Radial symmetric stationary solutions for a MEMS type reaction–diffusion equation with spatially dependent nonlinearity

Abstract

We consider the radial symmetric stationary solutions of $u_{t}=\varDelta u-|x|^{q}u^{-p}$. We first give a result on the existence of the negative value functions that satisfy the radial symmetric stationary problem on a finite interval for $p \in 2{\mathbb{N}}$, $q\in{\mathbb{R}}$. Moreover, we give the asymptotic behavior of u(r) and $u'(r)$ at both ends of the finite interval. Second, we obtain the existence of the positive radial symmetric stationary solutions with the singularity at $r=0$ for $p\in{\mathbb{N}}$ and $q\ge -2$. In fact, the behavior of solutions for $q>-2$ and $q=-2$ are different. In each case of $q=-2$ and $q>-2$, we derive the asymptotic behavior for $r \rightarrow 0$ and $r \rightarrow \infty$. These facts are studied by applying the Poincaré compactification and basic theory of dynamical systems.