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Existence of groundstates for a class of nonlinear Choquard equations
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 necessaryExpand
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Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
We consider a semilinear elliptic problem \[ - \Delta u + u = (I_\alpha \ast \abs{u}^p) \abs{u}^{p - 2} u\quad\text{in \(\R^N\),} \] where \(I_\alpha\) is a Riesz potential and \(p>1\). This familyExpand
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Semi-classical states for the Choquard equation
We study the nonlocal equation $$\begin{aligned} -\varepsilon ^2 \Delta u_\varepsilon +V u_\varepsilon = \varepsilon ^{-\alpha }\bigl (I_\alpha *|u_\varepsilon |^p\bigr ) |u_\varepsilon |^{p - 2}Expand
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Symmetrization and minimax principles
We develop a method to prove that some critical levels for functionals invariant by symmetry obtained by minimax methods without any symmetry constraint are attained by symmetric critical points. ItExpand
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Desingularization of Vortices for the Euler Equation
We study the existence of stationary classical solutions of the incompressible Euler equation in the planes that approximate singular stationary solutions of this equation. The construction isExpand
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Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials
We study the existence of positive solutions for a class of nonlinear Schrödinger equations of the type $$-{\varepsilon}^2\Delta u + Vu = u^p\quad{{\rm in}\,{\mathbb R}^N},$$where N ≥ 3, p > 1 isExpand
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Periodic homogenization of monotone multivalued operators
Using the unfolding method of Cioranescu, Damlamian and Griso [D. Cioranescu, A. Damlanuan, G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris Math. 335 (1) (2002) 99-104], weExpand
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Subelliptic Bourgain-brezis Estimates On Groups
We show that divergence-free $\mathrm{L}^1$ vector fields on a nilpotent homogeneous group of homogeneous dimension $Q$ are in the dual space of functions whose gradient is in $\mathrm{L}^Q$. ThisExpand
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A guide to the Choquard equation
We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$\begin{aligned} -\Delta u + V(x)u = \left( |x|^{-(N-\alpha )} *|u |^p\right)Expand
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Limiting Sobolev inequalities for vector fields and canceling linear differential operators
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 operatorExpand
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