Weak solutions and energy estimates for a class of nonlinear elliptic Neumann problems

We establish existence results and energy estimates of solutions for a homogeneous Neumann problem involving the p-Laplace operator. The case of large dimensions, corresponding to the lack of compactness of W1,p(Ω) in C0(Ω) 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. In the case of (p - 1)-sublinear terms at the origin, 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, a multiplicity result is obtained and concrete examples of applications are provided. A basic ingredient in our arguments is a recent local minimum theorem for differentiable functionals