Blow-up phenomena and asymptotic profiles passing from $H^1$-critical to super-critical quasilinear Schr\"{o}dinger equations

Abstract

We study the asymptotic profile, as $\hbar\rightarrow 0$, of positive solutions to $$-\hbar^2\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^2=K(x)|u|^{p-2}u,\ \ x\in \mathbb{R}^N $$ where $\gamma\geq 0$ is a parameter with relevant physical interpretations, $V$ and $K$ are given potentials and $N\geq 5$. 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 $\gamma=0$.