Concentrating solutions for an anisotropic planar elliptic Neumann problem with Hardy-H\'{e}non weight and large exponent

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-H\'{e}non weight $$ \begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, \Omega,\\[2mm] \frac{\partial u}{\partial\nu}=0\,\, \qquad\quad\qquad\qquad\qquad \qquad\qquad\qquad\qquad \,\ \ \,\,\,\, \textrm{on}\,\,\, \partial\Omega, \end{cases} $$ where $\nu$ denotes the outer unit normal vector to $\partial\Omega$, $q\in\overline{\Omega}$, $\alpha\in(-1,+\infty)\setminus\mathbb{N}$, $p>1$ is a large exponent and $a(x)$ is a positive smooth function. We investigate the effect of the interaction between anisotropic coefficient $a(x)$ and singular source $q$ on the existence of concentrating solutions. We show that if $q\in\Omega$ is a strict local maximum point of $a(x)$, there exists a family of positive solutions with arbitrarily many interior spikes accumulating to $q$; while if $q\in\partial\Omega$ is a strict local maximum point of $a(x)$ and satisfies $\langle\nabla a(q),\,\nu(q)\rangle=0$, such a problem has a family of positive solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$. In particular, we find that concentration at singular source $q$ is always possible whether $q\in\overline{\Omega}$ is an isolated local maximum point of $a(x)$ or not.