Multiple Solutions for {varvec{(p,2)}}-Equations with Resonance and Concave Terms

Abstract

We consider parametric Dirichlet problems driven by the sum of a p-Laplacian (p>2) and a Laplacian ((p, 2)-equation) and with a reaction term which exhibits competing nonlinearities. We prove two multiplicity theorems. In the first the competing terms are not decoupled, the dependence on the parameter is not necessarily linear and the reaction term has a general polynomial growth, possibly supercritical. We produce three nontrivial solutions for small values of the parameter. We provide sign information for all solutions (two of constant sign and the third nodal). Then we decouple the competing nonlinearities and allow for resonance to occur at pm ,infty . We produce six nontrivial smooth solutions for small values of the parameter. We provide sign information for five of these solutions.