Existence of entire solutions for a class of quasilinear elliptic equations

The paper deals with the existence of entire solutions for a quasilinear equation (Eλ) 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 (Eλ) 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 (Eλ) is proved under a further natural assumption on A.