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Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent
Jean Van Schaftingen, Jiankang Xia
We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -Δu+ u=\big(I_α*|u|^{\fracα{N}+1}\big)|…
Groundstates of the Choquard equations with a sign-changing self-interaction potential
Luca Battaglia, Jean Van Schaftingen
We consider a nonlinear Choquard equation when the self-interaction potential is unbounded from below. Under…
Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent
Daniele Cassani, Jean Van Schaftingen, Jianjun Zhang
For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, , in where $N\g…
Uniform boundedness principles for Sobolev maps into manifolds
Antonin Monteil, Jean Van Schaftingen
Given a connected Riemannian manifold , an \(m\)--dimensional Riemannian manifold which is either compact or the Euclidean space, and…
Gauge-measurable functions
Augusto C. Ponce, Jean Van Schaftingen
In 1973, E.J. McShane proposed an alternative definition of the Lebesgue integral based on Riemann sums, where gauges are used decide what tagged partitions are allowed. Such an ap…
Sobolev mappings: from liquid crystals to irrigation via degree theory
Jean Van Schaftingen
Sobolev spaces are a natural framework for the analysis of problems in partial differential equations and calculus of variations. Some physical and geometric contexts, such as liqu…
Weak approximation by bounded Sobolev maps with values into complete manifolds
Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen
We have recently introduced the trimming property for a complete Riemannian manifold as a necessary and sufficient condition for bounded maps to be strongly dense in $W^{1,…