Multiplicity and concentration of nontrivial nonnegative solutions for a fractional Choquard equation with critical exponent

Chen, Shaoxiong; Li, Yue; Yang, Zhipeng

doi:10.1007/s13398-019-00768-4

Cited by 10 publications

(3 citation statements)

References 45 publications

(43 reference statements)

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“…It seems that the only works concerning the concentration behavior of solutions are due to [13,51]. Assuming the global condition on V:…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…which was rstly introduced by Rabinowitz [39] in the study of the nonlinear Schrödinger equations. By using the method of Nehari manifold developed by Szulkin and Weth [46], authors in [13,51] obtained the multiplicity and concentration of positive solutions for the following fractional Choquard equation…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…where ε > , < µ < , F is the primitive function of f . Di erent to [13,51], in this paper, we are devote to establishing the existence and concentration of positive solutions for the fractional Choquard equation (1.8) when the potential function satis es the following local conditions [18]: To go on studying the problem (1.8), the following Hardy-Littlewood-Sobolev inequality [28] is the starting point. Lemma 1.1.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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

Yang

Zhao

2020

Advances in Nonlinear Analysis

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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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“…It seems that the only works concerning the concentration behavior of solutions are due to [13,51]. Assuming the global condition on V:…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%