Abstract
Using blow-up analysis, the author proves the existence of extremal functions for Trudinger–Moser inequalities with nonnegative weights on bounded Euclidean domains or compact Riemannian surfaces. This extends recent results of Yang (J. Differ. Equ. 258:3161–3193, 2015) and Yang–Zhu (Proc. Am. Math. Soc. 145:3953–3959, 2017).
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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
As the limit case of the Sobolev inequality, the famous Trudinger–Moser inequality [3, 4] states
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
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, \(0\leq \alpha <\lambda _{1}(\Omega )\),
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
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
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
where
and
It was also proved by Yang [1] that, for any α, \(0\leq \alpha <\lambda _{l+1}(\Omega )\), we have
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.
Theorem 1
Let Ω be a smooth bounded domain in \(\mathbb{R}^{2}\), \(\lambda _{1}(\Omega )\) be the first eigenvalue of the Laplace operator with Dirichlet boundary condition, and h be in \(C^{0}(\overline{\Omega })\) with \(h\geq0\) and \(h\not\equiv0\). Then we have, for any \(0\leq \alpha <\lambda _{1}(\Omega )\), the supremum
can be attained by some \(u_{0}\in W^{1,2}_{0}(\Omega ) \cap C^{1}(\overline{\Omega })\) satisfying \(\Vert u_{0} \Vert _{1,\alpha }=1\), where \(\Vert \cdot \Vert _{1,\alpha }\) is defined as in (1.4).
When the high order eigenvalues are involved, we have a similar result.
Theorem 2
Let Ω be a smooth bounded domain in \(\mathbb{R}^{2}\), \(\lambda _{l+1}(\Omega )\) be the \((l+1)\)th eigenvalue of the Laplace operator with Dirichlet boundary condition, and h be in \(C^{0}(\overline{\Omega })\) with \(h\geq0\) and \(h\not \equiv0\). Then we see that, for any \(0\leq \alpha <\lambda _{l+1}(\Omega )\), the supremum
can be attained by some \(u_{0}\in E_{l}^{\perp}\cap C^{1}(\overline {\Omega })\) satisfying \(\Vert u_{0} \Vert _{1,\alpha }=1\), where \(E_{l}^{\perp}\) is defined as in (1.7) and \(\Vert \cdot \Vert _{1,\alpha }\) defined as in (1.4).
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
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.
Theorem 3
Let \((\Sigma ,g)\) be a compact Riemannian surface without boundary, h be in \(C^{0}(\Sigma )\) with \(h\geq0\) and \(h\not\equiv0\). Then, for any α, \(0\leq \alpha <\lambda _{1}(\Sigma )\), the supremum
can be attained by some \(u_{0}\in W^{1,2}(\Sigma )\cap C^{1}{(\Sigma ) }\) satisfying \(\Vert u_{0} \Vert _{1,\alpha }=1\) and \(\int_{\Sigma }u_{0}\,dv_{g}=0\).
Corollary 4
Let \((\Sigma ,g)\) be a compact Riemannian surface without boundary, h be in \(C^{0}(\Sigma )\) with \(h\geq0\) and \(h\not\equiv0\). For any α, \(0\leq \alpha <\lambda _{1}(\Sigma )\), \(\forall u\in W^{1,2}(\Sigma )\textit{ with }\int_{\Sigma}u \,dv_{g}=0\), define
Then we have the weak Trudinger–Moser inequality \(J(u)\geq-C\), where C is a positive constant depending only on \((\Sigma,g)\) and α.
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
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
and
Similar to Theorem 2, we obtain the following.
Theorem 5
Let \((\Sigma ,g)\) be a compact Riemannian surface without boundary, h be in \(C^{0}(\Sigma )\) with \(h\geq0\) and \(h\not\equiv0\). Then, for any α, \(0\leq \alpha <\lambda _{l+1}(\Sigma )\), the supremum
can be attained by some \(u_{0}\in E_{l}^{\perp}\cap C^{1}{(\Sigma ) }\) satisfying \(\Vert u_{0} \Vert _{1,\alpha }=1\) and \(\int_{\Sigma }u_{0}\,dv_{g}=0\).
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).
