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.