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

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.

Multiple solutions for a fractional Choquard problem with slightly subcritical exponents on bounded domains

“…We also want to mention some recent work about multiplicity for Choquard equations dealing with concentration properties. Yang and Zhao [28] applied penalization techniques and Ljusternik-Schnirelmann theory to investigate the relation between the number of positive solutions and the topology of the set where the potential attains its minimum values for a singularly perturbed fractional Choquard equation, and showed that these positive solutions locate at the minimum point of the potential. In addition, Cingolani and Tanaka [8] studied the multiplicity and concentration of positive single-peak solutions for a nonlinear Choquard equation, and they used relative cup-length to estimate the topological changes and then related the number of positive solutions to the topology of the critical set of the potential.…”

Existence and multiplicity of normalized solutions for a class of fractional Schrödinger–Poisson equations

“…The first equation in (1.2) was introduced by Laskin (see [21,22]) and comes from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths. This class of fractional Schrödinger equations with a repulsive nonlocal Coulombic potential is obtained by approximation of the Hartree-Fock equation describing a quantum mechanical system of many particles; see, for instance, [14,23,24,38]. It also appeared in several areas such as optimization, finance, phase transitions, stratified materials, crystal dislocation, flame propagation, conservation laws, materials science and water waves (see [9]).…”

On fractional Schrödinger equations with Hartree type nonlinearities

“…Mathematically, doubly nonlocal equations have been treated in [25,26] in the case of pure power nonlinearities (see also [15] for some orbital stability results and [14] for a Strichartz estimates approach), obtaining existence and qualitative properties of the solutions. Other results can be found in [72,5,61] for superlinear nonlinearities, in [40] for L 2 -supercritical Cauchy problems, in [38] for bounded domains and in [77] for concentration phenomena with strictly noncritical and monotone sources.…”