Existence of Multiple Solution for a Singular p(x)-Laplacian Problem

We will study the following singular problem - d i v ( | ∇ φ | p ( x ) - 2 ∇ φ ) + Ψ ( x ) | φ | p ( x ) - 2 φ = a ( x ) φ - γ ( x ) + λ f ( x , φ ) , in Ω , φ = 0 , on ∂ Ω . Here Ω ⊂ R N , ( N > 2 ) is a bounded domain with smooth boundary ∂ Ω , λ > 0 , γ ∈ C ( Ω ¯ , ( 0 , 1 ) ) , the function a belongs to some L r ( x ) ( Ω ) . Under suitable hypotheses on the functions p , Ψ , and f , and by using variational methods, the existence of solutions is proved. Furthermore, we also prove the existence of infinitely many solutions to the problem by applying the Clarke’s and the fountain’s theorems. The novelty in this research is the combination of the sub-super solutions method with the Clarke’s and fountain theorem to solve singular problems.