Blow-up phenomena and asymptotic profiles passing from h1-critical to super-critical quasilinear Schrödinger equations

We study the asymptotic profile, as h → 0, of positive solutions to where γ≥0 is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite L2-energy solutions. We investigate the concentrating behavior of solutions when γ>0 and, differently from the case γ=0 where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for γ>0 we find a different concentration behavior of solutions in the case p=2NN-2 and when 2NN-24 NN-2. This phenomenon does not occur when γ=0.