An interior boundedness result for an elliptic equation

TL;DR

This paper establishes a local uniform boundedness result for solutions to elliptic equations that exhibit interior singularities, advancing understanding of their behavior.

Contribution

It introduces a new boundedness result specifically for elliptic equations with interior singularities, which was not previously addressed.

Findings

Proves local boundedness of solutions near interior singularities

Provides conditions under which solutions remain bounded

Enhances theoretical understanding of elliptic equations with singularities

Abstract

We derive a local uniform boundedness result for an elliptic equation having interior singularity.

Equations

$$(P)\left\{\begin{split}-\Delta u&=\dfrac{1}{-\log\dfrac{|x|}{2d}}Ve^{u}\,\,&\text{in}\,\,&\Omega\subset{\mathbb{R}}^{2},\\ u&=0\,\,&\text{in}\,\,&\partial\Omega.\end{split}\right.$$

$$0\le V\le b,{\int }_{\Omega }\frac{1}{-\log \frac{|x|}{2d}}{e}^{u}dx\le C,u\in W_0^{1,1}(\Omega ),$$

$$d=diam(\Omega ),0\in \Omega$$

$$\underset{K}{\sup }{u}_i\le c,$$

$$0\le V_i\le b<+\infty \mathrm{and}{\int }_{\Omega }{e}^{u_i}dy\le C,$$

$$\underset{K}{\sup }{u}_i\le c,$$

$$||u_i|{|}_{L^{\infty }(K)}\le c(b,C,K,\Omega ),$$

$$||\nabla u_i|{|}_q\le C_q,\forall 1\le q<2.$$

$$||\nabla V_i|{|}_{\infty }\le A,$$

$$0<a\le V\le b<+\infty ,$$

$$\underset{K}{\sup }u+C_1\underset{\Omega }{\inf }u\le C_2,$$

$$\underset{K}{\sup }u+\underset{\Omega }{\inf }u\le c(a,b,A,K,\Omega )?$$

$$u_i\in W_0^{1,1}(\Omega ),\mathrm{and}\frac{1}{-\log \frac{|x|}{2d}}{e}^{u_i}\in L^1(\Omega ).$$

$$e^{u_i}\in L^k(\Omega ),\forall k>2.$$

$$u_i\in W^{2,k}(\Omega )\cap C^{1,ϵ}(\bar{\Omega }).$$

$$||\nabla u_i|{|}_q\le C_q,1\le q<2.$$

$${\int }_{\Omega }\frac{1}{-\log \frac{|x|}{2d}}{V}_i{e}^{u_i}dx\le C,$$

$${\int }_{\Omega }\frac{1}{-\log \frac{|x|}{2d}}{V}_i{e}^{u_i}\varphi dx\to {\int }_{\Omega }\varphi d\mu ,\forall \varphi \in C_c(\Omega ).$$

$$\int \psi d\mu <4\pi .$$

$$\exists R_0>0suchthatonecanbound(u_i)=(u_i^{+})in{L}^{\infty }(B_{R_0}(x_0)).$$

$$||u_i^{+}|{|}_1=||u_i|{|}_1\le c_q||u_i|{|}_{q^{\ast }}\le C_q^{\prime }||\nabla u_i|{|}_q\le C_q,1\le q<2.$$

$$\forall R>0,\lim ||u_i^{+}|{|}_{L^{\infty }(B_R(x_0))}=+\infty .$$

$$||u_i^{+}|{|}_{L^{\infty }(B_{R_0}(x_0))}\le C.$$

$$||e^{u_{i_k}}|{|}_{L^{\infty }(B_{R_0}(x_0))}\le C,\mathrm{and}$$

$${\int }_{B_R(x_0)}\frac{1}{-\log \frac{|x|}{2d}}{V}_{i_k}{e}^{u_{i_k}}=o(1).$$

$$u_i^{+}(x_i)=\underset{B_R(x_0)}{\max }{u}_i^{+}\to +\infty .$$

$$z_i\in W_0^{1,2}(B_R(x_0)),$$

$$-\Delta z_i=f_i=\frac{1}{-\log \frac{|x|}{2d}}{V}_i{e}^{u_i}\mathrm{in}{B}_R(x_0),\mathrm{et}{z}_i=0\mathrm{on}\partial B_R(x_0).$$

$$z_i\in W_0^{1,q}(B_R),||\nabla z_i|{|}_q\le C_q.$$

$$\int \frac{1}{-\log \frac{|x|}{2d}}{e}^{z_i}\le \int \frac{1}{-\log \frac{|x|}{2d}}{e}^{u_i}\le C.$$

