Bubbles clustered inside for almost critical problems

We investigate the existence of blowing-up solutions of the following almost critical problem $$-\mathrm{\Delta }u+V(x)u=u^{p-\text{e}},u>0\text{quadin}\text{Ø},u=0\text{quadon}\mathrm{\partial }\text{Ø},$$ where $\text{Ø}$ is a bounded regular domain in ${\mathbb{R}}^n$, $n\ge 4$, $\epsilon$ is a small positive parameter, $p+1=(2n)\mathrm{/}(n-2)$ is the critical Soblolev exponent and the potential $V$ is a smooth positive function. We find solutions which exhibit bubbles clustered inside as $\text{e}$ 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.