Nonexistence of solutions for singular quasilinear differential inequalities with a gradient nonlinearity (Q412761)

It is considered the nonexistence of nonnegative weak solutions of the following two singular quasilinear differential inequalities NEWLINE\[NEWLINE -\text{div}\, (A(x,u,\nabla u)\nabla u)\geq a(x)u^q|\nabla u|^s \qquad \text{ in } \Omega NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \text{div}\, (A(x,u,\nabla u)\nabla u)\geq a(x)u^q|\nabla u|^s \qquad \text{ in } \Omega, NEWLINE\]NEWLINE where \(q>0\) or \(q<0\), \(s\geq 0.\) The proofs are based on the test function method developed by \textit{E. Mitidieri} and \textit{S. I. Pohozaev} [Milan J. Math. 72, 129--162 (2004; Zbl 1115.35157)].