Quasilinear Schr\"odinger equations: ground state and infinitely many normalized solutions

TL;DR

This paper investigates normalized solutions for quasilinear Schrödinger equations, establishing existence results for ground states and infinitely many solutions in supercritical and critical cases, and analyzing their concentration behavior.

Contribution

It introduces new existence results for normalized solutions in the mass-supercritical case using perturbation and index theory, and studies solutions in the mass-critical case.

Findings

Existence of ground state normalized solutions in supercritical case

Infinitely many normalized solutions in supercritical case

Concentration behavior of ground state solutions

Abstract

In the present paper, we study the normalized solutions for the following quasilinear Schr\"odinger equations: $$-\Delta u-u\Delta u^2+\lambda u=|u|^{p-2}u \quad \text{in}~\mathbb R^N,$$ with prescribed mass $$\int_{\mathbb R^N} u^2=a^2.$$ We first consider the mass-supercritical case $p>4+\frac{4}{N}$, which has not been studied before. By using a perturbation method, we succeed to prove the existence of ground state normalized solutions, and by applying the index theory, we obtain the existence of infinitely many normalized solutions. Then we turn to study the mass-critical case, i.e., $p=4+\frac{4}{N}$, and obtain some new existence results. Moreover, we also observe a concentration behavior of the ground state solutions.