A semilinear elliptic equation with competing powers and a radial potential

We verify the existence of radial positive solutions for the semi-linear equation $$-\mathrm{\Delta }u=u^p-V(y)u^q,u>0,in{\mathbb{R}}^N$$ where $N\ge 3$, $p$ is close to $p^{\ast }:=(N+2)\mathrm{/}(N-2)$, 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 $N\mathrm{/}(N-2)<q<p^{\ast }$, we prove that this Problem has a radial solution behaving like a super-position of {\it flat} bubbles with different rates of concentration, provided $\lim_{r \to \infty} V® <0$.