Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions

In this paper, mixed elliptic problems involving the p-Laplacian and with nonhomogeneous boundary conditions are investigated. At first, the existence of one non-trivial solution, under a suitable behaviour on the nonlinearity and without requiring neither conditions at zero nor conditions at infinity, is established. Then, by adding a condition at infinity on the nonlinearity, also a second non-trivial solution is guaranteed. Some special cases are pointed out as, in particular, the existence of one non-trivial solution when the datum is $(p-1)-$sublinear at zero and the existence of two non-trivial solutions when the nonlinear term is again $(p-1)-$sublinear at zero and, in addition, more than $(p-1)-$superlinear at infinity. As a consequence, the existence of two non-trivial solutions for concave-convex nonlinearities is emphasized. Finally, the case of a simple $(p-1)-$superlinearity at infinity is considered and it is also observed that the same results hold when the nonlinear behaviour, described before for the datum, is instead assumed by the nonhomogeneous Neumann boundary conditions. Concrete examples of applications are also given. The approach is based on variational methods and critical point theory. Precisely, a non-zero local minimum theorem and a two non-zero critical points theorem are applied.