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 $$ \begin{cases} -\nabla\big(a(x)\nabla \upsilon\big)= a(x)\big[e^{\upsilon}-s\phi_1-4\pi\alpha\delta_q-h(x)\big]\,\,\,\, \,\textrm{in}\,\,\,\,\,\Omega,\\[2mm] \upsilon=0 \qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\quad \textrm{on}\,\ \,\partial\Omega, \end{cases} $$ 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 $\phi_1$ is a positive first eigenfunction of the problem $-\nabla\big(a(x)\nabla \phi\big)=\lambda a(x)\phi$ under Dirichlet boundary condition in $\Omega$. We show that if $q$ is both a local maximum point of $\phi_1$ and an isolated local maximum point of $a(x)\phi_1$, this problem has a family of solutions $\upsilon_s$ with arbitrary $m$ bubbles accumulating to $q$ and the quantity $\int_{\Omega}a(x)e^{\upsilon_s}\rightarrow8\pi(m+1+\alpha)a(q)\phi_1(q)$ as $s\rightarrow+\infty$, which give a positive answer to the Lazer-McKenna conjecture for this case.