Applications Of Ricceri Variational Methods In P(x)-Laplacian Equations

we establish existence results and energy estimates of solutions for a homogeneous Neumann problem,Dirichlet problem and nonhomogeneous Neumann problem involving the p?x?-Laplace operator.The case of high dimensions,corresponding to the lack of compactness of W^1,p?x???? in C⁰??? is also considered.In particular,for a precise localization of the parameter,the existence of a non-zero solution is established without requiring any asymptotic condition at zero or at infinity of the nonlinear term.Under a suitable condition on the behavior of the potential at?10?0,we deduce the existence of solutions for small positive values of the parameter and we obtain that the corresponding solutions have smaller and smaller energies as the parameter goes to zero.Finally,some examples of application is provided.A basic ingredient in our arguments is a recent local minimum theorem.