Équations de champs scalaires euclidiens non linéaires dans le plan. (French) Zbl 0544.35042

We investigate the equation (1) \(-\Delta u=g(u)in {\mathbb{R}}^ 2\), \(u\in H^ 1({\mathbb{R}}^ 2)\), \(u\not\equiv 0\). We assume that \(g\in C({\mathbb{R}},{\mathbb{R}})\) is an odd function, that there exists \(\zeta>0\) such that \(G(\zeta)=\int^{\zeta}_{0}g(s)ds>0,\) that \(\lim_{s\to 0}g(s)/s=-m<0,\) and that for all \(\alpha>0\) there exists \(C_{\alpha}>0\) such that \(| g(s)| \leq C_{\alpha}\exp(\alpha s^ 2).\) Under these assumptions on g we prove the existence of a ground state (a solution with minimal action) and of infinitely many solutions with spherical symmetry. Analogous results were already proved by the first author and P. L. Lions in the case \(N\geq 3.\)
Roughly speaking, the proof of the result in the case \(N\geq 3\) consists in finding critical points of \(E(u)=\int | \nabla u|^ 2dx\) on the set \(M=\{u,V(u)=\int G(u)dx=1\},\) through a Ljusternik-Schnirelman type argument. This method does not apply to the case \(N=2\); where M should be replaced by \(N=\{u,V(u)\geq 0,\quad \int u^ 2dx=1\}.\) We then prove that the Lagrange multiplier introduced by the constraint \(\int u^ 2dx=1\) is equal to zero. The proof uses Pokhozaev identity.


35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces