scholarly journals GROUP INVARIANCE AND POHOZAEV IDENTITY IN MOSER-TYPE INEQUALITIES

2013 ◽  
Vol 15 (02) ◽  
pp. 1250054 ◽  
Author(s):  
DANIELE CASSANI ◽  
BERNHARD RUF ◽  
CRISTINA TARSI
Keyword(s):  
Bounded Domain ◽  
Best Constants ◽  
Pohozaev Identity ◽  
Lagrange Equations ◽  
Group Invariance ◽  
Type Identity ◽  
Zygmund Spaces ◽  
Pohozaev Type Identity

We study the so-called limiting Sobolev cases for embeddings of the spaces [Formula: see text], where Ω ⊂ ℝn is a bounded domain. Differently from J. Moser, we consider optimal embeddings into Zygmund spaces: we derive related Euler–Lagrange equations, and show that Moser's concentrating sequences are the solutions of these equations and thus realize the best constants of the corresponding embedding inequalities. Furthermore, we exhibit a group invariance, and show that Moser's sequence is generated by this group invariance and that the solutions of the limiting equation are unique up to this invariance. Finally, we derive a Pohozaev-type identity, and use it to prove that equations related to perturbed optimal embeddings do not have solutions.

Reverse Stein–Weiss Inequalities on the Upper Half Space and the Existence of Their Extremals

2019 ◽  
Vol 19 (3) ◽  
pp. 475-494 ◽  
Author(s):  
Lu Chen ◽  
Guozhen Lu ◽  
Chunxia Tao
Keyword(s):  
Half Space ◽  
Hardy Inequality ◽  
Necessary Conditions ◽  
Stereographic Projection ◽  
Pohozaev Identity ◽  
Lagrange Equations ◽  
High Dimensions ◽  
Spherical Form ◽  
Best Constant ◽  
Alpha Beta

Abstract The purpose of this paper is four-fold. First, we employ the reverse weighted Hardy inequality in the form of high dimensions to establish the following reverse Stein–Weiss inequality on the upper half space: \int_{\mathbb{R}^{n}_{+}}\int_{\partial\mathbb{R}^{n}_{+}}\lvert x|^{\alpha}|x% -y|^{\lambda}f(x)g(y)|y|^{\beta}\,dy\,dx\geq C_{n,\alpha,\beta,p,q^{\prime}}\|% f\/\|_{L^{q^{\prime}}(\mathbb{R}^{n}_{+})}\|g\|_{L^{p}(\partial\mathbb{R}^{n}_% {+})} for any nonnegative functions {f\in L^{q^{\prime}}(\mathbb{R}^{n}_{+})} , {g\in L^{p}(\partial\mathbb{R}^{n}_{+})} , and {p,q^{\prime}\in(0,1)} , {\beta<\frac{1-n}{p^{\prime}}} or {\alpha<-\frac{n}{q}} , {\lambda>0} satisfying \frac{n-1}{n}\frac{1}{p}+\frac{1}{q^{\prime}}-\frac{\alpha+\beta+\lambda-1}{n}% =2. Second, we show that the best constant of the above inequality can be attained. Third, for a weighted system analogous to the Euler–Lagrange equations of the reverse Stein–Weiss inequality, we obtain the necessary conditions of existence for any positive solutions using the Pohozaev identity. Finally, in view of the stereographic projection, we give a spherical form of the Stein–Weiss inequality and reverse Stein–Weiss inequality on the upper half space {\mathbb{R}^{n}_{+}} .

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 Existence, multiplicity and nonexistence results for Kirchhoff type equations

2020 ◽  
Vol 10 (1) ◽  
pp. 616-635 ◽  
Author(s):  
Wei He ◽  
Dongdong Qin ◽  
Qingfang Wu
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Existence Result ◽  
Critical Case ◽  
Type Equation ◽  
General Assumption ◽  
Pohozaev Identity ◽  
Kirchhoff Type ◽  
Whole Space ◽  
Space Case

Abstract In this paper, we study following Kirchhoff type equation: $$\begin{array}{} \left\{ \begin{array}{lll} -\left(a+b\int_{{\it\Omega}}|\nabla u|^2 \mathrm{d}x \right){\it\Delta} u=f(u)+h~~&\mbox{in}~~{\it\Omega}, \\ u=0~~&\mbox{on}~~ \partial{\it\Omega}. \end{array} \right. \end{array}$$ We consider first the case that Ω ⊂ ℝ3 is a bounded domain. Existence of at least one or two positive solutions for above equation is obtained by using the monotonicity trick. Nonexistence criterion is also established by virtue of the corresponding Pohožaev identity. In particular, we show nonexistence properties for the 3-sublinear case as well as the critical case. Under general assumption on the nonlinearity, existence result is also established for the whole space case that Ω = ℝ3 by using property of the Pohožaev identity and some delicate analysis.

scholarly journals Best Constants for Moser type Inequalities in Zygmund Spaces

2008 ◽  
Vol 36 (6) ◽  
Author(s):  
Daniele Cassani ◽  
Bernhard Ruf ◽  
Cristina Tarsi
Keyword(s):  
Best Constants ◽  
Zygmund Spaces

Simulations of an Agent-Based Model that Consists of a Large Number of Agents Moving Stochastically in a Wide Bounded Domain

2001 ◽  
Author(s):  
Minoru Tabata ◽  
Akira Ide ◽  
Nobuoki Eshima ◽  
Kyushu Takagi ◽  
Yasuhiro Takei ◽  
...  
Keyword(s):  
Bounded Domain ◽  
Agent Based Model ◽  
Agent Based

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

scholarly journals Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave–convex nonlinearities

2020 ◽  
pp. 2050008
Author(s):  
Shaya Shakerian
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Variational Approach ◽  
Nehari Manifold ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Smooth Bounded Domain ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave–convex nonlinearities: [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] containing [Formula: see text] in its interior, and [Formula: see text] with [Formula: see text] which may change sign in [Formula: see text]. We use the variational methods and the Nehari manifold decomposition to prove that this problem has at least two positive solutions for [Formula: see text] sufficiently small. The variational approach requires that [Formula: see text] [Formula: see text] [Formula: see text], and [Formula: see text], the latter being the best fractional Hardy constant on [Formula: see text].

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