On a class of logarithmic Schrödinger equations via perturbation method

In this paper, we consider the following logarithmic Schrödinger equation $$-\mathrm{\Delta }u+V(x)u=u\log u^2,x\in {\mathbb{R}}^N\mathrm{.}$$ Assuming that $V\in C({\mathbb{R}}^N,\mathbb{R})$, $V$ is bounded away from zero, and $V(x)\to +\mathrm{\infty }$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$, we develop a new perturbative variational approach to overcome the lack of $C^1$-smoothness of the associated functional and prove the existence and multiplicity of nontrivial weak solutions.