Stationnary states for the nonlinear Schrödinger equation are solutions of the elliptic equation \[ -\varepsilon^2 \Delta u + V u = u^p, \] in \(\mathbb{R}^n\) where \(V : \mathbb{R}^n \to \mathbb{R}\) is a given potential and \(\varepsilon\) is the adimensionalized Planck constant. In the semi-classical limit where \(\varepsilon \to 0\), one expects solutions to concentrate around critical points of \(V\).
When \(\inf V > 0\), the existence of solutions for small \(\varepsilon\) concentrating around critical point of \(V\) as \(\varepsilon \to 0\) has been shown by many authors. I have been interested in the critical-frequency case where \(V\) is positive but \(\inf V =0\). With Denis Bonheure, we have adapted the penalization method of Manuel del Pino et Patricio Felmer and we have shown the existence of solutions for potentials that do not decay too fast at infinity. With Vitaly Moroz, we have obtained some optimal results for fast decaying potentials, including compactly supported potentials. With Denis Bonheure and Jonathan Di Cosmo, we have applied these methode to obtain solutions concentrating on spheres of dimension \(k \in \{1, \dotsc, n-1\}\).
With J. Di Cosmo, we have studied the existence of solution in the semiclassical strong magnetic field régime.
With Denis Bonheure, we have also studied the equation \[ -\Delta u + V u = Ku^p, \] in \(\mathbb{R}^n\) where \(V : \mathbb{R}^n \to \mathbb{R}\) and \(K : \mathbb{R}^n \to \mathbb{R}\) are bounded potentials. We have given conditions under which this problem has a groundstate and studied the asymptotic behaviour at infinity of these solutions.
Jonathan Di Cosmo and Jean Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field, J. Differential Equations 259 (2015), no. 2, 596–627.
doi:10.1016/j.jde.2015.02.016 DIAL:158580 arXiv:1312.5467
Jonathan Di Cosmo and Jean Van Schaftingen, Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima, Calc. Var. Partial Differential Equations 47 (2013), no. 1–2, 243–271.
doi:10.1007/s00526-012-0518-z SharedIt DIAL:131965 arXiv:1109.6773
Denis Bonheure, Jonathan Di Cosmo and Jean Van Schaftingen, Nonlinear Schrödinger equation with unbounded or vanishing potentials: solutions concentrating on lower dimensional spheres, J. Differential Equations 252 (2012), no. 1, 941–968.
doi:10.1016/j.jde.2011.10.004 DIAL:92294 arXiv:1009.2600
Denis Bonheure and Jean Van Schaftingen, Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity, Ann. Mat. Pura Appl. (4) 189 (2010), 273–301.
doi:10.1007/s10231-009-0109-6 SharedIt DIAL:34078 preprint
Vitaly Moroz and Jean Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials, Calc. Var. Partial Differential Equations 37 (2010), no. 1, 1—27.
doi:10.1007/s00526-009-0249-y SharedIt DIAL:35138 arXiv:0902.0722
Vitaly Moroz and Jean Van Schaftingen, Existence and concentration for nonlinear Schrödinger equations with fast decaying potentials, C. R. Math. Acad. Sci. Paris 347 (2009), no. 15–16, 921–926.
doi:10.1016/j.crma.2009.05.009 DIAL:35386 preprint
Denis Bonheure and Jean Van Schaftingen, Bound state solutions for a class of nonlinear Schrödinger equations, Rev. Mat. Iberoam. 24 (2008), no. 1, 297–351.
doi:10.4171/RMI/537 euclid.rmi/1216247103 MR:2435974 DIAL:36445 preprint
Denis Bonheure and Jean Van Schaftingen, Nonlinear Schrödinger equations with potentials vanishing at infinity, C. R. Math. Acad. Sci. Paris 342 (2006), no. 12, 903–908.
doi:10.1016/j.crma.2006.04.011 MR:2235608 DIAL:38398 preprint
