Existence and multiplicity results for p-laplacian Dirichlet problem with combined nonlinearities

A class of p-laplacian equations with combined nonlinearities is studied. First, when the nonlinear term is asymptotically linear at infinity, existence result of four non-trivial solutions is obtained by Ekeland variational principle and mountain pass theorem. Second, when the nonlinear term is superlinear at positive infinity but asymptotically linear at negative infinity, some existence results for non-trivial solution are established by mountain pass theorem and a variant version of mountain pass theorem.