Normalized solutions for logarithmic Schr\"{o}dinger equation with a perturbation of power law nonlinearity

TL;DR

This paper investigates the existence and behavior of normalized solutions to a logarithmic Schrödinger equation with power law perturbation, covering various parameter regimes including less-studied cases with negative coefficients.

Contribution

It establishes existence results for ground and excited states under diverse conditions and analyzes the asymptotic behavior as the perturbation parameter approaches zero.

Findings

Existence of ground state solutions under multiple parameter conditions.

Existence of excited state solutions with different assumptions.

Asymptotic analysis of solutions as the perturbation parameter tends to zero.

Abstract

We study the existence of normalized solutions to the following logarithmic Schr\"{o}dinger equation \begin{equation*}\label{eqs01} -\Delta u+\lambda u=\alpha u\log u^2+\mu|u|^{p-2}u, \ \ x\in\R^N, \end{equation*} under the mass constraint \[ \int_{\R^N}u^2\mathrm{d}x=c^2, \] where $\alpha,\mu\in \R$, $N\ge 2$, $p>2$, $c>0$ is a constant, and $\lambda\!\in\!\R$ appears as Lagrange multiplier. Under different assumptions on $\alpha,\mu,p$ and $c$, we prove the existence of ground state solution and excited state solution. The asymptotic behavior of the ground state solution as $\mu\to 0$ is also investigated. Our results including the case $\alpha<0$ or $\mu<0$, which is less studied in the literature.