Blow-up phenomena and asymptotic profiles passing from $H^1$-critical to super-critical quasilinear Schrödinger equations

Abstract

We study the asymptotic profile, as $\hslash \to 0$, of positive solutions to $$-{\hslash }^2\Delta u+V(x)u-{\hslash }^{2+\gamma }u\Delta u^2=K(x)|u{|}^{p-2}u,x\in {\mathbb{R}}^N$$ where $\gamma \ge 0$ is a parameter with relevant physical interpretations, $V$ and $K$ are given potentials and $N\ge 5$. We investigate the concentrating behavior of solutions when $\gamma >0$ and, differently form the case $\gamma =0$ where the leading potential is $V$, the concentration is here localized by the source potential $K$. Moreover, surprisingly for $\gamma >0$ we find a different concentration behavior of solutions in the case $p=\frac{2N}{N-2}$ and when $\frac{2N}{N-2}<p<\frac{4N}{N-2}$. This phenomenon does not occur when $\gamma =0$.