Nonlinear Choquard Equations Research Articles

Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations

In the present paper, we consider the following magnetic nonlinear Choquard equation \begin{eqnarray} \big(-i\nabla+A(x)\big)^{2}u+\big( g_{0}(x)+\mu g(x)\big)u=\big(|x|^{-\alpha}*|u|^{p}\big)|u|^{p-2}u,\\ u\in H^{1}(\mathbb{R}^{N},\mathbb{C}), \end{eqnarray} where $N\geq 3$, $\alpha\in (0,N)$, $p\in(\frac{2N-\alpha}{N}, \frac{2N-\alpha}{N-2})$, $A(x): {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}^{N}$ is a magnetic vector potential, $\mu>0$ is a parameter, $g_{0}(x)$ and $g(x)$ are real valued electric potential functions on ${\mathbb{R}}^{N}$. Under some suitable conditions, we show that there exists $\mu^{*}>0$ such that the above equation has at least one ground state solution for $\mu\geq\mu^{*}$. Moreover, the concentration behavior of solutions is also studied as $\mu\rightarrow +\infty$.

Ground states for nonlinear fractional Choquard equations with general nonlinearities

We study the existence of ground states for the nonlinear Choquard equation driven by fractional Laplacian: urn:x-wiley:mma:media:mma3849:mma3849-math-0001 where the nonlinearity satisfies the general Berestycki–Lions‐type assumptions. Copyright © 2016 John Wiley & Sons, Ltd.

Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent

We consider nonlinear Choquard equation [Formula: see text] where N ≥ 3, V ∈ L∞(ℝN) is an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N). The power [Formula: see text] in the nonlocal part of the equation is critical with respect to the Hardy–Littlewood–Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if [Formula: see text] then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis–Lieb lemma.

Existence of groundstates for a class of nonlinear Choquard equations

We prove the existence of a nontrivial solution (u \in H^1 (\R^N)) to the nonlinear Choquard equation [- \Delta u + u = \bigl(I_\alpha \ast F (u)\bigr) F' (u) \quad \text{in (\R^N),}] where (I_\alpha) is a Riesz potential, under almost necessary conditions on the nonlinearity (F) in the spirit of Berestycki and Lions. This solution is a groundstate; if moreover (F) is even and monotone on ((0,\infty)), then (u) is of constant sign and radially symmetric.

The existence of positive solutions with prescribed L2-norm for nonlinear Choquard equations

In this paper, we study the existence of positive solutions with prescribed L2-norm to a class of nonlinear Choquard equation −Δu − λu = (Iα∗F(u)) F′(u) in ℝN, where λ ∈ ℝ, N ≥ 3, α ∈ (0, N), Iα:ℝN → ℝ is the Riesz potential. Under some conditions imposed on F, by using a minimax procedure and the concentration compactness of P. L. Lions, we show that for any c > 0, the equation possesses at least a couple (uc, λc) ∈ H1(ℝN) × ℝ− of weak solution such that ‖u‖L2(RN)2=c.

Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics

We consider a semilinear elliptic problem−Δu+u=(Iα⁎|u|p)|u|p−2uinRN, where Iα is a Riesz potential and p>1. This family of equations includes the Choquard or nonlinear Schrödinger–Newton equation. For an optimal range of parameters we prove the existence of a positive groundstate solution of the equation. We also establish regularity and positivity of the groundstates and prove that all positive groundstates are radially symmetric and monotone decaying about some point. Finally, we derive the decay asymptotics at infinity of the groundstates.