Sign-changing concentration phenomena of an anisotropic sinh-Poisson type equation with a Hardy or H\'{e}non term

Abstract

We consider the following anisotropic sinh-Poisson tpye equation with a Hardy or H\'{e}non term: $$-\mathrm{div} (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2\alpha}(e^u-e^{-u}) \quad\mathrm{in}\quad \Omega,$$ $$\frac{\partial u}{\partial n}=0,\quad \mathrm{on}\quad \partial\Omega,$$ where $\varepsilon>0$, $q\in \bar\Omega\subset \mathbb{R}^2$, $\alpha \in(-1,\infty)- \mathbb{N}$, $\Omega\subset \mathbb{R}^2$ is a smooth bounded domain, $n$ is the unit outward normal vector of $\partial \Omega$ and $a(x)$ is a smooth positive function defined on $\bar\Omega$. From finite dimensional reduction method, we proved that this problem has a sequence of sign-changing solutions with arbitrarily many interior spikes accumulating to $q$, provided $q\in \Omega$ is a local maximizer of $a(x)$. However, if $q\in \partial \Omega$ is a strict local maximum point of $a(x)$ and satisfies $\langle \nabla a(q),n \rangle=0$, we proved that this problem has a family of sign-changing solutions with arbitrarily many mixed interior and boundary spikes accumulating to $q$. Under the same condition, we could also construct a sequence of blow-up solutions for the following problem $$ -\mathrm{div} (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2\alpha}e^u\quad \mathrm{in} \quad\Omega,$$ $$\frac{\partial u}{\partial n}=0, \quad \mathrm{on}\quad \partial\Omega.$$