Existence and concentration result for fractional Choquard equations in $$\pmb {{\mathbb {R}}^{N}}$$ R N

Guofeng Che (), Yu Su () and Haibo Chen ()
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Guofeng Che: Guangdong University of Technology
Yu Su: Anhui University of Science and Technology
Haibo Chen: Central South University

Indian Journal of Pure and Applied Mathematics, 2023, vol. 54, issue 1, 182-199

Abstract: Abstract In this paper, we consider the following fractional Choquard equations with competing potentials: $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \varepsilon ^{2\alpha }(-\triangle )^{\alpha }u+V(x)u=\varepsilon ^{\mu -N}f(x,u) , \ \ \text{ in } {\mathbb {R}}^{N}\\ u\in H^{\alpha }({\mathbb {R}}^{N}), \end{array} \right. \end{aligned}$$ ε 2 α ( - ▵ ) α u + V ( x ) u = ε μ - N f ( x , u ) , in R N u ∈ H α ( R N ) , where $$f(x,u)=\left( \int _{{\mathbb {R}}^{N}}\frac{K(y)|u(y)|^{p}}{|x-y|^{\mu }}\mathrm {d}y\right) K(x)|u|^{p-2}u+ \left( \int _{{\mathbb {R}}^{N}}\frac{Q(y)|u(y)|^{q}}{|x-y|^{\mu }}\mathrm {d}y\right) Q(x)|u|^{q-2} u,$$ f ( x , u ) = ∫ R N K ( y ) | u ( y ) | p | x - y | μ d y K ( x ) | u | p - 2 u + ∫ R N Q ( y ) | u ( y ) | q | x - y | μ d y Q ( x ) | u | q - 2 u , $$\varepsilon >0$$ ε > 0 is a parameter, $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) , $$0 2\alpha $$ 0 2 α , $$\frac{2N-\mu }{N}

Keywords: Fractional Choquard equations; Nehari manifold; Ground state; Concentration; Variational methods; 35J60; 35J99 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s13226-022-00242-9

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