Existence of a positive bound state solution for logarithmic Schrödinger equation

Abstract

We investigate the existence of bound state solutions for the logarithmic Schrödinger equation−Δu+V(x)u=ulog⁡u2inRN,N≥1, where V(x)∈C(RN) has a limit at infinity. In the case when the ground state is not attained, we construct a higher critical value for the variational functional. The classical variational methods cannot be applied directly to deal with the above problem due to nonsmoothness. To recover the smoothness, we use the superlinear power-law perturbation of the logarithmic nonlinearity.