1 Introduction
Let Ω be a smooth bounded domain in
\(\mathbb{R}^{2}\),
\(W^{1,p}(\Omega )\) be the usual Sobolev space and
\(W_{0}^{1,p}(\Omega )\) be the closure of
\(C_{0}^{\infty }(\Omega )\) in
\(W^{1,p}(\Omega )\). For
\(1\leq p<2\), the classical Sobolev theorem says that
$$\begin{aligned} W_{0}^{1,p}(\Omega )\hookrightarrow L^{q}(\Omega )\quad \text{for any } 1< q\leq2p/(2-p). \end{aligned}$$
As the limit case of the Sobolev inequality, the famous Trudinger–Moser inequality [
3,
4] states
$$ \sup_{u\in W^{1,2}_{0} (\Omega ), \Vert \nabla u \Vert _{2}\leq1} \int_{\Omega }e^{4\pi u^{2}}\,dx < \infty . $$
(1.1)
This inequality is sharp in the sense that, for any
\(p>4\pi\), there exists a sequence
\(\{u_{j}\}\subset W_{0}^{1,2}(\Omega )\) with
\(\Vert \nabla u_{j} \Vert _{2}= 1\) such that
\(\int_{\Omega } e^{p u_{j}^{2}}\,dx\rightarrow \infty \) as
\(j\rightarrow\infty\). Furthermore, let
\(\{ u_{k}\}\) be a sequence of function in
\(W^{1,2}_{0}(\Omega )\) with
\(\Vert \nabla u_{k} \Vert _{2}=1\) such that
\(u_{k}\rightharpoonup u\) weakly in
\(W^{1,2}_{0}(\Omega )\). Lions [
5] proved that, for any
\(p<1/(1- \Vert \nabla u \Vert ^{2}_{2})\), we have
$$ \limsup_{k\rightarrow \infty } \int_{\Omega }e^{4\pi pu_{k}^{2}}\,dx< \infty . $$
(1.2)
If
\(u\not\equiv0\), the inequality (
1.2) gives more information than the Trudinger–Moser inequality (
1.1). If
\(u\equiv0\), (
1.2) is a consequence of (
1.1). Motivated by this, Adimurthi and Druet [
6] proved that, for any
α,
\(0\leq \alpha <\lambda _{1}(\Omega )\),
$$ \sup_{u\in W^{1,2}_{0} (\Omega ), \Vert \nabla u \Vert _{2}\leq1} \int_{\Omega }e^{4\pi u^{2}(1+\alpha \Vert u \Vert ^{2}_{2})}\,dx < \infty , $$
(1.3)
where
\(\lambda _{1}({\Omega })\) is the first eigenvalue of the Laplace operator with respect to Dirichlet boundary condition. If
\(\alpha \geq \lambda _{1}(\Omega )\), then the supremum in (
1.3) is infinity. The inequality (
1.3) provides valuable supplementary information on (
1.2). Note that if
\(\alpha =0\), (
1.3) becomes the classical Trudinger–Moser inequality. Adimurthi and Druet’s result was extended by Yang to high dimensions [
7] and compact Riemannian surfaces [
8], and by Tintarev to a stronger version [
9].
Denote
$$ \Vert u \Vert _{1,\alpha }= \biggl( \int_{\Omega } \vert \nabla u \vert ^{2}\,dx-\alpha \int_{\Omega }u^{2}\,dx \biggr)^{1/2} $$
(1.4)
for any
\(u\in W^{1,2}_{0}(\Omega )\) with
\(\int_{\Omega } \vert \nabla u \vert ^{2}\,dx-\alpha \int _{\Omega }u^{2}\,dx\geq0\). In [
1], Yang proved that, for any
α,
\(0\leq \alpha <\lambda _{1}(\Omega )\), we have
$$ \sup_{u\in W^{1,2}_{0}(\Omega ), \Vert u \Vert _{1,\alpha }\leq1} \int_{\Omega }e^{4\pi u^{2}}\,dx< \infty $$
(1.5)
and the supremum can be attained by some function
\(u_{0}\in W^{1,2}_{0}(\Omega )\cap C^{1}(\overline{\Omega })\) with
\(\Vert u_{0} \Vert _{1,\alpha }=1\). Let
\(\lambda _{1}(\Omega )<\lambda _{2}(\Omega )<\cdots\) be all distinct eigenvalues of the Laplace operator with respect to Dirichlet boundary condition and
\(E_{\lambda _{j}(\Omega )}\) be the eigenfunction space associated to
\(\lambda _{j}(\Omega )\). Noting that
\(W^{1,2}_{0}(\Omega )\) is a Hilbert space, for any positive integer
l, we have
$$\begin{aligned} W^{1,2}_{0}(\Omega )=E_{l}\oplus E_{l}^{\perp}, \end{aligned}$$
where
$$ E_{l}=E_{\lambda _{1}(\Omega )}\oplus E_{\lambda _{2}(\Omega )}\oplus\cdots\oplus E_{\lambda _{l}(\Omega )} $$
(1.6)
and
$$ E_{l}^{\perp}= \biggl\{ u\in W^{1,2}_{0}( \Omega ): \int_{\Omega }uv\,dx=0, \forall v\in E_{l} \biggr\} . $$
(1.7)
It was also proved by Yang [
1] that, for any
α,
\(0\leq \alpha <\lambda _{l+1}(\Omega )\), we have
$$ \sup_{u\in E_{l}^{\perp}, \Vert u \Vert _{1,\alpha }\leq1} \int_{\Omega }e^{4\pi u^{2}}\,dx< \infty $$
(1.8)
and the supremum can be attained by some
\(u_{0}\in E_{l}^{\perp}\cap C^{1}(\overline{\Omega })\) with
\(\Vert u_{0} \Vert _{1,\alpha }=1\). The analogs of (
1.5) and (
1.8) still hold on compact Riemannian surfaces.
