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

TL;DR

This paper analyzes the asymptotic behavior of solutions to a quasilinear Schrödinger equation as a parameter approaches zero, revealing different concentration phenomena depending on the nonlinearity and a parameter , with implications for physical models.

Contribution

It uncovers new concentration behaviors of solutions in the super-critical regime, contrasting with the critical case, and highlights the influence of the source potential on localization.

Findings

Solutions concentrate differently depending on  and p.

Distinct concentration profiles are observed for >0 versus =0.

The concentration behavior varies notably between critical and super-critical nonlinearities.

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 $\gamma=0$.