We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations 

$$\begin{aligned} -\Delta u + V(x)u = \left( |x|^{-(N-\alpha )} *|u |^p\right) |u |^{p - 2} u \quad \text {in} \ \mathbb {R}^N, \end{aligned}$$

and some of its variants and extensions.

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A guide to the Choquard equation

Singularly perturbed critical Choquard equations

On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents

Abstract The paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p-equation. As a consequence, the results can be applied to the special case ( a + b ⁢ ∥ u ∥ s p ⁢ ( θ - 1 ) ) ⁢ [ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u ] = λ ⁢ f ⁢ ( x , u ) + ( ∫ ℝ N | u | p μ , s * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | p μ , s * - 2 ⁢ u   in ⁢ ℝ N , (a+b\\|u\\|_{s}^{p(\\theta-1)})[(-\\Delta)^{s}_{p}u+V(x)|u|^{p-2}u]=\\lambda f(x,u)% +\\Bigg{(}\\int_{\\mathbb{R}^{N}}\\frac{|u|^{p_{\\mu,s}^{*}}}{|x-y|^{\\mu}}\\,dy% \\Biggr{)}|u|^{p_{\\mu,s}^{*}-2}u\\quad\\text{in }\\mathbb{R}^{N}, where ∥ u ∥ s = ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ) 1 p , \\|u\\|_{s}=\\Bigg{(}\\iint_{\\mathbb{R}^{2N}}\\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}% \\,dx\\,dy+\\int_{\\mathbb{R}^{N}}V(x)|u|^{p}\\,dx\\Biggr{)}^{\\frac{1}{p}}, a , b ∈ ℝ 0 + {a,b\\in\\mathbb{R}^{+}_{0}} , with a + b > 0 {a+b>0} , λ > 0 {\\lambda>0} is a parameter, s ∈ ( 0 , 1 ) {s\\in(0,1)} , N > p ⁢ s {N>ps} , θ ∈ [ 1 , N / ( N - p ⁢ s ) ) {\\theta\\in[1,N/(N-ps))} , ( - Δ ) p s {(-\\Delta)^{s}_{p}} is the fractional p-Laplacian, V : ℝ N → ℝ + {V:\\mathbb{R}^{N}\\rightarrow\\mathbb{R}^{+}} is a potential function, 0 < μ < N {0<\\mu<N} , p μ , s * = ( p ⁢ N - p ⁢ μ / 2 ) / ( N - p ⁢ s ) {p_{\\mu,s}^{*}=(pN-p\\mu/2)/(N-ps)} is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, and f : ℝ N × ℝ → ℝ {f:\\mathbb{R}^{N}\\times\\mathbb{R}\\rightarrow\\mathbb{R}} is a Carathéodory function. First, via the Mountain Pass theorem, existence of nonnegative solutions is obtained when f satisfies superlinear growth conditions and λ is large enough. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when f is sublinear at infinity and λ is small enough. More intriguingly, the paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero. Hence the results of the paper are new even for the standard stationary Kirchhoff problems.

Existence results for Schrödinger–Choquard–Kirchhoff equations involving the fractional p-Laplacian

Abstract We are concerned with the existence of ground states and qualitative properties of solutions for a class of nonlocal Schrödinger equations. We consider the case in which the nonlinearity exhibits critical growth in the sense of the Hardy–Littlewood–Sobolev inequality, in the range of the so-called upper-critical exponent. Qualitative behavior and concentration phenomena of solutions are also studied. Our approach turns out to be robust, as we do not require the nonlinearity to enjoy monotonicity nor Ambrosetti–Rabinowitz-type conditions, still using variational methods.

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Choquard-type equations with Hardy–Littlewood–Sobolev upper-critical growth

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$-\Delta u =\left(\int_{\Omega}\frac{|u|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dy\right)|u|^{2_{\mu}^{\ast}-2}u+\lambda u\4.14mm\mbox{in}\1.14mm \Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, with Lipschitz boundary, $\lambda$ is a real parameter, $N\geq3$, $2_{\mu}^{\ast}=(2N-\mu)/(N-2)$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation

We study the existence of ground states for the nonlinear Choquard equation driven by fractional Laplacian: 

(−Δ)su+u=|x|−μ∗F(u)f(u)inRN,

where the nonlinearity satisfies the general Berestycki–Lions-type assumptions. Copyright © 2016 John Wiley & Sons, Ltd.

Ground states for nonlinear fractional Choquard equations with general nonlinearities

In this paper, we are concerned with the following nonlinear Choquard equation −Δu + V (x)u = ∫ℝN G(y,u) |x − y|μdyg(x,u)in ℝN, where N ≥ 4, 0 < μ < N and G(x,u) =∫0ug(x,s)ds. If 0 lies in a gap of the spectrum of − Δ + V and g(x,u) is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrodinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579].

Fashun Gao

Papers

Singularly perturbed critical Choquard equations

On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents

On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation

Ground states for nonlinear fractional Choquard equations with general nonlinearities

A strongly indefinite Choquard equation with critical exponent due to the Hardy–Littlewood–Sobolev inequality