Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term

Abstract

We consider a nonlinear Dirichlet problem driven by the$p$-Laplace differential operator. We assume that theCarathéodory reaction term $f(z,x)$ exhibits an asymmetricbehavior on the two semiaxes of $\mathbb{R}$. Namely, $f(z,\cdot)$ is$(p-1)$-linear near $-\infty$ and $(p-1)$-superlinear near$+\infty$, but without satisfying the well-knownAmbrosetti--Rabinowitz condition (AR-condition). Combiningvariational methods based on critical point theory, with suitabletruncation techniques and Morse theory, we show that the problemhas at least three nontrivial smooth solutions, two of which haveconstant sign (one positive, the other negative).