Bubbles Clustered Inside for Almost Critical Problems

We investigate the existence of blowing-up solutions of the following almost-critical problem: −Δu+V(x)u=up−ε,u>0inΩ,u=0on∂Ω,-\Delta u+V\left(x)u={u}^{p-\varepsilon },\hspace{1.0em}u\gt 0\hspace{0.25em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,u=0\hspace{0.25em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.25em}\partial \Omega , where Ω\Omega is a bounded regular domain in Rn{{\mathbb{R}}}^{n}, n≥4n\ge 4, ε\varepsilon is a small positive parameter, p+1=(2n)∕(n−2)p+1=\left(2n)/\left(n-2) is the critical Soblolev exponent, and the potential VV is a smooth positive function. We find solutions that exhibit bubbles clustered inside as ε\varepsilon 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.