A local mountain pass theorem and applications to a double perturbed p(x)p(x)-Laplacian equations

In this paper, we prove a local mountain pass theorem without (P.S)(P.S) condition. Using this theorem and Ricceri’s variational principle, we consider a double perturbed Neumann problem with nonlinear boundary condition of the form-Δp(x)u+a(x)|u|p(x)-2u=f(x,u)+λh1(x,u)inΩ,|∇u|p(x)-2∂u∂γ=g(x,u)+μh2(x,u)on∂Ω.At least seven solutions are obtained under different assumptions.