Existence of strong solutions of a p(x)p(x)-Laplacian Dirichlet problem without the Ambrosetti–Rabinowitz condition

In this paper, we consider the existence of strong solutions of the following p(x)p(x)-Laplacian Dirichlet problem via critical point theory: {−div(∣∇u∣p(x)−2∇u)=f(x,u),  in  Ω,u=0,  on  ∂Ω. We give a new growth condition, under which, we use a new method to check the Cerami compactness condition. Hence, we prove the existence of strong solutions of the problem as above without the growth condition of the well-known Ambrosetti–Rabinowitz type and also give some results about multiplicity of the solutions.