Bubbles clustered inside for almost-critical problems

Abstract We investigate the existence of blowing-up solutions of the following almost-critical problem: − Δ u + V (x) u = u p − ε, u > 0 in Ω, u = 0 on ∂ Ω, - u+V (x) u=u^p-, 1. 0emu 00. 25em0. 1emin0. 1em0. 33em, u=00. 25em0. 1emon0. 1em0. 25em, where Ω is a bounded regular domain in R n {R}^n, n ≥ 4 n 4, ε is a small positive parameter, p + 1 = (2 n) ∕ (n − 2) p+1= (2n) / (n-2) is the critical Soblolev exponent, and the potential V V is a smooth positive function. We find solutions that exhibit bubbles clustered inside as ε 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.