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

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 $ε>0, N \geq 1$ 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.

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