Fractional Choquard Research Articles

Sign-changing solutions for a class of fractional Choquard equation with the Sobolev critical exponent in [formula omitted]

In this paper, we study the following fractional Choquard equation with critical exponent(−Δ)σu+u=(Iα⁎|u|p)|u|p−2u+|u|2σ⁎−2uinR3, where σ∈(34,1), α∈(2σ,3) and α+2σ3−2σ<p<2α,σ⁎:=3+α3−2σ, 2α,σ⁎ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, 2σ⁎:=63−2σ is the fractional Sobolev critical exponent and the operator (−Δ)σ stands for the fractional Laplacian of order σ. Based on the above assumptions, we establish the existence of positive ground state solutions, and if p∈(α+2σ3−2σ,2α3−2σ), we also have the corresponding regular property. Subsequently, by introducing some other additional hypotheses on σ, α, p and with the help of quantitative deformation lemma, we employ constrained minimization arguments on the sign-changing Nehari manifold to obtain the existence of ground state sign-changing solutions.

Multiplicity and Concentration Properties for Fractional Choquard Equations with Exponential Growth

Multiplicity and Concentration Properties for Fractional Choquard Equations with Exponential Growth

Semi-classical states for fractional Choquard equations with decaying potentials

Semi-classical states for fractional Choquard equations with decaying potentials

Regularity for critical fractional Choquard equation with singular potential and its applications

Abstract We study the following fractional Choquard equation ( − Δ ) s u + u ∣ x ∣ θ = ( I α * F ( u ) ) f ( u ) , x ∈ R N , {\left(-\Delta )}^{s}u+\frac{u}{{| x| }^{\theta }}=({I}_{\alpha }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ⩾ 3 N\geqslant 3 , s ∈ 1 2 , 1 s\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) , α ∈ ( 0 , N ) \alpha \in \left(0,N) , θ ∈ ( 0 , 2 s ) \theta \in \left(0,2s) , and I α {I}_{\alpha } is the Riesz potential. The main purpose of this article is twofold. We first study the regularity of weak solutions for the aforementioned equation with critical nonlinearity, which extends the results of θ = 0 \theta =0 in Moroz-Van Schaftingen [Existence of groundstates for a class of nonlinear Choquardequations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579]. Then, as an application of the regularity results, we establish the existence of ground state solutions for above equation with the nonlinearity involving embedding top and bottom indices, which is related to the Hardy-Littlewood-Sobolev inequality and singular term 1 ∣ x ∣ θ \frac{1}{{| x| }^{\theta }} . It is worth noting that our approach is not involving the concentration-compactness principle.

Normalized Ground States for a Fractional Choquard System in $$\mathbb {R}$$

Normalized Ground States for a Fractional Choquard System in $$\mathbb {R}$$

Asymptotic decay of solutions for sublinear fractional Choquard equations

Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation (−Δ)su+μu=(Iα∗F(u))f(u)onRNwhere s∈(0,1), N≥2, α∈(0,N), μ>0, Iα denotes the Riesz potential and F(t)=∫0tf(τ)dτ is a general nonlinearity with a sublinear growth in the origin. The found decay is of polynomial type, with a rate possibly slower than ∼1|x|N+2s. The result is new even for homogeneous functions f(u)=|u|r−2u, r∈[N+αN,2), and it complements the decays obtained in the linear and superlinear cases in Cingolani et al. (2022); D’Avenia et al. (2015). Differently from the local case s=1 in Moroz and Van Schaftingen (2013), new phenomena arise connected to a new “s-sublinear” threshold that we detect on the growth of f. To gain the result we in particular prove a Chain Rule type inequality in the fractional setting, suitable for concave powers.

The higher regularity and decay estimates for positive solutions of fractional Choquard equations

AbstractIn the paper, we study the higher regularity and decay estimates for positive solutions of the following fractional Choquard equations: where , is a constant, , , , . Let be a positive solution of (). We obtain an optimal decay estimate for and reveal the relation between and the decay rate of for all permitting . We skillfully use the properties of Bessel kernel to deduce our main result by iteration. Moreover, by the decay estimates, the stronger regular estimates for are obtained.

