A semilinear elliptic equation with competing powers and a radial potential

Abstract

We verify the existence of radial positive solutions for the semi-linear equation $$ -\,Δu=u^{p}\,-\,V(y)\,u^{q},\,\quad\quad u>0,\quad\quad\mbox{ in }\mathbb{R}^N$$ where $N\ge 3$, $p$ is close to $p^{\ast }:=(N+2)/(N-2)$, and $V$ is a radial smooth potential. If $q$ is super-critical, namely $q>p^{\ast }$, we prove that this Problem has a radial solution behaving like a super-position of bubbles blowing-up at the origin with different rates of concentration, provided $V(0)<0$. On the other hand, if $N/(N-2)<q<p^{\ast }$, we prove that this Problem has a radial solution behaving like a super-position of {\it flat} bubbles with different rates of concentration, provided ${\lim }_{r\to \infty }V(r)<0$.

19 pages