Quasilinear elliptic systems in [formula omitted] with multipower forcing terms depending on the gradient

Abstract

In this paper we deal with noncoercive 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 prove that any nonnegative nontrivial entire weak solution (nonnecessarily radial) is constant. For nontrivial solutions we intend that both components are nontrivial. The paper improves former results due to Clément, Fleckinger, Mitidieri, de Thélin (2000) in [4], to Bidaut-Véron and Pohozaev (2001) in [3], where no gradient terms are considered. On the other hand the paper contains, as subcases, also some recent results of Filippucci (2011) [12] in which systems with gradient dependence for the forcing terms are treated but not of multipower type.The key ingredient of the proof technique is the method of test functions, employed by Mitidieri and Pohozaev for studying systems in Mitidieri and Pohozaev (1999) [17]. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions is required.