Unsolvability conditions for some inequalities and systems with functional parameters and singular coefficients on boundary

Abstract

We consider the problem on nonexistence of positive solutions for some nonlinear elliptic inequalities in a bounded domain. The principal parts of the considered inequalities are $p(x)$-Laplacians with variable exponents. The lower terms of the considered inequalities can depend both on the unknown function and its gradient. We assume that the coefficients at the lower terms have singularities at the boundary. To the best of the authors' knowledge, the conditions for nonexistence of solutions to inequalities with variable exponents were not considered before. We obtain the sufficient conditions for nonexistence of positive solutions in terms of the exponent $p(x)$, of the singularities order and of parameters in the problem. To prove the obtained conditions, we employ an original modification of the nonlinear capacity method proposed by S.I. Pokhozhaev. The method is based on a special choice of test functions in the generalized formulation of the problem and on algebraic transformations of the obtained expression. This allows us to obtain asymptotically sharp apriori estimates for the solutions leading to a contradiction under a certain choice of the parameters. This implies the nonexistence of the solutions. We generalize the obtained results for the case of nonlinear systems with similar conditions for the operators and coefficients.