Concentrating solutions for an anisotropic planar elliptic Neumann problem with Hardy-Hénon weight and large exponent

Let $Ω$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary, we study the following anisotropic elliptic Neumann problem with Hardy-Hénon weight $\begin{cases} -\nabla(a(x)\nabla u)+a(x)u=a(x)|x-q|^{2α}u^p,\,\,\,\, u>0\,\,\,\,\, \textrm{in}\,\,\,\,\, Ω,\\[2mm] \frac{\partial u}{\partialν}=0\,\, \qquad\quad\qquad\qquad\qquad \qquad\qquad\qquad\qquad \,\ \ \,\,\,\, \textrm{on}\,\,\, \partialΩ, \end{cases}$ where $ν$ denotes the outer unit normal vector to $\partialΩ$, $q\in\overlineΩ$, $α\in(-1,+\infty)\setminus\mathbb{N}$, $p>1$ 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 $q\inΩ$ 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 $q\in\partialΩ$ is a strict local maximum point of $a(x)$ and satisfies $\langle\nabla a(q),\,ν(q)\rangle=0$, 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 $q\in\overlineΩ$ is an isolated local maximum point of $a(x)$ or not.

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