Quasilinear Schrödinger equations: ground state and infinitely many normalized solutions

Houwang Li and Wenming Zou

Vol. 322 (2023), No. 1, 99–138

DOI: 10.2140/pjm.2023.322.99

Abstract

We study the normalized solutions for the following quasilinear Schrödinger equations:

$$-\mathrm{\Delta }u-u\mathrm{\Delta }{u}^2+\lambda u=|u{|}^{p-2}u\text{ in }{ℝ}^N,$$

with prescribed mass

$${\int \; <!--nolimits-->}_{{ℝ}^N}{u}^2=a^2.$$

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. We also obtain new existence results for the mass-critical case $p=4+\frac{4}{N}$ and remark on a concentration behavior for ground state solutions.

Keywords

quasilinear Schrödinger equation, normalized solution, perturbation method, index theory

Mathematical Subject Classification

Primary: 35J15, 35J50, 35J60

Milestones

Received: 11 November 2021

Revised: 1 November 2022

Accepted: 15 January 2023

Published: 3 May 2023