Nonlinear Choquard Equations Research Articles

Existence of Solutions for Nonlinear Choquard Equations with (p, q)-Laplacian on Finite Weighted Lattice Graphs

In this paper, we consider the (p,q)-Laplacian Choquard equation on a finite weighted lattice graph G=(KN,E,μ,ω), namely for any 1<p<q<N, r>1 and 0<α<N, −Δpu−Δqu+V(x)(|u|p−2u+|u|q−2u)=∑y∈KN,y≠x|u(y)|rd(x,y)N−α|u|r−2u, where Δν is the discrete ν-Laplacian on graphs, and ν∈{p.q}, V(x) is a positive function. Under some suitable conditions on r, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of some known studies.

Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations

Abstract This article is concerned with the existence of multiple normalized solutions for a class of Choquard equations with a parametric perturbation − ε 2 Δ u + V ( x ) u = λ u + ε − α ( I α * F ( u ) ) f ( u ) , x ∈ R N , ∫ R N ∣ u ∣ 2 d x = a 2 ε N , \left\{\begin{array}{ll}-{\varepsilon }^{2}\Delta u+V\left(x)u=\lambda u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }* F\left(u))f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2}{\varepsilon }^{N},\hspace{1.0em}& \end{array}\right. where a > 0 a\gt 0 is a constant, ε > 0 \varepsilon \gt 0 is a parameter, N ≥ 3 N\ge 3 , α ∈ ( 0 , N ) \alpha \in \left(0,N) , λ ∈ R \lambda \in {\mathbb{R}} is unknown and appears as a Lagrange multiplier, f f is a continuous function with L 2 {L}^{2} -subcritical growth, and V : R N → [ 0 , ∞ ) V:{{\mathbb{R}}}^{N}\to \left[0,\infty ) is a continuous function, satisfying del Pino and Felmer’s local conditions. With the help of the penalization method, and Lusternik-Schnirelmann theory, we investigate the relationship between the number of positive normalized solutions and the topology of the set, where the potential V V attains its minimum value if the parameter ε > 0 \varepsilon \gt 0 is small.

Classification, non-degeneracy and existence of solutions to nonlinear Choquard equations

Classification, non-degeneracy and existence of solutions to nonlinear Choquard equations

Existence of solutions for nonlinear biharmonic Choquard equations on weighted lattice graphs

In this paper, let G=(ZN,E,μ,ω) be a weighted lattice graph, where μ is a uniformly bounded measure and ω is a positive symmetric weight. We are concerned with the following biharmonic Choquard equation with a nonlocal nonlinearity on ZN, for any α∈(0,N),Δ2u−Δu+hu=(∑y∈ZN,y≠xF(u(y))d(x,y)N−α)f(u), where h denotes a function on ZN, f:R→R is a continuous function, F(u)=∫0uf(s)ds represents the primitive function of f, and d(x,y) stands for the distance between x and y. In particular, under some suitable conditions on h, we prove that if the nonlinearity f satisfies distinct growth conditions, then there exists a mountain-pass solution and a ground state solution respectively. As far as we know, there are no existing results for the biharmonic Choquard equation with the nonlinearity f on weighted lattice graphs.

Normalized ground states to the nonlinear Choquard equations with local perturbations

<abstract><p>In this paper, we considered the existence of ground state solutions to the following Choquard equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{aligned} &-\Delta u = \lambda u + (I_{\alpha}\ast F(u))f(u) + \mu|u|^{q-2}u \hskip0.5cm \mbox{in} \hskip0.2cm\mathbb{R}^{N}, \\ & \int\limits_{\mathbb{R}^{N}}|u|^{2}dx = a >0, \end{aligned} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p> <p>where $ N \geq 3 $, $ I_{\alpha} $ is the Riesz potential of order $ \alpha \in (0, N) $, $ 2 < q \leq 2+ \frac{4}{N} $, $ \mu > 0 $ and $ \lambda \in \mathbb{R} $ is a Lagrange multiplier. Under general assumptions on $ F\in \mathcal{C}^{1}(\mathbb{R}, \mathbb{R}) $, for a $ L^{2} $-subcritical and $ L^{2} $-critical of perturbation $ \mu|u|^{q-2}u $, we established several existence or nonexistence results about the normalized ground state solutions.</p></abstract>

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.

Regularity and Pohozaev identity for the Choquard equation involving the [formula omitted]-Laplacian operator

In this paper, we study the regularity of weak solutions for a class of nonlinear Choquard equations driven by the p-Laplacian operator. We also establish a Pohozaev type identity.

The Ground State Solutions to Discrete Nonlinear Choquard Equations with Hardy Weights

The Ground State Solutions to Discrete Nonlinear Choquard Equations with Hardy Weights

Normalized solutions for nonlinear Choquard equations with general nonlocal term

Normalized solutions for nonlinear Choquard equations with general nonlocal term

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.

