scholarly journals Nonlinear perturbations of a periodic magnetic Choquard equation with Hardy–Littlewood–Sobolev critical exponent

2020 ◽  
Vol 71 (4) ◽  
Author(s):  
H. Bueno ◽  
N. da Hora Lisboa ◽  
L. L. Vieira
Keyword(s):  
Critical Exponent ◽  
Nonlinear Perturbations ◽  
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.

scholarly journals Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth

2020 ◽  
Vol 10 (1) ◽  
pp. 732-774
Author(s):  
Zhipeng Yang ◽  
Fukun Zhao
Keyword(s):  
Small Parameter ◽  
Variational Methods ◽  
Critical Exponent ◽  
Laplace Operator ◽  
Sobolev Inequality ◽  
Fractional Laplacian ◽  
Singularly Perturbed ◽  
Choquard Equation ◽  
Fractional Choquard Equation ◽  
Fractional Laplace Operator

Abstract In this paper, we study the singularly perturbed fractional Choquard equation $$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$ where ε > 0 is a small parameter, (−△)s denotes the fractional Laplacian of order s ∈ (0, 1), 0 < μ < 3, $2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator. F is the primitive of f which is a continuous subcritical term. Under a local condition imposed on the potential V, we investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.

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