Nonlinear Choquard Equation Research Articles

Asymptotic profiles for Choquard equations with combined attractive nonlinearities

We study asymptotic behaviour of positive ground state solutions of the nonlinear Choquard equation(Pε)−Δu+εu=(Iα⁎|u|p)|u|p−2u+|u|q−2u,inRN,where N≥3 is an integer, p∈[N+αN,N+αN−2], q∈(2,2NN−2), Iα is the Riesz potential of order α∈(0,N) and ε>0 is a parameter. We show that as ε→0 (resp. ε→∞), the ground state solutions of (Pε), after appropriate rescalings dependent on parameter regimes, converge in H1(RN) to particular solutions of five different limit equations. We also establish a sharp asymptotic characterisation of such rescalings, and the precise asymptotic behaviour of uε(0), ‖∇uε‖22, ‖uε‖22, ∫RN(Iα⁎|uε|p)|uε|p and ‖uε‖qq, which depend in a non-trivial way on the exponents p, q and the space dimension N. Further, we discuss a connection of our results with a mass constrained problem, associated to (Pε) with normalization constraint ∫RN|u|2=c2. As a consequence of the main results, we obtain the existence, multiplicity and precise asymptotic behaviour of positive normalized solutions of such a problem as c→0 and c→∞.

Sign-changing solutions for critical Choquard equation on bounded domain

We consider the existence of sign-changing solutions for the following nonlinear Choquard equation{−Δu=(Iα⁎|u|p)|u|p−2u+λu,x∈Ω,u∈H01(Ω), where N≥3,α∈(0,N),λ∈R, p∈(1,N+αN−2] and Ω⊂RN is a smooth bounded domain. Iα:RN→R is the Riesz potential. In critical case p=N+αN−2, if Ω is symmetric about the axis x1, we firstly develop the limit profiles for the symmetric Palais-Smale sequence by the concentration compactness principle. Then we conclude that the problem admits an odd solution with exactly two nodal domains for λ∈(0,λ1),N≥4andα∈(2,N). In contrast, the local Brezis-Nirenberg type problem −Δu=|u|4N−2u+λu,u∈H01(Ω) can not permit such type of odd symmetry solution.

Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities

Abstract In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation ( − Δ ) s u + μ u = ( I α * F ( u ) ) F ′ ( u ) in R N , ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u ∈ H s ( R N ) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫ R N u 2 = m > 0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.

Existence and orbital stability results for the nonlinear Choquard equation with rotation

Existence and orbital stability results for the nonlinear Choquard equation with rotation

Positive solutions to the planar logarithmic Choquard equation with exponential nonlinearity

In this paper we study the following nonlinear Choquard equation −Δu+u=ln1|x|∗F(u)f(u),inR2,where f∈C1(R,R) and F is the primitive of the nonlinearity f vanishing at zero. We use an asymptotic approximation approach to establish the existence of positive solutions to the above problem in the standard Sobolev space H1(R2). We give a new proof and at the same time extend part of the results established in (Cassani-Tarsi, Calc.Var.PDE, 2021) [11].

Ground state solutions of a magnetic nonlinear Choquard equation with lower critical exponent

In this article, we establish the existence of ground state solutions for a magnetic nonlinear Choquard equation with lower critical exponent by variational methods.

Existence of Ground State Solutions for Choquard Equation with the Upper Critical Exponent

In this article, we investigate the existence of a nontrivial solution for the nonlinear Choquard equation with upper critical exponent see Equation (6). The Riesz potential in this case has never been studied. We establish the existence of the ground state solution within bounded domains Ω⊂RN. Variational methods are used for this purpose. This method proved to be instrumental in our research, enabling us to address the problem effectively. The study of the existence of ground state solutions for the Choquard equation with a critical exponent has applications and relevance in various fields, primarily in theoretical physics and mathematical analysis.

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.

