Existence and concentration of positive solutions for a logarithmic Schrödinger equation via penalization method (1906.07454v2)

Abstract: In this article we are concerned with the following logarithmic Schr\"{o}dinger equation $$ \left{ \begin{array}{lc} -{\epsilon}^2\Delta u+ V(x)u=u \log u^2, & \mbox{in} \,\, \mathbb{R}^{N}, \ %u(x)>0, & \mbox{in} \quad \mathbb{R}^{N} \ u \in H^1(\mathbb{R}^{N}), & \; \ \end{array} \right. $$ where $\epsilon >0, N \geq 1$ and $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ 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.