An Investigation into the Nonexistence of Nontrivial Weak Solutions of Nonlinear Inequalities with Gradient Nonlinearities

Abstract

In this article, we extend and refine the work of Mitidieri and Pohozhaev concerning the nonexistence of nontrivial weak solutions to nonlinear inequalities and systems. Specifically, we focus on problems involving integer powers of the Laplace operator and nonlinear terms of the form a(x)∣∇(Δmu)∣q+b(x)∣∇u∣sa(x) |\nabla(\Delta^m u)|^q + b(x) |\nabla u|^s. By leveraging the nonlinear capacity method and carefully selecting suitable test functions, we derive optimal a priori estimates. These estimates allow us to prove, via contradiction, the nonexistence of nontrivial weak solutions to the given nonlinear inequalities and systems. Our results provide new insights into the conditions under which solutions fail to exist, contributing to the broader understanding of nonlinear partial differential equations with gradient nonlinearity.