The Lazer-McKenna conjecture for an anisotropic planar exponential nonlinearity with a singular source

Abstract

Given a bounded smooth domain $\Omega$ in ${\mathbb{R}}^2$, we study the following anisotropic elliptic problem $$\{\begin{array}{l}-\nabla (a(x)\nabla \upsilon )=a(x)[e^{\upsilon }-s{\varphi }_1-4\pi \alpha {\delta }_q-h(x)]\text{in}\Omega ,\\ \upsilon =0\text{on}\partial \Omega ,\end{array}$$ where $a(x)$ is a positive smooth function, $s>0$ is a large parameter, $h\in C^{0,\gamma }(\overline{\Omega })$, $q\in \Omega$, $\alpha \in (-1,+\infty )\setminus \mathbb{N}$, ${\delta }_q$ denotes the Dirac measure with pole at point $q$ and ${\varphi }_1$ is a positive first eigenfunction of the problem $-\nabla (a(x)\nabla \varphi )=\lambda a(x)\varphi$ under Dirichlet boundary condition in $\Omega$. We show that if $q$ is both a local maximum point of ${\varphi }_1$ and an isolated local maximum point of $a(x){\varphi }_1$, this problem has a family of solutions ${\upsilon }_s$ with arbitrary $m$ bubbles accumulating to $q$ and the quantity ${\int }_{\Omega }a(x)e^{{\upsilon }_s}\to 8\pi (m+1+\alpha )a(q){\varphi }_1(q)$ as $s\to +\infty$, which give a positive answer to the Lazer-McKenna conjecture for this case.

arXiv commentarXiv admin note: text overlap with arXiv:1908.05532