Sign-changing concentration phenomena of an anisotropic sinh-Poisson type equation with a Hardy or Hénon term

Abstract

We consider the following anisotropic sinh-Poisson tpye equation with a Hardy or Hénon term: $$-\mathrm{div}(a(x)\nabla u)+a(x)u={\epsilon }^{2}a(x)|x-q{|}^{2\alpha }(e^u-e^{-u})\mathrm{in}\Omega ,$$ $$\frac{\partial u}{\partial n}=0,\mathrm{on}\partial \Omega ,$$ where $\epsilon >0$, $q\in \bar{\Omega }\subset {\mathbb{R}}^2$, $\alpha \in (-1,\infty )-\mathbb{N}$, $\Omega \subset {\mathbb{R}}^2$ 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 $⟨\nabla a(q),n⟩=0$, 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 $$-\mathrm{div}(a(x)\nabla u)+a(x)u={\epsilon }^{2}a(x)|x-q{|}^{2\alpha }{e}^u\mathrm{in}\Omega ,$$ $$\frac{\partial u}{\partial n}=0,\mathrm{on}\partial \Omega .$$

54 pages

References in corpus (1)

• On the Local Behavior of Local Weak Solutions to some Singular Anisotropic Elliptic Equations