890 citations · 3.5k across the 69 of their papers we have counts for
8 papers · 1 filter
The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate
Denis Bonheure, Silvia Cingolani, Jean Van Schaftingen
We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation $$ - Δu + a u = \frac{1}{2 π} \Bigl[\ln \frac{1}{|x|}* |u|^2…
Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces
Jacopo Bellazzini, Marco Ghimenti, Carlo Mercuri +2
We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$\|φ\|_{L^p(\mathbb{R}^d)}\le C\|φ\|_{\dot H^{s}(\mathbb{R}^d)}^β \left(\iint_{\mathbb{R}^d \times \ma…
Standing waves with a critical frequency for nonlinear Choquard equations
Jean Van Schaftingen, Jiankang Xia
In this paper, we study the nonlocal Choquard equation where $N\ge…
Bourgain-Brezis Estimates on Symmetric Spaces of Non-compact Type
Sagun Chanillo, Jean Van Schaftingen, Po-lam Yung
Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of vector fields with divergence zero and and in L^1 with vector fields in a critical S…
Properties of groundstates of nonlinear Schrödinger equations under a weak constant magnetic field
Denis Bonheure, Manon Nys, Jean Van Schaftingen
We study the qualitative properties of groundstates of the time-independent magnetic semilinear Schrödinger equation \[ - (\nabla + i A)^2 u + u = |u|^{p-2} u, \qquad \text{ in } \…
Choquard equations under confining external potentials
Jean Van Schaftingen, Jiankang Xia
We consider the nonlinear Choquard equation where , is the Riesz potentia…
Odd symmetry of least energy nodal solutions for the Choquard equation
David Ruiz, Jean Van Schaftingen
We consider the Choquard equation (also known as stationary Hartree equation or Schrödinger--Newton equation) \[ -Δu + u = (I_α\star |u|^p) |u|^{p - 2}u. \] Here stands for t…
A guide to the Choquard equation
Vitaly Moroz, Jean Van Schaftingen
We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$ -Δu + V(x)u = \bigl(|x|^{-(N-α)} * |u|^p\bigr)|u|^{p - 2}…