Our first result is the following.
When the high order eigenvalues are involved, we have a similar result.
Similar results hold on compact Riemannian surfaces. Denote by
\((\Sigma , g)\) a compact Riemannian surface without boundary, by
\(\nabla _{g}\) its gradient operator and by
\(\Delta _{g}\) the Laplace–Beltrami operator, respectively. Let
\(\lambda _{1}(\Sigma)\) be the first eigenvalue of
\(\Delta _{g}\). Denote
$$ \Vert u \Vert _{1,\alpha }= \biggl( \int_{\Sigma} \vert \nabla _{g} u \vert ^{2} \,dx-\alpha \int_{\Sigma }u^{2}\,dv_{g} \biggr)^{1/2} $$
(1.9)
for all
\(u\in W^{1,2}(\Sigma )\) with
\(\int_{\Sigma} \vert \nabla _{g} u \vert ^{2}\,dx-\alpha \int _{\Sigma}u^{2}\,dv_{g}\geq0\). Then we have the following theorem.
If
h is strictly positive and
\(J(u)\) has no minimizer on
\(\mathcal {H}=\{ u\in W^{1,2}(\Sigma ): \int_{\Sigma}u\,dv_{g}=0\}\), Yang and Zhu [
10] calculated the infimum of
\(J(u)\) on
\(\mathcal{H}\) by using the method of blow-up analysis. One may refer to [
11] for earlier results on the functional
$$\frac{1}{2} \int_{\Sigma} \vert \nabla _{g} u \vert ^{2} \,dv_{g}+8\pi \int_{\Sigma }u\,dv_{g}-8\pi\log \int_{\Sigma}he^{u} \,dv_{g}. $$
Let
\(\lambda _{1}(\Sigma )<\lambda _{2}(\Sigma )<\cdots\) be all distinct eigenvalues of
\(\Delta _{g}\) and
\(E_{\lambda _{i}(\Sigma )}\) be the eigenfunction space associated to
\(\lambda _{i}(\Sigma )\). For any positive integer
l, denote
$$\begin{aligned} E_{l}=E_{\lambda _{1}(\Sigma )}\oplus E_{\lambda _{2}(\Sigma )}\oplus\cdots \oplus E_{\lambda _{l}(\Sigma )} \end{aligned}$$
and
$$\begin{aligned} E_{l}^{\perp}= \biggl\{ u\in W^{1,2}(\Sigma ): \int_{\Sigma }uv\,dv_{g}=0, \forall v\in E_{l} \biggr\} . \end{aligned}$$
Similar to Theorem
2, we obtain the following.
Existence of extremal functions for Trudinger–Moser inequality can be traced back to Carleson and Chang [
12], where the unit ball case was treated. Later contributions in this direction include M. Struwe [
13], Flucher [
14], Lin [
15], Ding–Jost–Li–Wang [
11], Adimurthi–Struwe [
16], Li [
17], Adimurthi–Druet [
6], and so on. In our proof, we use the blow-up method. Compared with [
1], there are some different key points. First, we derive the different Euler–Lagrange equation on which the analysis is performed. Then we prove that
h must be positive at the blow-up point. Hence we use the different scaling when define the maximizing sequences of functions. We also obtain the different upper bound of the subcritical functionals. Finally, when proving the existence of the extremal function, we obtain the different lower bounds for the integrals of test functions constructed in Sects.
