Variational analysis of anisotropic Schrödinger equations without Ambrosetti–Rabinowitz-type condition

Abstract

This article is concerned with the qualitative analysis of weak solutions to nonlinear stationary Schrodinger-type equations of the form $$\begin{aligned} \left\{ \begin{array}{ll} - \displaystyle \sum _{i=1}^N\partial _{x_i} a_i(x,\partial _{x_i}u)+b(x)|u|^{P^+_+-2}u =\lambda f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array}\right. \end{aligned}$$ without the Ambrosetti–Rabinowitz growth condition. Our arguments rely on the existence of a Cerami sequence by using a variant of the mountain-pass theorem due to Schechter.