New results concerning a Schrödinger equation involving logarithmic nonlinearity - Yaqing Cai; Yulin Zhao; Chaoliang Luo (Electronic Journal of Qualitative Theory of Differential Equations)
In this paper, we investigate the existence of ground state solution to a class of Schrödinger equation involving logarithmic nonlinearity. To overcome the lack of smoothness, the corresponding functional $J$ is first decomposed into the sum of a $\mathcal{C}^{1}$ functional and a convex lower semicontinuous functional by adapting to the approach of Squassina–Szulkin in [Calc. Var. Partial Differential Equations 54(2015), 585–597]. Secondly, the existence of a ground state solution to the studied equation is proved by using the Mountain Pass Theorem under the weakened Ambrosetti–Rabinowitz conditions.