Multiple boundary bubbling phenomenon of solutions to a Neumann problem

Abstract

We consider the following anisotropic problem $$-\div\big( a(x)\nabla u\big)+a(x)u=0 \quad \mbox{in ${\Omega }$,}\qquad \frac{{\partial} u} {{\partial}\nu}={\varepsilon } e^u \quad\mbox{on ${\partial\Omega }$,} $$ where ${\Omega }\subseteq \mathbb{R}^2$ is a bounded smooth domain, ${\varepsilon }$ is a small parameter and $a(x)$ is a positive smooth function. First, we establish a decomposition result for the regular part of a relative Green's function, which yields its Hölder continuous character and the smoothness of its diagonal. Next, we employ this result to derive the accumulation of bubbles at given local maximum points of $a(x)$ on the boundary, which verifies the existence of large energy solutions to the problem in [17].