nonexistence result for p-Laplacian systems in a ball

Abstract

We consider the \(p\)-Laplacian system $$ \displaylines{ -\Delta_p u = \lambda f(v) \quad \text{in } \Omega; \cr -\Delta_p v = \lambda g(u) \quad \text{in } \Omega; \cr u = v=0 \quad \text{on }\partial \Omega, }$$ where \(\lambda >0\) is a parameter, \(\Delta_p u:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator for \(p > 1\) and \(\Omega\) is a unit ball in \(\mathbb{R}^N\) (\(N \geq 2)\). The nonlinearities \(f, g: [0,\infty) \to \mathbb{R}\) are assumed to be \(C^1\) non-decreasing semipositone functions (\(f(0)< 0\) and \(g(0)<0\)) that are \(p\)-superlinear at infinity. By analyzing the solution in the interior of the unit ball as well as near the boundary, we prove that the system has no positive radially symmetric and radially decreasing solution for \(\lambda\) large. See also https://ejde.math.txstate.edu/special/02/a1/abstr.html