scholarly journals Finite Morse index solutions of the Hénon Lane–Emden equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Abdellaziz Harrabi ◽  
Cherif Zaidi
Keyword(s):  
Morse Index ◽  
Monotonicity Formula ◽  
Liouville Type ◽  
Emden Equation ◽  
Type Identity ◽  
Liouville Type Theorems ◽  
Whole Space ◽  
Stable Solutions ◽  
Pohozaev Type Identity

Abstract In this paper, we are concerned with Liouville-type theorems of the Hénon Lane–Emden triharmonic equations in whole space. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence.

scholarly journals Liouville Theorems for Fractional Parabolic Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenxiong Chen ◽  
Leyun Wu
Keyword(s):  
Half Space ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Maximum Principles ◽  
Qualitative Properties ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Narrow Region ◽  
Liouville Type Theorems ◽  
Whole Space

Abstract In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space R n - 1 × R \mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.

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

Mathematics ◽  
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.

Liouville type theorems for stable solutions ofp-Laplace equation inRN

2017 ◽  
Vol 160 ◽  
pp. 44-52 ◽  
Author(s):  
Caisheng Chen ◽  
Hongxue Song ◽  
Hongwei Yang
Keyword(s):  
Laplace Equation ◽  
Liouville Type ◽  
Liouville Type Theorems ◽  
Stable Solutions

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

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.

Liouville-type theorems for elliptic equations in half-space with mixed boundary value conditions

2016 ◽  
Vol 8 (1) ◽  
pp. 193-202 ◽  
Author(s):  
Abdellaziz Harrabi ◽  
Belgacem Rahal
Keyword(s):  
Elliptic Equations ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Nonlinear Elliptic Equations ◽  
Monotonicity Formula ◽  
Compact Set ◽  
Liouville Type ◽  
Nonlinear Elliptic ◽  
Liouville Type Theorems ◽  
Nonexistence Of Solutions

Abstract In this paper we study the nonexistence of solutions, which are stable or stable outside a compact set, possibly unbounded and sign-changing, of some nonlinear elliptic equations with mixed boundary value conditions. The main methods used are the integral estimates and the monotonicity formula.

Sign in / Sign up

Export Citation Format

Share Document