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

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