Existence of entire solutions for a class of quasilinear elliptic equations

Abstract

The paper deals with the existence of entire solutions for a quasilinear equation (E)λ in ℝN, depending on a real parameter λ, which involves a general elliptic operator in divergence form A and two main nonlinearities. The competing nonlinear terms combine each other, being the first subcritical and the latter supercritical. We prove the existence of a critical value λ* > 0 with the property that (E)λ admits nontrivial non-negative entire solutions if and only if λ ≥ λ*. Furthermore, when λ > ̄λ ≥ λ*, the existence of a second independent nontrivial non-negative entire solution of (E)λ is proved under a further natural assumption on A. © 2012 Springer Basel AG.