2‐
5. It should be remarked that our analysis on the weight
h is essentially different from that of Yang and Zhu [
2], where a weak version of Trudinger–Moser inequality was studied.
The rest of the paper is arranged as follows. In Sects.
2 and
3, we prove the main results in the Euclidean case (Theorems
1 and
2). In Sects.
4 and
5, we prove the main results in the Riemannian surface case (Theorems
3 and
5).
4 Proof of Theorem 3
First, we prove that, for any
\(0<\epsilon <4\pi\), there exists some
\(u_{\epsilon }\in C^{1}(\Sigma )\) such that
$$ \int_{\Sigma}he^{(4\pi-\epsilon ) u_{\epsilon }^{2}}\,dv_{g}=\sup _{u\in W^{1,2}(\Sigma ),\int_{\Sigma }u \,dv_{g}=0, \Vert u \Vert _{1,\alpha }\leq1} \int_{\Sigma }he^{(4\pi-\epsilon ) u^{2}}\,dv_{g} $$
(4.1)
with
\(\Vert u_{\epsilon }\Vert _{1,\alpha }=1\) and
\(\int_{\Sigma}u_{\epsilon }\,dv_{g}=0\).
The main procedure of the proof is as follows. Since
\(0\leq \alpha <\lambda _{1}(\Sigma )\), we may choose a bounded sequence
\(u_{j}\) in
\(W^{1,2}(\Sigma )\) such that
$$\begin{aligned} \int_{\Sigma}he^{(4\pi-\epsilon )u_{j}^{2}}\,dv_{g}\rightarrow\sup _{u\in W^{1,2}(\Sigma ),\int_{\Sigma }u \,dv_{g}=0, \Vert u \Vert _{1,\alpha }\leq1} \int_{\Sigma }he^{(4\pi-\epsilon )u^{2}}\,dv_{g}. \end{aligned}$$
There exists some
\(u_{\epsilon }\in W^{1,2}(\Sigma )\) such that up to a subsequence,
$$\begin{aligned} &u_{j}\rightharpoonup u_{\epsilon } \quad\text{weakly in } W^{1,2}(\Sigma ), \\ &u_{j}\rightarrow u_{\epsilon } \quad \text{strongly in } L^{2}(\Sigma ), \\ &u_{j}\rightarrow u_{\epsilon } \quad \text{a.e. in }\Sigma . \end{aligned}$$
Using the same argument as in the proof of Theorem 3 in [
1], we get
\(he^{(4\pi-\epsilon )u_{j}^{2}}\) is bounded in
\(L^{q}\) for some
\(q>1\). Hence
\(he^{(4\pi-\epsilon )u_{j}^{2}}\rightarrow he^{(4\pi-\epsilon )u_{\epsilon }^{2}}\) strongly in
\(L^{1}(\Sigma )\). Hence (
4.1) holds. The fact that
\(\int_{\Sigma }u_{j}\,dv_{g}=0\) implies
\(\int_{\Sigma }u_{\epsilon }\,dv_{g}=0\). We also have
\(\Vert u_{\epsilon }\Vert _{1,\alpha }=1\) by contradiction as in the proof of Lemma
6.
Moreover,
\(u_{\epsilon }\) satisfies the Euler–Lagrange equation
$$\begin{aligned} \textstyle\begin{cases}\Delta_{g} u_{\epsilon}-\alpha u_{\epsilon }=\frac{1}{\lambda _{\epsilon}}hu_{\epsilon}e^{(4\pi-\epsilon ) u_{\epsilon}^{2}}-\frac{\mu_{\epsilon }}{\lambda _{\epsilon }},\\ \lambda_{\epsilon}=\int_{\Sigma } h u_{\epsilon}^{2}e^{(4\pi-\epsilon) u_{\epsilon}^{2}}\,dv_{g},\\ \mu_{\epsilon }=\frac{1}{{\mathrm{ Vol}_{g}}(\Sigma )}\int_{\Sigma}h u_{\epsilon }e^{(4\pi -\epsilon )u_{\epsilon }^{2}}\,dv_{g}, \end{cases}\displaystyle \end{aligned}$$
where
\(\Delta _{g}\) denotes the Laplace–Beltrami operator.
