Existence and multiplicity of normalized solutions for the quasi-linear Schr\"{o}dinger equations with mixed nonlinearities

TL;DR

This paper investigates the existence and multiplicity of normalized solutions for a quasi-linear Schrödinger equation with mixed nonlinearities, providing new results on ground states and mountain pass solutions in various dimensions.

Contribution

It establishes the existence of normalized ground state and mountain pass solutions for the quasi-linear Schrödinger equation with specific mixed nonlinearities, extending previous work.

Findings

Existence of normalized ground state solutions.

Existence of mountain pass type solutions.

Results applicable for dimensions 1 to 4.

Abstract

In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -\Delta u-\Delta(|u|^2)u+\lambda u=|u|^{p-2}u+\tau|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4, \end{equation*} with prescribed mass $$\int_{\mathbb{R}^N}|u|^2dx=a ,$$ where $\lambda\in\mathbb{R}$ appears as a Lagrange multiplier and the parameters $a,\tau$ are all positive constants. We are concerned about the mass-mixed case $2<q<2+\frac{4}{N}$ and $4+\frac{4}{N}<p<2\cdot2^*$, where $2^*:=\frac{2N}{N-2}$ for $N\geq3$, while $2^*:=\infty$ for $N=1,2$. We show the existence of normalized ground state solution and normalized solution of mountain pass type. Our results can be regarded as a supplement to Lu et al. ( Proc. Edinb. Math. Soc., 2024) and Jeanjean et al. ( arXiv:2501.03845).