The Choquard equation in \(\mathbb{R}^N\) writes as \[ -\Delta u + V u = (I_\alpha \ast u^p)u^{p - 1}, \] where \(I_\alpha : \mathbb{R}^n \to \mathbb{R}\) is a Riesz potential defined for \(x \in \mathbb{R}^n \setminus \{0\}\) by \[ I_\alpha (x) = \frac{A_\alpha}{\vert x \vert^{N - \alpha}}. \] This equation was introduced in 1976 by P. Choquard in order to describe an electron trapped in its own hole. It also appears in the theory of the polaron at rest and in models of interaction between non relativistic quantum mechanics and gravitation.
For this equation,
- with V. Moroz, we have given nonexistence results for supersolutions in outer domains,
- with V. Moroz, in the autonomous case \(V = 1\), we have studied properties of solutions and existence for general nonlinearities, the corresponding problem in the plane was considered in collaboration with L. Battaglia,
- with V. Moroz, we have studied associated nonlocal Hardy inequalities,
- with V. Moroz, we have constructed solutions in the semi-classical limit,
- with Xia Jiankang, we have studied the case of confining potentials and the semi-classicali limit in the critical frequency case,
- with V. Moroz, we have studied the existence for the
lower critical exponent \(p = 1 + \frac{\alpha}{N}\), - with V. Moroz and M. Ghimenti, we have studied the existence of minimal action nodal solutions,
- with D. Ruiz, we have studied the odd symmetry of these minimal action nodal solutions,
- with D. Bonheure and S. Cingolani, we have proved the nondegeneracy of the solution to the two-dimensional logarithmic Choquard equation.
David Ruiz and Jean Van Schaftingen, Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differential Equations 264 (2018), no. 2, 1231–1262.
doi:10.1016/j.jde.2017.09.034 DIAL:191995 arXiv:1606.05668
Jean Van Schaftingen and Jiankang Xia, Standing waves with a critical frequency for nonlinear Choquard equations, Nonlinear Anal. 161 (2017), 87–107.
doi:10.1016/j.na.2017.05.014 DIAL:190306 arXiv:1611.08952
Denis Bonheure, Silvia Cingolani and Jean Van Schaftingen, The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal. 272 (2017), no. 12, 5255–5281.
doi:10.1016/j.jfa.2017.02.026 DIAL:186089 arXiv:1612.02194
Luca Battaglia and Jean Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations in the plane, Adv. Nonlinear Stud. 17 (2017), no. 3, 581–594.
doi:10.1515/ans-2016-0038 DIAL:192017 arXiv:1604.03294
Vitaly Moroz and Jean Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl. 19 (2017), no. 1, 773–813.
doi:10.1007/s11784-016-0373-1 SharedIt DIAL:184670 arXiv:1606.02158
Jean Van Schaftingen and Jiankang Xia, Choquard equations under confining external potentials, NoDEA Nonlinear Differential Equations Appl. 24 (2017), no. 1, 1–24.
doi:10.1007/s00030-016-0424-8 SharedIt DIAL:179262 arXiv:1607.00151
Marco Ghimenti, Vitaly Moroz and Jean Van Schaftingen, Least action nodal solutions for the quadratic Choquard equation, Proc. Amer. Math. Soc. 145 (2017), no. 2, 737–747.
doi:10.1090/proc/13247 DIAL:179188 arXiv:1511.04779
Marco Ghimenti and Jean Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal. 271 (2016), no. 1, 107–135.
doi:10.1016/j.jfa.2016.04.019 DIAL:173887 arXiv:1503.06031
Vitaly Moroz and Jean Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), no. 5, 1550005 (12 pages).
doi:10.1142/S0219199715500054 DIAL:165641 arXiv:1403.7414
Vitaly Moroz and Jean Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579.
doi:10.1090/S0002-9947-2014-06289-2 DIAL:163152 arXiv:1212.2027
Vitaly Moroz and Jean Van Schaftingen, Semi-classical states for the Choquard equations, Calc. Var. Partial Differential Equations 52 (2015), no. 1, 199–235.
doi:10.1007/s00526-014-0709-x SharedIt DIAL:155963 arXiv:1308.1571
Vitaly Moroz and Jean Van Schaftingen, Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), no. 2, 153–184.
doi:10.1016/j.jfa.2013.04.007 DIAL:131969 arXiv:1205.6286
Vitaly Moroz and Jean Van Schaftingen, Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains, J. Differential Equations 254 (2013), no. 8, 3089–3145.
doi:10.1016/j.jde.2012.12.019 DIAL:128740 arXiv:1203.3154
Vitaly Moroz and Jean Van Schaftingen, Nonlocal Hardy type inequalities with optimal constants and remainder terms, Ann. Univ. Buchar. Math. Ser. 3 (LXI) (2012), no. 2, 187–200.
