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   \(-\Delta u+\lambda u=\alpha u\log u^2+\mu|u|^{p-2}u\), \(x\in\mathbb{R}^N\),   under the mass constraint   \(\int_{\mathbb{R}^N}u^2\,\mathrm{d}x=c^2\),   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 solutions and excited state solutions. The asymptotic behavior of the ground state solution as \(\mu\to 0\) is also investigated. Our results include the case \(\alpha<0\) or \(\mu<0\), which is less studied in the literature.