Existence for semilinear equations on exterior domains
Joseph A. Iaia · Electronic journal of qualitative theory of differential equations · 2016
In this paper we study radial solutions of $\Delta u + K(r)f(u)= 0$ on the exterior of the ball of radius $R>0$ centered at the origin in ${\mathbb R}^{N}$ where $f$ is odd with $f0$ on $(\beta, \infty),$ and $f$ superlinear. The function $K(r)$ is assumed to be positive and $K(r)\to 0$ as $r \to \infty$. We prove existence of an infinite number of radial solutions with $u \to 0$ as $r \to \infty$ when $K(r)\sim r^{-\alpha}$ with $N < \alpha < 2(N-1)$.