Some Nonexistence Results of Systems of Quasilinear Elliptic Inequalities Involving the \(\Delta_\lambda\)-Laplacian

Abstract

In this paper, we establish some nonexistence results of a quasilinear elliptic inequality

and a system of quasilinear elliptic inequalities

where \(\nabla_\lambda=(\lambda_1\partial_{x_1},\lambda_2\partial_{x_2},\dots,\lambda_N\partial_{x_N})\) and \(\lambda_i\), \(i=1,2,\dots,N\) are some functions. Here, \(p,\widetilde{p}>1\), \(q, \widetilde{q}\geq 0\), and \(w,h, \widetilde{w}, \widetilde{h} \in L^1_{\mathrm{loc}}(\mathbb R^N)\) are nonnegative functions satisfying some growth conditions at infinity. In particular, our results can be seen as a generalization of some results of Mitidieri and Pokhozhaev from the Laplace operator to the \(\Delta_\lambda\)-Laplacian.