Normalized solutions for logarithmic Schrödinger equation with a perturbation of power law nonlinearity

Abstract

We study the existence of normalized solutions to the following logarithmic Schrödinger equation \begin{equation*}\label{eqs01} -Δu+λu=αu\log u^2+μ|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 $\alpha ,\mu \in \mathbb{R}$, $N\ge 2$, $p>2$, $c>0$ is a constant, and $\lambda \in \mathbb{R}$ 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.

33 pages