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: {\begin{array}{ll} -\Div (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2\alpha}(e^u-e^{-u}) &\mbox{in $\Omega$,} \\ \frac{\partial u}{\partial n}=0, &\mbox{on $\Omega$,} \end{array} where $\varepsilon>0$, $q\in \bar\Omega\subset \R^2$, $\alpha \in(-1,\infty)\char92 \N$, $\Omega\subset \R^2$ 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 the problem \eqref{115} 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 \eqref{115} 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 $$ {\begin{array}{ll} -\Div (a(x)\nabla u)+ a(x)u=\varepsilon^2a(x)|x-q|^{2\alpha}e^u &\mbox{in $\Omega$,} \\ \frac{\partial u}{\partial n}=0, &\mbox{on $\partial\Omega$.} \end{array}$