Semi-classical Analysis Around Local Maxima and Saddle Points for Degenerate Nonlinear Choquard Equations

We study existence of semi-classical states for the nonlinear Choquard equation: -ε2Δv+V(x)v=1εα(Iα∗F(v))f(v)inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\varepsilon ^2\\Delta v+ V(x)v = {1\\over \\varepsilon ^\\alpha }(I_\\alpha *F(v))f(v) \\quad \ ext {in}\\ {\\mathbb {R}}^N, \\end{aligned}$$\\end{document}where Nge 3, alpha in (0,N), I_alpha (x)=A_alpha /|{x}|^{N-alpha } is the Riesz potential, Fin C^1({mathbb {R}},{mathbb {R}}), F'(s)=f(s) and varepsilon >0 is a small parameter. We develop a new variational approach, in which our deformation flow is generated through a flow in an augmented space to get a suitable compactness property and to reflect the properties of the potential. Furthermore our flow keeps the size of the tails of the function small and it enables us to find a critical point without introducing a penalization term. We show the existence of a family of solutions concentrating to a local maximum or a saddle point of V(x)in C^N({mathbb {R}}^N,{mathbb {R}}) under general conditions on F(s). Our results extend the results by Moroz and Van Schaftingen (Calc Var Partial Differ Equ 52:199–235, 2015) for local minima (see also Cingolani and Tanaka (Rev Mat Iberoam 35(6):1885–1924, 2019)) and Wei and Winter (J Math Phys 50:012905, 2009) for non-degenerate critical points of the potential.

Semiclassical states to the nonlinear Choquard equation with critical growth

Semiclassical states to the nonlinear Choquard equation with critical growth

Normalized saddle solutions for a mass supercritical Choquard equation

We investigate the existence of saddle type normalized solutions for the nonlinear Choquard equation:{−Δu−λu=(Iα⁎F(u))F′(u), in RN∫RN|u|2dx=a2,u∈H1(RN). Here N≥1, a>0 is given in advance, Iα is the Riesz potential of order α∈(0,N) and the unknown parameter λ appears as a Lagrange multiplier. In a mass supercritical setting on F, we prove the existence of a couple (uaG,λaG)∈H1(RN)×R− of saddle solutions for any a>0 and for given finite Coxeter group G with its rank k≤N. Our method is to combine the concentration compactness principle with a minimax procedure in the saddle type symmetric subspace, which gives a variational framework of constructing normalized saddle solutions for the Choquard equation.

Liouville‐type theorems for a nonlinear fractional Choquard equation

AbstractIn this paper, we are concerned with the fractional Choquard equation on the whole space with , and . We first prove that the equation does not possess any positive solution for . When , we establish a Liouville type theorem saying that if then the equation has no positive stable solution. This extends, in particular, a result in [27] to the fractional Choquard equation.

Normalized solutions to the nonlinear Choquard equations with Hardy-Littlewood-Sobolev upper critical exponent

We study the effect of lower order nonlocal perturbations in the existence of positive solutions to the following nonlinear Choquard equation−Δu=λu+μ(Iα⁎|u|p)|u|p−2u+(Iα⁎|u|2α⁎)|u|2α⁎−2ux∈RN, having prescribed L2−norm∫RN|u|2dx=a>0, where N≥3, α∈(0,N), N+αN<p<2α⁎, μ>0 and λ∈R. Under different assumptions on p, we establish several existence results and characterize the behavior of solutions as μ→0.

The existence of positive solutions to the Choquard equation with critical exponent and logarithmic term

We investigate the existence of solutions for the following nonlinear critical Choquard equation with the logarithmic term:{−Δu=(∫Ω|u(y)|2μ⁎|x−y|μdy)|u|2μ⁎−2u+βu+λulog⁡u2,x∈Ω,u=0,x∈∂Ω, where Ω is a bounded domain of RN with smooth boundary. λ, β are real parameters and 2μ⁎=2N−μN−2(N≥3) is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We will show that there exists at last one positive solution to the above problem under some appropriate assumptions of λ and β.

Ground state solution for a class of Choquard equation with indefinite periodic potential

This paper is concerned with a class of nonlinear Choquard equation with potential. We assume the potential satisfies general indefinite periodic condition, so the Schrödinger operator has purely continuous spectrum and the associate energy functional is strongly indefinite. Applying the generalized Nehari manifold method developed by Szulkin and Weth, we prove the existence of ground state solution.

Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities

We prove existence of infinitely many solutions u in H^1_r({mathbb {R}}^N) for the nonlinear Choquard equation -Δu+μu=(Iα∗F(u))f(u)inRN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} - {\\varDelta } u + \\mu u =(I_\\alpha *F(u)) f(u) \\quad \\hbox {in}\\ {\\mathbb {R}}^N, \\end{aligned}$$\\end{document}where Nge 3, alpha in (0,N), I_alpha (x) := frac{{varGamma }(frac{N-alpha }{2})}{{varGamma }(frac{alpha }{2}) pi ^{N/2} 2^alpha } frac{1}{|x|^{N- alpha }}, x in {mathbb {R}}^N setminus {0} is the Riesz potential, and F is an almost optimal subcritical nonlinearity, assumed odd or even. We analyze the two cases: mu is a fixed positive constant or mu is unknown and the L^2-norm of the solution is prescribed, i.e. int _{{mathbb {R}}^N} |u|^2 =m>0. Since the presence of the nonlocality prevents to apply the classical approach, introduced by Berestycki and Lions (Arch Ration Mech Anal 82(4):347–375, 1983), we implement a new construction of multidimensional odd paths, where some estimates for the Riesz potential play an essential role, and we find a nonlocal counterpart of their multiplicity results. In particular we extend the existence results due to Moroz and Van Schaftingen (Trans Am Math Soc 367(9):6557–6579, 2015).

Ground State Solutions of a Magnetic Nonlinear Choquard Equation with Lower Critical Exponent

Ground State Solutions of a Magnetic Nonlinear Choquard Equation with Lower Critical Exponent

Ground state solutions and infinitely many solutions for a nonlinear Choquard equation

In this paper we study the existence and multiplicity of solutions for the following nonlinear Choquard equation: \t\t\t−Δu+V(x)u=[|x|−μ∗|u|p]|u|p−2u,x∈RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} -\\Delta u+V(x)u=\\bigl[ \\vert x \\vert ^{-\\mu }\\ast \\vert u \\vert ^{p}\\bigr] \\vert u \\vert ^{p-2}u,\\quad x \\in \\mathbb{R}^{N}, \\end{aligned}$$ \\end{document} where Ngeq 3, 0<mu <N, frac{2N-mu }{N}leq p<frac{2N-mu }{N-2}, ∗ represents the convolution between two functions. We assume that the potential function V(x) satisfies general periodic condition. Moreover, by using variational tools from the Nehari manifold method developed by Szulkin and Weth, we obtain the existence results of ground state solutions and infinitely many pairs of geometrically distinct solutions for the above problem.

Ground state solutions for nonlinear Choquard equation with singular potential and critical exponents

In this paper, we study the ground state solutions to the following Choquard equation involving singular potential:−Δu+V(|x|)u=(Iα⁎F(u))f(u),x∈RN, where N⩾3, α∈(0,N), Iα is the Riesz potential, V is a singular potential with parameter θ∈(max⁡{0,4−N},2)∪(2,2N−2)∪(2N−2,∞), F is the primitive of f and f satisfies critical growth in sense of the Hardy-Littlewood-Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki-Lions type conditions, we divide this paper into three parts. By virtue of two different kinds of Lions-type theorem and Nehari manifold, some existence results are established.

Multi-bump solutions for the nonlinear magnetic Choquard equation with deepening potential well

In this paper, using variational methods, we study multiplicity of multi-bump solutions for the following nonlinear magnetic Choquard equation{−(∇+iA(x))2u+(λV(x)+1)u=(1|x|μ⁎|u|p)|u|p−2ux∈RN,u∈H1(RN,C), where N≥2, λ>0 is a real parameter, 0<μ<2, i is the imaginary unit, p∈(2,2⁎(2(N−μ)2N)), where 2⁎=2NN−2 if N≥3, 2⁎=+∞, if N=2. The magnetic potential A∈Lloc2(RN,RN) and V:RN→R is a nonnegative continuous function. We show that if the zero set of V has several isolated connected components Ω1,⋯,Ωk such that the interior of Ωj is non-empty and ∂Ωj is smooth, then for λ>0 large enough, the above equation has at least 2k−1 multi-bump solutions.