Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

In this article, we modify the results obtained by Mitidieri and Pohozhaev on sufficient conditions for the nonexistence of nontrivial weak solutions to nonlinear inequalities and systems with integer powers of the Laplace operator and with a nonlinear term of the form a(x) |∇(Δmu)|q + b(x)|∇u|s. We obtain optimal a priori estimates by applying the nonlinear capacity method with an appropriate choice of test functions. As a result, we prove the nonexistence of nontrivial weak solutions to nonlinear inequalities and systems by contradiction.