Existence and Multiplicity of Solutions for Resonant (p,2)-Equations

Abstract

We consider Dirichlet elliptic equations driven by the sum of a p-Laplacian (2<p){(2<p)} and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both ±∞{\pm\infty} and at zero. We prove an existence theorem (producing one nontrivial smooth solution) and a multiplicity theorem (producing five nontrivial smooth solutions, four of constant sign and the fifth nodal; the solutions are ordered). Our approach uses variational methods and critical groups.

Keywords

• resonance
• variational eigenvalues
• critical groups
• nonlinear regularity
• multiple solutions
• nodal solutions
• 35j20
• 35j60
• 58e05