Full text

Keywords: $C^{0}$ weight, interior singularity, a priori estimate, maximum principle.

MSC: 35J60, 35B44, 35B45, 35B50

1 Introduction and Main Results

We set $\Delta=\partial_{11}+\partial_{22}$ on open set $\Omega$ of ${\mathbb{R}}^{2}$ with a smooth boundary.

We consider the following equation:

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Here:

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and,

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Equations of the previous type were studied by many authors, with or without the boundary condition, also for Riemannian surfaces, see  [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], where one can find some existence and compactness results.

Among other results, we can see in [11] the following important Theorem

Theorem A*(Brezis-Merle [11])*.If $(u_{i})$ is a sequence of solutions of problem $(P)$ with $(V_{i})$ satisfying $0<a\leq V_{i}\leq b<+\infty$ and without the term $\dfrac{1}{-\log\dfrac{|x|}{2d}}$, then, for any compact subset $K$ of $\Omega$, it holds:

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with c depending on $a,b,K,\Omega$

One can find in [11] an interior estimate if we assume $a=0$, but we need an assumption on the integral of $e^{u_{i}}$, namely, we have:

Theorem B*(Brezis-Merle [11])*.For $(u_{i})_{i}$ and $(V_{i})_{i}$ two sequences of functions relative to the problem $(P)$ without the term $\dfrac{1}{-\log\dfrac{|x|}{2d}}$ and with,

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then for all compact set $K$ of $\Omega$ it holds;

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with $c$ depending on $b,C,K$ and $\Omega$.

If we assume $V$ with more regularity, we can have another type of estimates, a $\sup+\inf$ type inequalities. It was proved by Shafrir see [18], that, if $(u_{i})_{i}$ is a sequence of functions solutions of the previous equation without assumption on the boundary with $V_{i}$ satisfying $0<a\leq V_{i}\leq b<+\infty$, then we have a $\sup+\inf$ inequality.

Here, we have:

** Theorem **

For sequences $(u_{i})_{i}$ and $(V_{i})_{i}$ of the Problem $(P)$, for all compact subsets $K$ of $\Omega$ we have:

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Remark: Remark that we have a $C^{0}$ weight $\dfrac{1}{-\log\dfrac{|x|}{2d}}$, the solutions are not $C^{2}$, but if we add some assumptions on $V_{i}$ one can consider $C^{2}$ solutions and $C^{2}$ convergence of sequences.

On can have the regularity $C^{2}$ of the solutions and the $C^{2}$ convergence of the solutions if we suppose for example $V_{i}\in C^{0,\epsilon},\epsilon>0$ and for the convergence $V_{i}\to V$ in the space $C^{0,\epsilon},\epsilon>0$. Indeed, one can reduce the problem to regularity and convergence of the Newtonian potential of a radial distribution $f(x)=f(|x|)=\dfrac{V_{i}(0)e^{u_{i}(0)}}{-\log\left(\dfrac{|x|}{2d}\right)}\eta(|x|)$, with $\eta$ a cutoff function ($\eta\equiv 1$ in a neighborhood of [math] with compact support and radial), see for example the book of Dautray-Lions, chapter 2, Laplace operator.

By a duality theorem one can prove that (see [12]):

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If we add the assumption that

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then by a result of Chen-Li of "moving-plane" we have a compactness of $(u_{i})_{i}$ near the boundary, see [13].

