Noncoercive Resonant (p, 2)-Equations

Abstract

We consider a nonlinear Dirichlet problem driven by the sum of p-Laplacian and a Laplacian (a (p, 2)-equation) which is resonant at $\pm \infty$ with respect to the principal eigenvalue $\hat{\lambda }_1(p)$ of $(-\Delta _p,W^{1,p}_{0}(\Omega ))$ and resonant at zero with respect to any nonprincipal eigenvalue of $(-\Delta ,H^1_0(\Omega ))$. At $\pm \infty$ the resonance occurs from the right of $\hat{\lambda }_1(p)$ and so the energy functional of the problem is indefinite. Using critical groups, we show that the problem has at least one nontrivial smooth solution. The result complements the recent work of Papageorgiou and Rădulescu (Appl Math Optim 69:393–430, 2014), where resonant (p, 2)-equations were examined with the resonance occurring from the left of $\hat{\lambda }_1(p)$ (coercive problem).