Nonexistence of Solutions for Nonlinear Differential Inequalities with Singularities on Unbounded Sets

Li, Xiaohong; Wan, Haitao; Li, Xiliang

doi:10.1134/s0001434620010113

Cited by 1 publication

(1 citation statement)

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“…Over past ten years the absence of solutions to various nonlinear differential inequalities and systems was studied by many mathematicians, in particular, for elliptic inequalities and systems of partial differential equations with singular coefficients and gradient terms, what is equivalent to the absence of stationary states for the corresponding parabolic inequalities and systems, see [1]- [25]. Here we can mention a result by B. Gidas and J. Spruck [13] on absence of positive solutions to the equation…”

Section: Introductionmentioning

confidence: 99%

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

Admasu,

Galakhov

2023

Ufimsk. Mat. Zh.

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In this paper we study Liouville type theorems for elliptic higher order inequalities with singular coefficients and gradient terms in 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.

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Section: Introductionmentioning

confidence: 99%