2 Proof of Theorem 1
2.1 The subcritical functionals
In this subsection, using the method in the calculus of variations, we prove the existence of maximizers for the subcritical functionals.
Lemma 6
For any \(0<\epsilon <4\pi\), any \(0\leq \alpha <\lambda _{1}(\Omega )\), there exists some \(u_{\epsilon }\in W^{1,2}_{0}(\Omega )\cap C^{1}(\overline{\Omega })\) with \(\Vert u_{\epsilon }\Vert _{1,\alpha }=1\) such that
where \(\Vert \cdot \Vert _{1,\alpha }\) is defined as in (1.4).
Proof
For \(0<\epsilon <4\pi\), we choose a function sequence \(u_{j}\in W^{1,2}_{0}(\Omega )\) such that
and
as \(j\rightarrow\infty\). Then there exists some \(u_{\epsilon }\in W^{1,2}_{0}(\Omega )\) such that up to a subsequence,
Using a similar argument in the spirit of the one in [1], we find that \(he^{(4\pi-\epsilon )u_{j}^{2}}\) is bounded in \(L^{q}(\Omega )\) for some \(q>1\). Then we get \(he^{(4\pi-\epsilon )u_{j}^{2}}\rightarrow he^{(4\pi-\epsilon )u_{\epsilon }^{2}}\) strongly in \(L^{1}(\Omega )\). This together with (2.2) immediately yields (2.1). We claim that \(\Vert u_{\epsilon }\Vert _{1,\alpha }=1\). Otherwise \(\Vert u_{\epsilon }\Vert _{1,\alpha } <1\). It follows that
There is a contradiction between in (2.1) and (2.3). Hence \(\Vert u_{\epsilon }\Vert _{1,\alpha }=1\). □
Moreover, the Euler–Lagrange equation for \(u_{\epsilon }\) is
Using elliptic estimates, we get \(u_{\epsilon }\in C^{1}(\overline{\Omega })\). Let \(c_{\epsilon }=u_{\epsilon }(x_{\epsilon })=\max_{\Omega }u_{\epsilon }\). If \(c_{\epsilon }\) is bounded, the existence of the extremal function is trivial by standard elliptic estimates. Thus we assume that \(c_{\epsilon }\rightarrow\infty\) and \(x_{\epsilon }\rightarrow x_{0}\in \overline{\Omega }\). A result of Gidas, Ni and Nirenberg on page 223 of [18] implies \(x_{0}\notin\partial \Omega \).
Using the same argument as the one in step 2 of [1], we get the energy concentration. For the function sequence \(u_{\epsilon }\), we have \(u_{\epsilon }\rightharpoonup0\) weakly in \(W^{1,2}_{0}(\Omega )\), \(u_{\epsilon }\rightarrow0\) strongly in \(L^{q}(\Omega )\) for any \(q>1\), and \(\vert \nabla u_{\epsilon }\vert ^{2}\,dx\rightharpoonup\delta_{x_{0}}\) in the sense of measure as \(\epsilon \rightarrow0\), where \(\delta_{x_{0}}\) denotes the Dirac measure centered at \(x_{0}\).
Next we prove that h is positive at the blow-up point \(x_{0}\). This property plays an important part in our analysis.
Lemma 7
There holds \(h(x_{0})>0\).
Proof
We prove it by contradiction. Suppose that \(h(x_{0})=0\). Note that up to a sequence
where η is a positive constant. Let ϵ be sufficiently small such that
Choose \(r>0\) such that \(B_{r}(x_{0})\subset \Omega \). Then
where \(o_{r}(1)\rightarrow0\) as \(r\rightarrow0\).
Choose r sufficiently small such that
Here we have used the Trudinger–Moser inequality (1.5).
Applying elliptic estimates to the Euler–Lagrange equation (2.4), we obtain \(u_{\epsilon }\rightarrow0\) in \(C^{1}_{\mathrm{loc}}(\Omega \setminus \{x_{0}\})\). Hence
Combining (2.6), (2.7) and (2.8), we find that if ϵ is sufficiently small,
There is a contradiction between (2.5) and (2.9). Hence \(h(x_{0})>0\). □
2.2 Blow-up analysis
We shall analyze the behavior of the maximizers by using a blow-up analysis. Let
Using the Hölder inequality and the classical Trudinger–Moser inequality, we have
where \(0<\delta<4\pi\), C depends only on h and δ. Thus we get
as \(\epsilon \rightarrow0\).
