Multiple Solutions for Noncoercive Problems with the $p$-Laplacian

Abstract

We consider a nonlinear elliptic equation driven by the $p$-Laplacian and with a Carathéodory right hand side nonlinearity which exhibits an asymmetric asymptotic behaviour at $+\infty$ and at $-\infty$. These hypotheses imply that the Euler functional of the problem is noncoercive (indefinite). Using critical point theory, we prove the existence of at least two nontrivial smooth solutions. Also in the last section for the asymmetric functionals considered here, we compute the critical groups at infinity.