Liouville type results for systems of elliptic inequalities with gradient terms

We discuss some recent results about non coercive elliptic multipower systems of divergence type, which include $p$-Laplacian type operators as well as mean curvature operators, and whose right hand sides depend on the product of both components of the solution and on a gradient factor. We are interested in Liouville type results, namely when any nonnegative nontrivial entire weak solution (non necessarily radial) is constant. For nontrivial solutions we intend that both components are nontrivial. The key ingredient of the proof technique is the method of test functions. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions is required.