Nonexistence of nontrivial weak solutions of some nonlinear inequalities with integer power Laplacian

In this paper, we make modification on the results obtained by Mitidieri and Pokhozhaev on sufficient conditions for the nonexistence of nontrivial weak solutions of nonlinear inequality and system with integer power of the Laplacian with nonlinearity term of the form $a(x)|\Delta^{m}u|^{q}+b(x)|u|^s$. We obtain the optimal a priori estimate by employing the nonlinear capacity method under the wise choice of test function. Finally, we prove the nonexistence of nontrivial weak solutions of the considered inequality and its system by contradiction.