Ground states for fractional Choquard equations with doubly critical exponents and magnetic fields

In this paper, we investigate the ground states for the fractional Choquard equations with doubly critical exponents and magnetic fields. We prove that the equation has a ground state solution by using the Nehari method and the Pohožaev identity. Bibliography: 12 titles.

Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent

Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent

Normalized Solutions for the Fractional Choquard Equations with Lower Critical Exponent and Nonlocal Perturbation

Normalized Solutions for the Fractional Choquard Equations with Lower Critical Exponent and Nonlocal Perturbation

Ground states for fractional Choquard equations with doubly critical exponents and magnetic fields

In this paper, we investigate the ground states for the fractional Choquard equations with doubly critical exponents and magnetic fields. We prove that the equation has a ground state solution by using the Nehari method and the Pokhozhaev identity.

Infinitely many solutions for p-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method

We prove the existence of infinitely many solutions to a fractional Choquard type equation(−Δ)psu+V(x)|u|p−2u=(K⁎g(u))g′(u)+εWW(x)f′(u)in RN involving the fractional p-Laplacian and a general convolution term with critical growth. In order to obtain infinitely many solutions, we use a type of the symmetric mountain pass lemma which gives sequences of critical values converging to zero for even functionals. To assure the (PS)c conditions, we also use a nonlocal version of the concentration compactness lemma.

Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation

Abstract In this article, we study the fractional critical Choquard equation with a nonlocal perturbation: ( − Δ ) s u = λ u + α ( I μ * ∣ u ∣ q ) ∣ u ∣ q − 2 u + ( I μ * ∣ u ∣ 2 μ , s * ) ∣ u ∣ 2 μ , s * − 2 u , in R N , {\left(-{\Delta })}^{s}u=\lambda u+\alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u+\left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{{2}_{\mu ,s}^{* }}){| u| }^{{2}_{\mu ,s}^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, having prescribed mass ∫ R N u 2 d x = c 2 , \mathop{\int }\limits_{{{\mathbb{R}}}^{N}}{u}^{2}{\rm{d}}x={c}^{2}, where s ∈ ( 0 , 1 ) , N &gt; 2 s , 0 &lt; μ &lt; N , α &gt; 0 , c &gt; 0 s\in \left(0,1),N\gt 2s,0\lt \mu \lt N,\alpha \gt 0,c\gt 0 , and I μ ( x ) {I}_{\mu }\left(x) is the Riesz potential given by I μ ( x ) = A μ ∣ x ∣ μ with A μ = Γ μ 2 2 N − μ π N ⁄ 2 Γ N − μ 2 , {I}_{\mu }\left(x)=\frac{{A}_{\mu }}{{| x| }^{\mu }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\mu }=\frac{\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{\mu }{2}\right)}{{2}^{N-\mu }{\pi }^{N/2}\Gamma \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{N-\mu }{2}\right)}, and 2 N − μ N &lt; q &lt; 2 μ , s * = 2 N − μ N − 2 s \frac{2N-\mu }{N}\lt q\lt {2}_{\mu ,s}^{* }=\frac{2N-\mu }{N-2s} is the fractional Hardy-Littlewood-Sobolev critical exponent. Under the L 2 {L}^{2} -subcritical perturbation α ( I μ * ∣ u ∣ q ) ∣ u ∣ q − 2 u \alpha \left({I}_{{\mu }^{* }}\hspace{-0.25em}{| u| }^{q}){| u| }^{q-2}u with exponent 2 N − μ N &lt; q &lt; 2 N − μ + 2 s N \frac{2N-\mu }{N}\lt q\lt \frac{2N-\mu +2s}{N} , we obtain the existence of normalized ground states and mountain-pass-type solutions. Meanwhile, for the L 2 {L}^{2} -critical and L 2 {L}^{2} -supercritical cases 2 N − μ + 2 s N ≤ q &lt; 2 N − μ N − 2 s \frac{2N-\mu +2s}{N}\le q\lt \frac{2N-\mu }{N-2s} , we also prove that the equation has ground states of mountain-pass-type.

