Existence of entire solutions for quasilinear elliptic equations in the Heisenberg group

The paper deals with the existence of entire solutions for a quasilinear equation (E) in the Heisenberg group, depending on a real parameter, which involves a general elliptic operator A in divergence form and two main nonlinearities. The competing nonlinear terms combine each other. Under some conditions, we prove the existence of a critical value with the property that (E) admits nontrivial nonnegative entire solutions if and only if the parameter is bigger or equal to the treshold. Furthermore, under the further assumption that the potential of A is uniform convex, we give the existence of a second independent nontrivial nonnegative entire solution of (E) , when the parameter is strictly bigger than the treshold.