Nonhomogeneous Nonlinear Dirichlet Problems with a p -Superlinear Reaction

Abstract

We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reaction $f\left(z,\zeta \right)$, whose primitive $f\left(z,\zeta \right)$ is $p$-superlinear near $\pm \infty$, but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like the $p$-Laplacian, the $\left(p,q\right)$-Laplacian, and the $p$-generalized mean curvature operator.