Nonexistence of solutions to systems of higher-order semilinear inequalities in cone-like domains

Abstract

In this paper, we obtain nonexistence results for global solutions to the system of higher-order semilinear partial differential inequalities $$displaylines{ frac{partial^k u_i}{partial t^k}-Delta (a_i (x,t) u_i (x,t)) geq t^{gamma_{i+1}}|x|^{sigma_{i+1}} |u_{i+1} (x,t) |^{p_{i+1}}, quad 1 leq i leq n, cr u_{n+1}=u_1, }$$ in cones and cone-like domains in $mathbb{R}^N$, $t>0$. Our results apply to nonnegative solutions and to solutions which change sign. Moreover, we provide a general formula of the critical exponent corresponding to this system. Our proofs are based on the test function method, developed by Mitidieri and Pohozaev.

Keywords

• nonexistence
• blow-up
• higher-order differential inequalities
• critical exponent.