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

- Vitaly Moroz, Jean Van Schaftingen
- Mathematics
- 29 May 2012

Existence of groundstates for a class of nonlinear Choquard equations

- Vitaly Moroz, Jean Van Schaftingen
- Mathematics
- 10 December 2012

We prove the existence of a nontrivial solution 𝑢 ∈ H¹ (ℝ^N) to the nonlinear Choquard equation -Δ 𝑢 + 𝑢 = (I_α * 𝐹 (𝑢)) 𝐹' (𝑢) in ℝ^N, where I_α is a Riesz potential, under almost necessary… Expand

Desingularization of Vortices for the Euler Equation

- D. Smets, Jean Van Schaftingen
- Mathematics
- 7 September 2009

We study the existence of stationary classical solutions of the incompressible Euler equation in the planes that approximate singular stationary solutions of this equation. The construction is… Expand

Symmetrization and minimax principles

- Jean Van Schaftingen
- Mathematics
- 1 August 2005

We develop a method to prove that some critical levels for functionals invariant by symmetry obtained by minimax methods without any symmetry constraint are attained by symmetric critical points. It… Expand

A guide to the Choquard equation

- Vitaly Moroz, Jean Van Schaftingen
- Mathematics
- 7 June 2016

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)… Expand

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

- Vitaly Moroz, Jean Van Schaftingen
- Mathematics
- 28 March 2014

We consider nonlinear Choquard equation where N ≥ 3, V ∈ L∞(ℝN) is an external potential and Iα(x) is the Riesz potential of order α ∈ (0, N). The power in the nonlocal part of the equation is… Expand

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

- Jean Van Schaftingen
- Mathematics
- 1 April 2011

The estimate \[ \norm{D^{k-1}u}_{L^{n/(n-1)}} \le \norm{A(D)u}_{L^1} \] is shown to hold if and only if \(A(D)\) is elliptic and canceling. Here \(A(D)\) is a homogeneous linear differential operator… Expand

Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials

- Vitaly Moroz, Jean Van Schaftingen
- Mathematics
- 4 February 2009

We study the existence of positive solutions for a class of nonlinear Schrödinger equations of the type $$-{\varepsilon}^2\Delta u + Vu = u^p\quad{{\rm in}\,{\mathbb R}^N},$$where N ≥ 3, p > 1 is… Expand

Semi-classical states for the Choquard equation

- Vitaly Moroz, Jean Van Schaftingen
- Mathematics, Computer Science
- 7 August 2013

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