Extremal functions for optimal Sobolev inequalities on compact manifolds

2001 ◽  
Vol 12 (1) ◽  
pp. 59-84 ◽  
Author(s):  
Zindine Djadli ◽  
Olivier Druet
Keyword(s):  
Compact Manifolds ◽  
Sobolev Inequalities ◽  
Extremal Functions ◽  
Optimal Sobolev Inequalities

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.

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