Existence of entire solutions for a class of quasilinear elliptic equations

The paper deals with the existence of entire solutions for a quasilinear equation in , depending on a real parameter lambda, 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 lambda* > 0 with the property that admits nontrivial non-negative entire solutions if and only if lambda a parts per thousand yen lambda*. Furthermore, when , the existence of a second independent nontrivial non-negative entire solution of is proved under a further natural assumption on A.