Abstract
AbstractIn this paper, we study the singularly perturbed fractional Choquard equationε2s(−Δ)su+V(x)u=εμ−3(∫R3|u(y)|2μ,s∗+F(u(y))|x−y|μdy)(|u|2μ,s∗−2u+12μ,s∗f(u))inR3,$$\begin{equation*}\varepsilon^{2s}(-{\it\Delta})^su+V(x)u=\varepsilon^{\mu-3}(\int\limits_{\mathbb{R}^3}\frac{|u(y)|^{2^*_{\mu,s}}+F(u(y))}{|x-y|^\mu}dy)(|u|^{2^*_{\mu,s}-2}u+\frac{1}{2^*_{\mu,s}}f(u)) \, \text{in}\, \mathbb{R}^3, \end{equation*}$$whereε> 0 is a small parameter, (−△)sdenotes the fractional Laplacian of orders ∈(0, 1), 0 <μ< 3,2μ,s⋆=6−μ3−2s$2_{\mu ,s}^{\star }=\frac{6-\mu }{3-2s}$is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator.Fis the primitive offwhich is a continuous subcritical term. Under a local condition imposed on the potentialV, 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.
Highlights
Introduction and the main resultsIn the present paper we are interested in the existence, multiplicity and concentration behavior of the semi-classical solutions of the singularly perturbed nonlocal elliptic equation ε s(−∆)s u + V(x)u = εμ−N ( G(u(y)) dy)g(u) in RN, (1.1)|x − y|μ where ε > is a small parameter, < μ < N, V, g = G are real continuous functions on RN and the fractional Laplacian (−∆)s is de ned by (−∆)s Ψ(x) = CN,s P.V .Ψ(x) − Ψ(y) dy, Ψ ∈ S(RN), |x − y|N+ s RNP.V . stands for the Cauchy principal value, CN,s is a normalized constant, S(RN) is the Schwartz space of rapidly decaying functions, s ∈ (, )
In this paper, we study the singularly perturbed fractional Choquard equation ε s(−∆)s u + V(x)u = εμ− (
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
Summary
For the critical Choquard equations in the sense of Hardy-LittlewoodSobolev, Cassani and Zhang [12] developed a robust method to get the existence of ground states and qualitative properties of solutions, where they do not require the nonlinearity to enjoy monotonicity nor AmbrosettiRabinowitz-type conditions. 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 ε s(−∆)s u + V(x)u = εμ− 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]: (V )V ∈ C(R , R) and < inf V(x). We show the critical point of the modi ed functional which satis es the original problem, and investigate its concentration behavior, which completes the proof Theorem 1.1
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