Bubbles clustered inside for almost critical problems

Abstract

We investigate the existence of blowing-up solutions of the following almost critical problem $$ -Δu +V(x)u =u^{p-\e},\quad u>0\quad\mbox{in}\quad Ø,\quad u=0\quad\mbox{on}\quad \partialØ, $$ where $\tilde{A}˜$ is a bounded regular domain in ${\mathbb{R}}^n$, $n\ge 4$, $\epsilon$ is a small positive parameter, $p+1=(2n)/(n-2)$ is the critical Soblolev exponent and the potential $V$ is a smooth positive function. We find solutions which exhibit bubbles clustered inside as $\e$ goes to zero. To the best of our knowledge, this is the first existence result for interior non-simple blowing-up positive solutions to Dirichlet problems in general domains. Our results are proven through delicate asymptotic estimates of the gradient of the associated Euler-Lagrange functional.