Concentrating solutions for an anisotropic planar elliptic Neumann problem with Hardy-Hénon weight and large exponent - Zhang, Yibin (Topological Methods in Nonlinear Analysis, 2022)
Let $\Omega$ be a bounded domain in $\mathbb{R}^2$ with smooth boundary,we study the following anisotropic elliptic Neumann problemwith Hardy-Hénon weight$$\begin{cases}- abla(a(x) abla u)+a(x)u=a(x)|x-q|^{2\alpha}u^p,\ u> 0& \text{in } \Omega,\\[1mm]\dfrac{\partial u}{\partial u}=0 & \text{on } \partial\Omega,\end{cases}$$%where $ u$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 exponentand $a(x)$ 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 manyinterior spikes accumulating to $q$;while, if $q\in\partial\Omega$ is a strict local maximum point of $a(x)$and satisfies $\langle abla a(q), u(q)\rangle=0$,such a problem has a family of positive solutions with arbitrarily many mixedinterior and boundary spikes accumulating to $q$. In particular, we findthat concentration at singular source $q$ is always possible whether$q\in\overline{\Omega}$is an isolated local maximum point of $a(x)$ or not.