Existence and uniqueness of solutions for Choquard equation involving Hardy–Littlewood–Sobolev critical exponent

2019 ◽  
Vol 58 (4) ◽  
Author(s):  
Lun Guo ◽  
Tingxi Hu ◽  
Shuangjie Peng ◽  
Wei Shuai
Keyword(s):  
Critical Exponent ◽  
Existence And Uniqueness ◽  
Choquard Equation ◽  
Uniqueness Of Solutions ◽  
Sobolev Critical Exponent

scholarly journals Critical growth elliptic problems involving Hardy-Littlewood-Sobolev critical exponent in non-contractible domains

2019 ◽  
Vol 9 (1) ◽  
pp. 803-835 ◽  
Author(s):  
Divya Goel ◽  
Konijeti Sreenadh
Keyword(s):  
Critical Exponent ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Problems ◽  
Annular Domain ◽  
Choquard Equation ◽  
Existence And Multiplicity ◽  
Lusternik Schnirelmann ◽  
Multiplicity Of Positive Solutions ◽  
Sobolev Critical Exponent

Abstract The paper is concerned with the existence and multiplicity of positive solutions of the nonhomogeneous Choquard equation over an annular type bounded domain. Precisely, we consider the following equation $$\begin{array}{} \displaystyle -{\it \Delta} u = \left(\int\limits_{{\it\Omega}}\frac{|u(y)|^{2^*_{\mu}}}{|x-y|^{\mu}}dy\right)|u|^{2^*_{\mu}-2}u+f \; \text{in}\; {\it\Omega},\quad u = 0 \; \text{ on } \partial {\it\Omega} , \end{array}$$ where Ω is a smooth bounded annular domain in ℝN(N ≥ 3), $\begin{array}{} 2^*_{\mu}=\frac{2N-\mu}{N-2} \end{array}$, f ∈ L∞(Ω) and f ≥ 0. We prove the existence of four positive solutions of the above problem using the Lusternik-Schnirelmann theory and varitaional methods, when the inner hole of the annulus is sufficiently small.

Integral Representations for the Solution of Dynamic Bending of a Plate with Displacement-Traction Boundary Data

2003 ◽  
Vol 10 (3) ◽  
pp. 467-480
Author(s):  
Igor Chudinovich ◽  
Christian Constanda
Keyword(s):  
Existence And Uniqueness ◽  
Boundary Data ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Boundary Integral ◽  
Integral Representations ◽  
Transverse Shear Deformation ◽  
Uniqueness Of Solutions ◽  
Nonstationary Boundary

Abstract The existence of distributional solutions is investigated for the time-dependent bending of a plate with transverse shear deformation under mixed boundary conditions. The problem is then reduced to nonstationary boundary integral equations and the existence and uniqueness of solutions to the latter are studied in appropriate Sobolev spaces.

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