Existence and Multiplicity of Positive Radial Solutions for Semilinear Elliptic Equations in Annular Domains

Abstract

The existence and multiplicity of positive radial solutions of equation $\Delta u + f(u) = 0$ is studied in annular domains in $\mathbb{R}^n $, $n \leqq 2$. It is proven that if $f(0) \geqq 0$, f is somewhere negative in $(0,\infty )$ and superlinear at $\infty $, then there is a large positive radial solutions on all annuli. If $f(0) < 0$ and satisfies certain conditions, then the equation has no solution if the annuli are too wide. Multiplicity results are also obtained when f has many humps with positive areas.