Noncoercive resonant (p,2)-equations with concave terms

Abstract

We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplace and a Laplacian (a (p,2){(p,2)}-equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory perturbation which is resonant with respect to the principle eigenvalue of the Dirichlet p-Laplacian. Using variational methods together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small values of the parameter, the problem has as least six nontrivial smooth solutions all with sign information (two positive, two negative and two nodal (sign changing)).

Keywords

• concave term
• resonance
• nonlinear regularity theory
• nonlinear maximum principle,strong comparison principle
• truncation
• critical groups
• constant sign and nodal solutions
• 35j20
• 35j60
• 58e05