Infinite multiplicity of positive solutions of an inhomogeneous supercritical elliptic equation on ${\mathbb{R}}^N$

Abstract

We are concerned with positive radial solutions of the inhomogeneous elliptic equation $\Delta u+K(|x|)u^p+\mu f(|x|)=0$ on ${\mathbb{R}}^N$, where $N\ge 3$, $\mu >0$ and $K$ and $f$ are nonnegative nontrivial functions. If $K(r)\sim r^{\alpha }$, $\alpha >-2$, near $r=0$, $K(r)\sim r^{\beta }$, $\beta >-2$, near $r=\infty$ and certain assumptions on $f$ are imposed, then the problem has a unique positive radial singular solution for a certain range of $\mu$. We show that existence of a positive radial singular solution is equivalent to existence of infinitely many positive bounded solutions which are not uniformly bounded, if $p$ is between the critical Sobolev exponent $p_S(\alpha )$ and Joseph-Lundgren exponent $p_{JL}(\alpha )$. Using these theorems, we establish existence of infinitely many positive bounded solutions which are not uniformly bounded, for $p_S(\alpha )<p<p_{JL}(\alpha )$ if $K(r)=r^{-\alpha }$, $\alpha >-2$.

33 pages