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

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