Bubbles clustered inside for almost critical problems

Abstract: We investigate the existence of blowing-up solutions of the following almost critical problem $$ -\Delta u +V(x)u =u^{p-\e},\quad u>0\quad\mbox{in}\quad \O,\quad u=0\quad\mbox{on}\quad \partial\O, $$ where $\O$ is a bounded regular domain in $\mathbb{R}^n$, $n\geq 4$, $\varepsilon$ 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.