scholarly journals The existence of extremal functions for discrete Sobolev inequalities on lattice graphs

2021 ◽  
Vol 305 ◽  
pp. 224-241
Author(s):  
Bobo Hua ◽  
Ruowei Li
Keyword(s):  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Lattice Graphs

Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

2018 ◽  
Vol 149 (04) ◽  
pp. 979-994 ◽  
Author(s):  
Daomin Cao ◽  
Wei Dai
Keyword(s):  
Critical Case ◽  
Classical Solutions ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Method Of Moving Planes ◽  
Nonnegative Solutions ◽  
Moving Planes ◽  
Image Position ◽  
Harmonic Equation

AbstractIn this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 < γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 < γ < 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

Weighted Hardy–Littlewood–Sobolev inequalities on the upper half space

2016 ◽  
Vol 18 (05) ◽  
pp. 1550067 ◽  
Author(s):  
Jingbo Dou
Keyword(s):  
Half Space ◽  
Type Inequality ◽  
Boundary Term ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Biharmonic Operator ◽  
Trace Inequality ◽  
Hardy Type Inequality ◽  
Hardy Type

In this paper, we establish a weighted Hardy–Littlewood–Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. As an application, we can show a weighted Sobolev–Hardy trace inequality with [Formula: see text]-biharmonic operator.

scholarly journals Sharp Sobolev Inequalities via Projection Averages

2020 ◽  
Author(s):  
Philipp Kniefacz ◽  
Franz E. Schuster
Keyword(s):  
Bounded Variation ◽  
Sobolev Inequality ◽  
Complete Classification ◽  
Functions Of Bounded Variation ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Affine Invariant ◽  
Entire Family ◽  
New Inequalities

Abstract A family of sharp $$L^p$$ L p Sobolev inequalities is established by averaging the length of i-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $$L^p$$ L p  Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them—the affine $$L^p$$ L p  Sobolev inequality of Lutwak, Yang, and Zhang. When $$p = 1$$ p = 1 , the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.

Homogeneous sharp Sobolev inequalities on product manifolds

2006 ◽  
Vol 136 (2) ◽  
pp. 277-300 ◽  
Author(s):  
Jurandir Ceccon ◽  
Marcos Montenegro
Keyword(s):  
Riemannian Manifolds ◽  
Sobolev Inequality ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Mass Transportation ◽  
Best Constant ◽  
First Order ◽  
Image Position ◽  
Optimal Sobolev Inequality ◽  
Homogeneous Convex

Let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. For p-homogeneous convex functions f(s, t) on [0,∞) × [0, ∞), we study the validity and non-validity of the first-order optimal Sobolev inequality on H1, p(M × N) where and Kf = Kf (m, n, p) is the best constant of the homogeneous Sobolev inequality on D1, p (Rm+n), The proof of the non-validity relies on the knowledge of extremal functions associated with the Sobolev inequality above. In order to obtain such extremals we use mass transportation and convex analysis results. Since variational arguments do not work for general functions f, we investigate the validity in a uniform sense on f and argue with suitable approximations of f which are also essential in the non-validity. Homogeneous Sobolev inequalities on product manifolds are connected to elliptic problems involving a general class of operators.

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