Existence and concentration of positive solutions for a logarithmic Schrödinger equation via penalization method

Abstract

In this article we are concerned with the following logarithmic Schrödinger equation $$ \left\{ \begin{array}{lc} -ε^2Δu+ V(x)u=u \log u^2, & \mbox{in} \,\, \mathbb{R}^{N}, \\ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \\ u \in H^1(\mathbb{R}^{N}), & \; \\ \end{array} \right. $$ where $\epsilon >0,N\ge 1$ and $V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous potential. Under a local assumption on the potential $V$, we use the variational methods to prove the existence and concentration of positive solutions for the above problem.

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• Logarithmic nonlinearity in theories of quantum gravity: Origin of time and observational consequences
• Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential
• Multi-bump positive solutions for a logarithmic Schrödinger equation with deepening potential well
• Bound states for logarithmic Schrodinger equations with potentials unbounded below

Cited by in corpus (1)

• Multi-bump positive solutions for a logarithmic Schrödinger equation with deepening potential well