We ask the following question about inequality of type $\sup+\inf$, as in the work of Tarantello, see [19] and Bartolucci-Tarantello, see [8]:

Problems. 1) Consider the Problem $(P)$ without the boundary condition (without Dirichlet condition) and assume that:

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Does exists constants $C_{1}=C_{1}(a,b,K,\Omega),C_{2}=C_{2}(a,b,K,\Omega)$ such that:

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for all solution $u$ of $(P)$ ?

2. If we add the condition $||\nabla V||_{\infty}\leq A$, can we have a sharp inequality:

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2 Proof of the Theorem

We have:

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Thus, by corollary 1 of Brezis and Merle we have:

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Using the elliptic estimates and the Sobolev embedding, we have:

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By the maximum principle $u_{i}\geq 0$.

Also, by a duality theorem or a result of Brezis-Strauss, we have:

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Since,

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We have a convergence to a nonegative measure $\mu$:

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We set $S$ the following set:

$S=\{x\in\Omega,\exists\,(x_{i})\in\Omega,\,x_{i}\to x,\,u_{i}(x_{i})\to+\infty\}$.

We say that $x_{0}$ is a regular point of $\mu$ if there function $\psi\in C_{c}(\Omega)$, $0\leq\psi\leq 1$, with $\psi=1$ in a neighborhood of $x_{0}$ such that:

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We can deduce that a point $x_{0}$ is non-regular if and only if $\mu({x_{0}})\geq 4\pi.$

A consequence of this fact is that if $x_{0}$ is a regular point then:

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We deduce $(\ref{(2)})$ from corollary 4 of Brezis-Merle paper, because we have by the Gagliardo-Nirenberg-Sobolev inequality:

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We denote by $\Sigma$ the set of non-regular points.

Step 1: S = $\Sigma$.

We have $S\subset\Sigma$. Let’s consider $x_{0}\in\Sigma$. Then we have:

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Suppose contrary that:

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Then:

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For $R$ small enough, which imply $(\ref{(1)})$ for a function $\psi$ and $x_{0}$ will be regular, contradiction. Then we have $(\ref{(3)})$. We choose $R_{0}>0$ small such that $B_{R_{0}}(x_{0})$ contain only $x_{0}$ as non -regular point. $\Sigma$. Let’s $x_{i}\in B_{R}(x_{0})$ scuh that:

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We have $x_{i}\to x_{0}$. Else, there exists $x_{i_{k}}\to\bar{x}\not=x_{0}$ and $\bar{x}\not\in\Sigma$, i.e. $\bar{x}$ is a regular point. It is impossible because we would have $(\ref{(2)})$.

Since the measure is finite, if there are blow-up points, or non-regular points, $S=\Sigma$ is finite.

Step 2: $\Sigma=\{\emptyset\}$.

Now: suppose contrary that there exists a non-regular point $x_{0}$. We choose a radius $R>0$ such that $B_{R}(x_{0})$ contain only $x_{0}$ as non-regular point. Thus outside $\Sigma$ we have local unfirorm boundedness of $u_{i}$, also in $C^{1}$ norm. Also, we have weak *-convergence of $V_{i}$ to $V\geq 0$ with $V\leq b$.

Let’s consider (by a variational method):

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By a duality theorem:

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By the maximum principle, $u_{i}\geq z_{i}$ in $B_{R}(x_{0})$.

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On the other hand, $z_{i}\to z$ a.e. (uniformly on compact sets of $B_{R}(x_{0})-\{x_{0}\}$) with $z$ solution of :

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Also, we have up to a subsequence, $z_{i}\to z$ in $W_{0}^{1,q}(B_{R}(x_{0})),1\leq q<2$ weakly, and thus $z\in W_{0}^{1,q}(B_{R}(x_{0}))$.

Then by Fatou lemma:

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As $x_{0}\in S$ is not regular point we have $\mu(\{x_{0}\})\geq 4\pi$, which imply that, $\mu\geq 4\pi\delta_{x_{0}}$ and by the maximum principle in $W^{1,1}_{0}(B_{R}(x_{0}))$ (obtainded by Kato’s inequality)

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Because,

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Thus,

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Both in the cases $x_{0}=0$ and $x_{0}\not=0$ we have:

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But, by $(\ref{(**)})$:

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which a contradiction.