Existence of entire solutions for a class of quasilinear elliptic equations

The paper deals with the existence of entire solutions for a quasilinear equation \({(\mathcal E)_\lambda}\) in \({\mathbb{R}^N}\) , 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 \({(\mathcal E)_\lambda}\) admits nontrivial non-negative entire solutions if and only if λ ≥ λ*. Furthermore, when \({\lambda > \overline{\lambda} \ge \lambda^*}\) , the existence of a second independent nontrivial non-negative entire solution of \({(\mathcal{E})_\lambda}\) is proved under a further natural assumption on A.