Existence, nonexistence and multiplicity of positive solutions for parametric nonlinear elliptic equations

Abstract

We consider a parametric nonlinear elliptic equation driven by the Dirichlet $p$-Laplacian. We study the existence, nonexistence and multiplicity of positive solutions as the parameter $\lambda$ varies in $\mathbb{R}^{+}_{0}$ and the potential exhibits a $p$-superlinear growth, without satisfying the usual in such cases Ambrosetti--Rabinowitz condition. We prove a bifurcation-type result when the reaction has ($p-1$)-sublinear terms near zero (problem with concave and convex nonlinearities). We show that a similar bifurcation-type result is also true, if near zero the right hand side is ($p-1$)-linear.