \(H=\tfrac{c^{2}}{4\varepsilon ^{2}}\). Then when \(\xi \ge 0\), \(\eta \le 0\), and \(|(\xi ,\,\eta )|>H\), by (4.4) we have Hence, f satisfies (F3) for \(a_{1}=a+2\varepsilon \), \(b_{1}=b+\varepsilon \).
$$\begin{aligned} f(x,\,\xi ,\,\eta ) &\le a\xi -b\eta + 2\cdot \tfrac{c}{2\sqrt{\varepsilon}}\cdot |\sqrt{\varepsilon \xi}| \\ & \le a\xi -b\eta + \varepsilon \xi +\tfrac{c^{2}}{4\varepsilon} \\ & \le a\xi -b\eta + \varepsilon \xi +\varepsilon |(\xi ,\,\eta )| \\ & \le a\xi -b\eta + \varepsilon \xi +\varepsilon (\xi -\eta ) \\ & = (a+2\varepsilon )\xi -(b+\varepsilon )\eta . \end{aligned}$$
This example illustrates the applicability of Theorem 4.1.
Acknowledgements
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