Set
We define two sequences of functions on \(\Omega _{\epsilon }\):
They satisfy the following equation:
It is clear that \(\Omega _{\epsilon }\rightarrow\mathbb{R}^{2}\) as \(\epsilon \rightarrow0\). Noting that \(\vert \psi_{\epsilon }\vert \leq1\) and \(\Delta \psi_{\epsilon }\rightarrow0\) uniformly in \(\Omega _{\epsilon }\) as \(\epsilon \rightarrow0\) and using the elliptic estimates, we get \(\psi_{\epsilon }\rightarrow\psi\) in \(C_{\mathrm{loc}}^{1}(\mathbb{R}^{2})\), where ψ is a bounded harmonic function in \(\mathbb{R}^{2}\). Since \(\psi(0)=\lim_{\epsilon \rightarrow 0}\psi_{\epsilon }(0)=1\), we have by the Liouville theorem
Similarly, we have by the elliptic estimates
where φ satisfies
and
We calculate
A result of Chen and Li [19] implies that
and
For the convergence behavior away from \(x_{0}\), we have \(c_{\epsilon }u_{\epsilon }\rightharpoonup G\) weakly in \(W^{1,p}_{0}(\Omega )\) for any \(1< p<2\), strongly in \(L^{q}(\Omega )\) for any \(q\geq1\) and in \(C_{{\mathrm{loc}}}^{1}(\overline{\Omega }\backslash\{x_{0}\})\), where G is a Green function satisfying
where \(\delta_{x_{0}}\) is the Dirac measure centered at \(x_{0}\).
G can be represented by
where \(A_{0}\) is a constant depending on \(x_{0}\) and α, \(\Phi\in C^{1}(\Omega )\) with \(\Phi(x_{0})=0\).
2.3 Upper bound estimates
Let δ be small such that \(B_{\delta}(x_{0})\subset \Omega \). Let \(s_{\epsilon }=\sup_{\partial B_{\delta}(x_{0})}u_{\epsilon }\) and \(\bar{u}_{\epsilon }=(u_{\epsilon }-s_{\epsilon })^{+}\). Then \(\bar{u}\in W_{0}^{1,2}(B_{\delta}(x_{0}))\). Let \(\tau_{\epsilon }=1-\frac {1}{c_{\epsilon }^{2}} (\frac{1}{2\pi}\log\frac{1}{\delta}+A_{0}+o_{\delta}(1)+o_{\epsilon }(1) )\). Then, by the calculation in step 4 of Sect. 3 in [1], we get
and
Hence
This together with (2.10) leads to
The argument in the proof of Lemma 3.3 in [20] yields
where \(\gamma=\int_{\Omega }h \,dx\). This implies that
2.4 Existence of extremal functions
Let \(r(x)= \vert x-x_{0} \vert \). Define
as in [1], where c and B are constants, \(R=-\log \epsilon \), \(\eta\in C^{\infty}_{0}(B_{2R\epsilon }(x_{0}))\) with \(\eta=1\) on \(B_{R\epsilon }(x_{0})\) and \(\Vert \nabla \eta \Vert _{L^{\infty}}=O(\frac{1}{R\epsilon })\). Choose
as in [1] such that \(\phi_{\epsilon }\in W^{1,2}_{0}(\Omega )\) and \(\Vert \phi _{\epsilon }\Vert _{1,\alpha }=1\). Then we get
and
Finally, we obtain
This contradicts (2.11). Hence \(c_{\epsilon }\) must be bounded and the elliptic estimates imply the existence of extremal functions. This completes the proof of Theorem 1.