On stable solutions of a weighted elliptic equation involving the fractional Laplacian

In this paper, we study the following fractional Choquard equation with weight where and is a positive weight function satisfying at infinity, for some . We establish, in this paper, a Liouville type theorem saying that if then the above equation has no nontrivial stable solution. Our result, in particular, extends the result in [Le, Phuong. Bull. Aust. Math. Soc. 102 (2020), no. 3, 471–478.] from the Laplace operator to the fractional Laplacian.

Normalized Solutions for the Fractional Choquard Equations with Hardy–Littlewood–Sobolev Upper Critical Exponent

Normalized Solutions for the Fractional Choquard Equations with Hardy–Littlewood–Sobolev Upper Critical Exponent

On nonlinear fractional Choquard equation with indefinite potential and general nonlinearity

In this paper, we consider a class of fractional Choquard equations with indefinite potential (−Δ)αu+V(x)u=[∫RNM(ϵy)G(u)|x−y|μdy]M(ϵx)g(u),x∈RN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ (-\\Delta )^{\\alpha}u+V(x)u= \\biggl[ \\int _{{\\mathbb{R}}^{N}} \\frac{M(\\epsilon y)G(u)}{ \\vert x-y \\vert ^{\\mu}}\\,\\mathrm{d}y \\biggr]M( \\epsilon x)g(u), \\quad x\\in {\\mathbb{R}}^{N}, $$\\end{document} where alpha in (0,1), N> 2alpha , 0<mu <2alpha , ϵ is a positive parameter. Here (-Delta )^{alpha} stands for the fractional Laplacian, V is a linear potential with periodicity condition, and M is a nonlinear reaction potential with a global condition. We establish the existence and concentration of ground state solutions under general nonlinearity by using variational methods.

Existence of solution for the (p, q)-fractional Laplacian equation with nonlocal Choquard reaction and exponential growth

In this paper, we study the existence of weak solution to ( p , q ) -fractional Choquard equation in R N as follows L p s u + L q s u + V ( x ) ( | u | p − 2 u + | u | q − 2 u ) = ( 1 | x | μ ∗ F ( u ) ) f ( u ) , where 2 ≤ N s = p < q , 0 ≤ μ < N , f has exponential growth. The function f and V are continuous which satisfy some suitable assumptions. In our best knowledge, sofar there is not any work about existence of weak solution to ( p , q ) -fractional Choquard equation with exponential growth. Our results are even new in the case ( N , q ) -Laplace equation which are complement the results of Fiscella and Pucci [ ( p , N ) equations with critical exponential nonlinearities in R N . J Math Anal Appl. 2021;501(1), Article 123379].

Ground State Solutions of Fractional Choquard Problems with Critical Growth

In this article, we investigate a class of fractional Choquard equation with critical Sobolev exponent. By exploiting a monotonicity technique and global compactness lemma, the existence of ground state solutions for this equation is obtained. In addition, we demonstrate the existence of ground state solutions for the corresponding limit problem.

Normalized solutions of fractional Choquard equation with critical nonlinearity

In this paper, we are concerned with normalized solutions of the fractional critical Choquard equation with a local perturbation and prescribed mass. For the $L^2$-subcritical case, we study the multiplicity of normalized solutions by applying the truncation technique, the concentration-compactness principle and the genus theory. For the $L^2$-supercritical case, we obtain a couple of normalized solutions by developing a fiber map and using the concentration-compactness principle.

Existence of Positive Ground State Solutions for Fractional Choquard Systems in Subcritical and Critical Cases

We investigate a class of fractional linearly coupled Choquard systems. For the subcritical case and all critical cases, we prove the existence, nonexistence and symmetry of positive ground state solutions of systems, by using the Nehari manifold method, the Pohožaev identity and the Schwartz symmetrization rearrangements. In particular, we overcome the lack of compactness of the critical nonlinearities by using the behaviour of sufficiently small Nehari energy levels.