Denote
\(c_{\epsilon }= \vert u_{\epsilon }(x_{\epsilon }) \vert =\max_{\Sigma } \vert u_{\epsilon }\vert \). If
\(c_{\epsilon }\) is bounded, the existence of the extremal function follows from the elliptic estimates. We assume that
\(c_{\epsilon }\rightarrow+\infty\) and
\(x_{\epsilon }\rightarrow p\in \Sigma \). Similar to Lemma
7, we have
\(h(p)>0\). Choosing an isothermal coordinate system
\((U,\phi)\) near
p such that the metric
g can be written as
\(g=e^{f}(dx_{1}^{2}+dx_{2}^{2})\), where
\(f\in C^{1}(\phi(U), \mathbb {R})\) and
\(f(0)=0\). Denote
\(\Omega =\phi(U)\),
\(\widetilde{u}_{\epsilon }=u_{\epsilon }\circ \phi^{-1}\) and
\(\widetilde{x}_{\epsilon }=\phi(x_{\epsilon })\). Let
$$\begin{aligned} r_{\epsilon }=\sqrt{ \lambda _{\epsilon }}\bigl[h(p)\bigr]^{-1/2}c_{\epsilon }^{-1}e^{-(2\pi-\epsilon /2)c_{\epsilon }^{2}}. \end{aligned}$$
Define
$$\begin{aligned} \psi_{\epsilon }(x)=c_{\epsilon }^{-1}\widetilde{u}_{\epsilon }( \widetilde{x}_{\epsilon }+r_{\epsilon }x) \end{aligned}$$
and
$$\begin{aligned} \varphi_{\epsilon }(x)=c_{\epsilon }\bigl(\widetilde{u}( \widetilde{x}_{\epsilon }+r_{\epsilon }x)-c_{\epsilon }\bigr) \end{aligned}$$
for
\(x\in \Omega _{\epsilon }=\{x\in\mathbb{R}^{2}: \widetilde{x}_{\epsilon }+r_{\epsilon }x\in \Omega \}\). Then we get
$$\begin{aligned} &{-}\Delta _{\mathbb{R}^{2}}\psi_{\epsilon }=e^{f(\widetilde{x}_{\epsilon }+r_{\epsilon }x)} \biggl(\alpha r_{\epsilon }^{2}\psi_{\epsilon }+\frac{h\psi_{\epsilon }e^{(4\pi-\epsilon )(\widetilde{u}_{\epsilon }^{2}-c_{\epsilon }^{2})}}{c_{\epsilon }^{2} h(p)}- \frac{\mu_{\epsilon }}{c_{\epsilon }^{3} e^{(4\pi-\epsilon )c_{\epsilon }^{2}}h(p)} \biggr), \\ &{-}\Delta _{\mathbb{R}^{2}}\varphi_{\epsilon }=e^{f(\widetilde{x}_{\epsilon }+r_{\epsilon }x)} \biggl(\alpha r_{\epsilon }^{2} c_{\epsilon }^{2} \psi_{\epsilon }+\frac{h\psi_{\epsilon }e^{(4\pi-\epsilon )(1+\psi _{\epsilon })\varphi _{\epsilon }}}{ h(p)}-\frac{\mu_{\epsilon }}{c_{\epsilon }e^{(4\pi-\epsilon )c_{\epsilon }^{2}}h(p)} \biggr), \end{aligned}$$
where
\(-\Delta _{\mathbb{R}^{2}}\) is the usual Laplace operator in
\(\mathbb {R}^{2}\). By the same argument as in Sect.
2.2, we obtain
$$\begin{aligned} \psi_{\epsilon }\rightarrow1 \quad\text{in } C^{1}_{\mathrm{loc}} \bigl(\mathbb{R}^{2}\bigr) \end{aligned}$$
and
$$\begin{aligned} \varphi_{\epsilon }\rightarrow\varphi\quad\text{in } C^{1}_{\mathrm{loc}} \bigl(\mathbb{R}^{2}\bigr), \end{aligned}$$
where
$$\begin{aligned} \varphi(x)=-\frac{1}{4\pi}\log \bigl(1+\pi \vert x \vert ^{2} \bigr) \end{aligned}$$
and
$$\begin{aligned} \int_{\mathbb{R}^{2}} e^{8\pi\varphi}\,dx=1. \end{aligned}$$
We also have
\(c_{\epsilon }u_{\epsilon }\rightharpoonup G\) weakly in
\(W^{1,q}(\Sigma )\) for all
\(1< q<2\), and
\(c_{\epsilon }u_{\epsilon }\rightarrow G\) in
\(C^{1}_{\mathrm{loc}}(\Sigma \backslash\{p\})\cap L^{2}(\Sigma )\), where
G is Green function satisfying
$$\begin{aligned} \Delta _{g} G-\alpha G=\delta_{p}-\frac{1}{\mathrm{Vol}_{g}(\Sigma )}\quad \text{in } \Sigma \end{aligned}$$
and
\(\int_{\Sigma}G\,dv_{g}=0\). As before,
G can be represented by
$$\begin{aligned} G=-\frac{1}{2\pi}\log r+A_{p} +\Phi_{p}, \end{aligned}$$
where
r is the geodesic distance from
p,
\(A_{p}\) is a constant and
\(\Phi_{p}\in C^{1}(\Sigma )\) with
\(\Phi_{p}(p)=0\).
