Conditions for absence of solutions to some higher order elliptic inequalities with singular coefficients in $\mathbb{R}^n$

Abstract

In this paper we study Liouville type theorems for elliptic higher order inequalities with singular coefficients and gradient terms in $\mathbb{R}^n$. Our approach is based on the Pokhozhaev nonlinear capacity method, which is widely used for studying various nonlinear elliptic inequalities. We obtain apriori estimates for solutions of an elliptic inequality using the method of test functions. An optimal choice of the test function leads us to a nonlinear minimax problem, which generates a nonlinear capacity induced by a corresponding nonlinear problem. The existence of the zero limit of the corresponding apriori estimate ensures the absence of a nontrivial solution to the problem. Our result provide a new view on the behavior of solutions of higher order elliptic inequalities with singular coefficients and gradient terms and this approach can be useful in studying nonlinear elliptic inequalities of other types.