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

Abstract In this paper, we consider the existence of strong solutions of the following p ( x ) -Laplacian Dirichlet problem via critical point theory: { − d i v ( ∣ ∇ 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.

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