Similar to (
2.11), we can get
$$ \sup_{u\in W^{1,2}(\Sigma ),\int_{\Sigma }u \,dv_{g}=0, \Vert u \Vert _{1,\alpha }\leq 1} \int _{\Sigma }he^{4\pi u^{2}}\,dv_{g}\leq \gamma _{1}+\pi h(p) e^{1+4\pi A_{p}}, $$
(4.2)
where
\(\gamma _{1}=\int_{\Sigma}h\,dv_{g}\).
For the extremal function, define
$$ \phi_{\epsilon }(x)=\textstyle\begin{cases}c+\frac{1}{c} (-\frac{1}{4\pi}\log(1+\pi \frac {r^{2}}{\epsilon ^{2}})+B ) & \text{for } x\leq R\epsilon ,\\ \frac{G-\eta\Phi_{p}}{c}& \text{for } R\epsilon \leq r\leq2R\epsilon ,\\ \frac{G}{c}& \text{for } r>2R\epsilon , \end{cases} $$
(4.3)
as in [
1], where
c and
B are constants,
\(R=-\log \epsilon \),
\(\eta\in C^{\infty}_{0}(B_{2R\epsilon }(p))\) with
\(\eta=1\) on
\(B_{R\epsilon }(p)\) and
\(\Vert \nabla _{g} \eta \Vert _{L^{\infty}}=O(\frac{1}{R\epsilon })\). Choose
$$\begin{aligned} &c= \biggl(\frac{-\log \epsilon -2\pi B+2\pi A_{p}+\frac{1}{2}\log\pi+O (\frac {1}{R^{2}} )}{2\pi} \biggr)^{1/2}, \\ &B=\frac{1}{4\pi}+O \biggl( \frac{1}{R^{2}} \biggr) +O \bigl(R\epsilon \log(R\epsilon ) \bigr), \end{aligned}$$
as in [
1] such that
\(\phi_{\epsilon }\in W^{1,2}(\Sigma )\) and
\(\Vert \phi _{\epsilon }-\bar{\phi}_{\epsilon } \Vert _{1,\alpha }=1\), where
$$\begin{aligned} \overline{\phi}_{\epsilon }=\frac{1}{\mathrm{ Vol}_{g}(\Sigma )} \int_{\Sigma}\phi _{\epsilon }\,dv_{g} . \end{aligned}$$
Then we have on
\(B_{R\epsilon }(p)\)
$$\begin{aligned} 4\pi (\phi_{\epsilon }-\overline{\phi}_{\epsilon })^{2}\geq4 \pi c^{2}-2\log \biggl(1+\pi\frac {r^{2}}{\epsilon ^{2}} \biggr)+8\pi B+O \bigl(R \epsilon \log(R\epsilon ) \bigr). \end{aligned}$$
It follows that
$$ \int_{B_{R\epsilon }(p)}h e^{4\pi(\phi_{\epsilon }-\overline{\phi}_{\epsilon })^{2}}\,dv_{g}\geq \pi h(p) e^{1+4\pi A_{p}} +O \biggl(\frac{1}{(\log \epsilon )^{2}} \biggr) $$
(4.4)
and
$$\begin{aligned} \int_{\Sigma \backslash B_{R\epsilon }(p)} he^{4\pi(\phi_{\epsilon }-\overline {\phi}_{\epsilon })^{2}}\,dv_{g} &\geq \int_{\Sigma \backslash B_{2R\epsilon }(p)}\bigl(1+4\pi\phi_{\epsilon }^{2}\bigr) \,dv_{g} +O \bigl(R\epsilon \log(R\epsilon ) \bigr) \\ &\geq \gamma _{1}+4\pi\frac{ \Vert \sqrt{h} G \Vert ^{2}_{2}}{c^{2}}+o \biggl(\frac {1}{c^{2}} \biggr). \end{aligned}$$
(4.5)
Combining (
4.4) and (
4.5), we find a contradiction with (
4.2). Hence
\(c_{\epsilon }\) must be bounded. Using the elliptic estimates, we have the existence of the extremal function.
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