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JOURNALS // Ufimskii Matematicheskii Zhurnal // Archive

Ufimsk. Mat. Zh., 2023 Volume 15, Issue 2, Pages 3–8 (Mi ufa648)

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

W. E. Admasu, E. I. Galakhov

RUDN University, Miklukho-Maklay str. 6, 117198, Moscow, Russia

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.

Keywords: Liouville type theorems, apriori estimate, nonlinear capacity, singular coefficients, gradient terms.

UDC: 517.957

MSC: 35J30, 35J62

Received: 28.02.2022


 English version:
Ufa Mathematical Journal, 2023, 15:2, 3–8


© Steklov Math. Inst. of RAS, 2025