Liouville-type theorems and existence results for stable-at-infinity solutions of higher-order m-polyharmonic problems

We study Navier problems involving the higher-order fractional Laplacians. We first obtain nonexistence of positive solutions, known as the Liouville-type theorems, in the upper half-space [Formula: see text] by studying an equivalent integral form of the fractional equation. Then we show symmetry for positive solutions on [Formula: see text] through a delicate iteration between lower-order differential/pseudo-differential equations split from the higher-order equation.

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Some Existence Results For Higher Order Fractional Differential Equations with Multi-Strip Riemann-Liouville Type Fractional Integral Boundary Conditions

Journal of Nonlinear Analysis and Application ◽

10.5899/2019/jnaa-00441 ◽

2019 ◽

Vol 2019 (1) ◽

pp. 1-12

Author(s):

P. Duraisamy ◽

T. Nandha Gopal

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Fractional Differential Equations ◽

Fractional Integral ◽

Integral Boundary Conditions ◽

Higher Order ◽

Existence Results ◽

Liouville Type ◽

Integral Boundary ◽

Fractional Differential

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Non-Existence Results for Stable Solutions to Weighted Elliptic Systems Including Advection Terms

Mathematics ◽

10.3390/math10020252 ◽

2022 ◽

Vol 10 (2) ◽

pp. 252

Author(s):

Suleman Alfalqi

Keyword(s):

Elliptic Systems ◽

Existence Results ◽

Liouville Type ◽

Non Linear ◽

Liouville Type Theorems ◽

Stable Solutions ◽

Bootstrap Iteration ◽

Comparison Property

In this paper, we study a non-linear weighted Grushin system including advection terms. We prove some Liouville-type theorems for stable solutions of the system, based on the comparison property and the bootstrap iteration. Our results generalise and improve upon some previous works.

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Liouville-type theorems and existence results for stable solutions to weighted Lane–Emden equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.160 ◽

2019 ◽

Vol 150 (3) ◽

pp. 1567-1579

Author(s):

Alberto Farina ◽

Shoichi Hasegawa

Keyword(s):

Euclidean Space ◽

Existence Result ◽

Large Family ◽

Existence Results ◽

Liouville Type ◽

Liouville Type Theorems ◽

Stable Solutions

AbstractWe devote this paper to proving non-existence and existence of stable solutions to weighted Lane-Emden equations on the Euclidean space ℝN, N ⩾ 2. We first prove some new Liouville-type theorems for stable solutions which recover and considerably improve upon the known results. In particular, our approach applies to various weighted equations, which naturally appear in many applications, but that are not covered by the existing literature. A typical example is provided by the well-know Matukuma's equation. We also prove an existence result for positive, bounded and stable solutions to a large family of weighted Lane–Emden equations, which indicates that our Liouville-type theorems are somehow sharp.

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