3 Proof of Theorem 1
Let l be a positive integer and \(0\leq \alpha < \lambda _{l+1}(\Omega )\). Following the same steps as in the proof of Theorem 1, we see that, for any ϵ, \(0<\epsilon <4\pi\), there exists some \(u_{\epsilon }\in E_{l}^{\perp}\cap C^{1}(\overline{\Omega })\) with \(\Vert u_{\epsilon }\Vert _{1,\alpha }=1\) such that
where \(\Vert \cdot \Vert _{1,\alpha }\) is defined as in (1.4). Moreover, the Euler–Lagrange equation for \(u_{\epsilon }\) is
Let \(c_{\epsilon }= \vert u_{\epsilon }(x_{\epsilon }) \vert =\max_{\overline{\Omega }} \vert u_{\epsilon }\vert \). We assume that \(c_{\epsilon }\rightarrow\infty\) and \(x_{\epsilon }\rightarrow x_{0}\in\overline{\Omega }\). Similar to (2.11), we obtain
where \(\gamma=\int_{\Omega }h\,dx\).
Let \(r(x)= \vert x-x_{0} \vert \). Define the same function
as in (2.12). Set
where \((e_{ij})\ (1\leq i\leq l, 1\leq j\leq n_{i})\) is the basis of \(E_{l}\). Then, by (75) and (76) of [1], we have
and
Thus
Set \(\widehat{\phi}_{\epsilon }=\frac{\widetilde{\phi}_{\epsilon }}{ \Vert \widetilde {\phi } \Vert _{1,\alpha }}\). Then \(\int_{\Omega } he^{4\pi\widehat{\phi}_{\epsilon }^{2} }\,dx>\gamma +\pi h(x_{0}) e^{1+4\pi A_{0}}\). This contradicts (3.1). Hence \(c_{\epsilon }\) must be bounded and the extremal function exists. We finish the proof of Theorem 2.
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
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
There exists some \(u_{\epsilon }\in W^{1,2}(\Sigma )\) such that up to a subsequence,
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
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
Define
and
for \(x\in \Omega _{\epsilon }=\{x\in\mathbb{R}^{2}: \widetilde{x}_{\epsilon }+r_{\epsilon }x\in \Omega \}\). Then we get
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
and
where
and
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
and \(\int_{\Sigma}G\,dv_{g}=0\). As before, G can be represented by
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
where \(\gamma _{1}=\int_{\Sigma}h\,dv_{g}\).
For the extremal function, define
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
as in [1] such that \(\phi_{\epsilon }\in W^{1,2}(\Sigma )\) and \(\Vert \phi _{\epsilon }-\bar{\phi}_{\epsilon } \Vert _{1,\alpha }=1\), where
Then we have on \(B_{R\epsilon }(p)\)
It follows that
and
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.
5 Proof of Theorem 5
Let l be a positive integer and \(0\leq \alpha <\lambda _{l+1}(\Sigma )\). First, by the same arguments, we obtain for any ϵ, \(0<\epsilon <4 \pi\), there exists some \(u_{\epsilon }\in E^{\perp}_{l}\cap C^{1}(\Sigma )\) satisfying \(\Vert u_{\epsilon }\Vert _{1,\alpha }=1\) and
Moreover, \(u_{\epsilon }\) satisfies the Euler–Lagrange equation
Let \(c_{\epsilon }=\max_{\Sigma } \vert u_{\epsilon }\vert \). We assume that \(c_{\epsilon }= \vert u_{\epsilon }(x_{\epsilon }) \vert \rightarrow +\infty\) and \(x_{\epsilon }\rightarrow p\in \Sigma \). We also get the upper bound estimate
where \(\gamma _{1}=\int_{\Sigma}h\,dv_{g}\).
For the existence of the extremal function, we define \(\phi_{\epsilon }\) as in (4.3). Then we have
Set
where
and \((e_{ij})\ (1\leq i\leq l, 1\leq j\leq n_{i})\) is the basis of \(E_{l}\).
Then, by (102) and (103) in [1], we have
and
Thus
Set \(\widehat{\phi}_{\epsilon }=\frac{\widetilde{\phi}_{\epsilon }}{ \Vert \widetilde {\phi}_{\epsilon }\Vert _{1,\alpha }}\). We have
This contradicts (5.2). Hence \(c_{\epsilon }\) must be bounded and the extremal function exists. We finish the proof.
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Hou, S. Extremal functions for Trudinger–Moser inequalities with nonnegative weights. J Inequal Appl 2018, 125 (2018). https://doi.org/10.1186/s13660-018-1718-7
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DOI: https://doi.org/10.1186/s13660-018-1718-7