scholarly journals Symmetry and nonexistence results for a fractional Choquard equation with weights

2021 ◽  
Vol 41 (2) ◽  
pp. 489-505
Author(s):  
Anh Tuan Duong ◽  
◽  
Phuong Le ◽  
Nhu Thang Nguyen ◽  
◽  
...  
Keyword(s):  
Choquard Equation ◽  
Fractional Choquard Equation

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.

Nonlocal perturbations of the fractional Choquard equation

2017 ◽  
Vol 8 (1) ◽  
pp. 694-706 ◽  
Author(s):  
Gurpreet Singh
Keyword(s):  
Nehari Manifold ◽  
Choquard Equation ◽  
Minimization Method ◽  
Nodal Set ◽  
Alpha Beta ◽  
Sign Changing Solutions ◽  
Least Energy ◽  
Fractional Choquard Equation

Abstract We study the equation (-\Delta)^{s}u+V(x)u=(I_{\alpha}*\lvert u\rvert^{p})\lvert u\rvert^{p-2}u+% \lambda(I_{\beta}*\lvert u\rvert^{q})\lvert u\rvert^{q-2}u\quad\text{in }{% \mathbb{R}}^{N}, where {I_{\gamma}(x)=\lvert x\rvert^{-\gamma}} for any {\gamma\in(0,N)} , {p,q>0} , {\alpha,\beta\in(0,N)} , {N\geq 3} , and {\lambda\in{\mathbb{R}}} . First, the existence of groundstate solutions by using a minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.

scholarly journals Ground State Solutions for Fractional Choquard Equations with Potential Vanishing at Infinity

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 151
Author(s):  
Huxiao Luo ◽  
Shengjun Li ◽  
Chunji Li
Keyword(s):  
Perturbation Method ◽  
Potential Function ◽  
Ground State ◽  
Fractional Laplacian ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Choquard Equation ◽  
Nonlinear Choquard Equation ◽  
Mass Case ◽  
Fractional Choquard Equation

In this paper, we study a class of nonlinear Choquard equation driven by the fractional Laplacian. When the potential function vanishes at infinity, we obtain the existence of a ground state solution for the fractional Choquard equation by using a non-Nehari manifold method. Moreover, in the zero mass case, we obtain a nontrivial solution by using a perturbation method. The results improve upon those in Alves, Figueiredo, and Yang (2015) and Shen, Gao, and Yang (2016).

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