Nonexistence of positive solutions for polyharmonic systems in $\mathbb{R}^N_+$

Abstract

In this paper, we study themonotonicity and nonexistence of positive solutions for polyharmonicsystems $\left\{\begin{array}{rlll}(-\Delta)^m u&=&f(u, v)\\(-\Delta)^m v&=&g(u, v)\end{array}\right.\;\hbox{in}\;\mathbb{R}^N_+.$ By using the Alexandrov-Serrin method of moving plane combined with integral inequalities and Sobolev's inequality in a narrow domain, we prove the monotonicity of positive solutions for semilinear polyharmonic systems in $\mathbb{R_+^N}.$ As a result, the nonexistence for positive weak solutions to the system are obtained.