Multiple solutions for nonhomogeneous Choquard equation involving Hardy–Littlewood–Sobolev critical exponent

2017 ◽  
Vol 68 (3) ◽  
Author(s):  
Zifei Shen ◽  
Fashun Gao ◽  
Minbo Yang
Keyword(s):  
Critical Exponent ◽  
Multiple Solutions ◽  
Choquard Equation ◽  
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.

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