Infinitely many solutions for The Brézis-Nirenberg problem with nonlinear Choquard equations

In this paper, we consider the following Brézis-Nirenberg problem with the Choquard equations:{−Δu=∫Ω|u(y)|2α⁎|x−y|N−αdy|u|2α⁎−2u+λuin Ω,u=0on ∂Ω, where α∈(N−4,N), Ω is a bounded smooth domain in RN(N≥7) and 2α⁎=N+αN−2 is the Hardy-Littlewood-Sobolev critical exponent. We show that, for each λ>0, this problem has infinitely many solutions by using truncation method.

Symmetric positive solutions to nonlinear Choquard equations with potentials

Existence results for a class of Choquard equations with potentials are established. The potential has a limit at infinity and it is taken invariant under the action of a closed subgroup of linear isometries of $$\mathbb {R}^N$$ . As a consequence, the positive solution found will be invariant under the same action. Power nonlinearities with exponent greater or equal than two or less than two will be handled. Our results include the physical case.

Ground state solutions for nonlinear Choquard equations with doubly critical exponents

This paper is concerned with nonlinear Choquard equations with doubly critical growth. We apply the Pohozăev-type identity to overcome loss of compactness caused by the doubly critical nonlinearities and obtain the existence of a ground state solution.

Existence and asymptotic behavior of positive least energy solutions for coupled nonlinear Choquard equations

In this article, we study the coupled nonlinear Schrodinger equations with Choquard type nonlinearities $$\displaylines{ -\Delta u+\nu_1u=\mu_1(\frac{1}{|x|^{\alpha}} *u^2)u +\beta (\frac{1}{|x|^{\alpha}} *v^2)u \quad\hbox{in } \mathbb{R}^{N},\cr -\Delta v+\nu_2v=\mu_2(\frac{1}{|x|^{\alpha}} *v^2)v + \beta (\frac{1}{|x|^{\alpha}} *u^2)v \quad\hbox{in } \mathbb{R}^{N},\cr u,v \geq 0\quad \hbox{in } \mathbb{R}^{N}, \quad u,v \in H^{1}(\mathbb{R}^{N}), }$$ where \(\nu_1,\nu_2,\mu_1,\mu_2\) are positive constants, \(\beta&gt;0\) is a coupling constant, \(N\geq 3\), \(\alpha\in(0,N)\cap (0,4)\), and ``*'' is the convolution operator We show that the nonlocal elliptic system has a positive least energy solution for positive small \(\beta\) and positive large \(\beta\) via variational methods. For the case in which \(\nu_1=\nu_2\), \(\mu_1\neq\mu_2\), \(N=3,4,5\) and \(\alpha=N-2\), we prove the uniqueness of positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as \(\beta\to 0^{+}\) are studied. For more information see https://ejde.math.txstate.edu/Volumes/2021/47/abstr.html

On damped non-linear Choquard equations

On damped non-linear Choquard equations

Publisher Correction to: Normalized solutions for a class of nonlinear Choquard equations

Publisher Correction to: Normalized solutions for a class of nonlinear Choquard equations

Infinitely many sign-changing solutions for Choquard equation with doubly critical exponents

In this paper, we consider the following Choquard equation: where , , is the Riesz potential, and , where and are lower and upper critical exponents in sense of the Hardy–Littlewood–Sobolev inequality. Based on perturbation method and the invariant sets of descending flow, we prove that the above equation possesses infinitely many sign-changing solutions. Our results extend the results in Seok [Nonlinear Choquard equations: doubly critical case. Appl Math Lett. 2018;76:148–156] and Su [New result for nonlinear Choquard equations: doubly critical case. Appl Math Lett. 2020;102(106092):0–7].

Normalized solutions for a class of nonlinear Choquard equations

We prove the existence of a least energy solution to the problem $$\begin{aligned} -\Delta u-(I_{\alpha }*F(u))f(u)=\lambda u\ \text { in }\ {\mathbb {R}}^{N},\quad \int _{{\mathbb {R}}^N}u^2(x)dx = a^2, \end{aligned}$$ where $$N\ge 1$$ , $$\alpha \in (0,N)$$ , $$F(s):=\int _{0}^{s}f(t)dt$$ , and $$I_{\alpha }:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$$ is the Riesz potential. If f is odd in u then we prove the existence of infinitely many normalized solutions.

The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

Based on recent works of Dodson-Murphy [ 12 ] and Arora-Dodson-Murphy [ 3 ], we give a unified approach for the energy scattering with radially symmetric initial data for nonlinear Schrodinger equations and nonlinear Choquard equations in any dimensions \begin{document}$ N\geq 2 $\end{document} . We also discuss its applications for other Schrodinger-type equations.

Ground state solutions for nonlinear Choquard equations with inverse-square potentials1

This paper is concerned with the following Choquard equation − Δ u + ( V ( x ) − μ | x | 2 ) u = ( ∫ R N F ( u ( y ) ) | x − y | α d y ) f ( u ) , u ∈ H 1 ( R N ) , where N ⩾ 3 , 0 ⩽ μ &lt; ( N − 2 ) 2 4 , 0 &lt; α &lt; N , V is 1-periodic in each of