A semilinear elliptic equation with competing powers and a radial potential

TL;DR

This paper proves the existence of radial positive solutions for a semilinear elliptic equation with competing powers and a radial potential, showing different bubble behaviors depending on the relation between q and the critical exponent.

Contribution

It establishes the existence of solutions with bubble superpositions for a class of semilinear elliptic equations with competing powers and radial potentials, depending on the criticality of q.

Findings

Solutions behave like bubbles blowing-up at the origin when q > p^* with V(0)<0.

Solutions behave like flat bubbles with different concentration rates when q < p^* and V(r) < 0 at infinity.

Existence results depend on the sign and limits of the radial potential V.

Abstract

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