On superlinear p(x)p(x)-Laplacian problems without Ambrosetti and Rabinowitz condition

In this paper, we consider the p(x)p(x)-Laplacian equations on the bounded domain. The nonlinearity is superlinear but does not satisfy the usual Ambrosetti–Rabinowitz condition near infinity, or its dual version near zero. Existence and multiplicity results are obtained via Morse theory and modified functional methods. In a sense, we expand a recent result of Gasiński and Papageorgiou [L. Gasiński, N.S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations 42 (2011) 323–354].