Existence of Solutions for $$p(x)$$ p ( x ) Laplacian Equations Without Ambrosetti–Rabinowitz Type Condition

Abstract

This paper investigates the existence and multiplicity of solutions for superlinear \(p(x)\)-Laplacian equations with Dirichlet boundary conditions. Under no Ambrosetti–Rabinowitz’s superquadraticity conditions, we obtain the existence and multiplicity of solutions using a variant Fountain theorem without Palais-Smale type assumptions.