Wednesday, March 28, 2012

1203.6002 (Louis Jeanjean et al.)

Sharp non-existence results of prescribed $L^2$-norm solutions for a
class of stationary Schrödinger-Poisson equations

Louis Jeanjean, Tingjian Luo
In this paper we are interested in the existence of minimizers for $$ F(u) = 1/2\int_{\R^3} |\nabla u|^2 dx + 1/4\int_{\R^3}\int_{\R^3}\frac{|u(x)|^2|u(y) |^2}{|x-y|}dxdy-\frac{1}{p}\int_{\R^3}|u|^p dx$$ on the constraint $$S(c) = \{u \in H^1(\R^3) : \int_{\R^3}|u|^2 dx = c \},$$ where $c>0$ is a given parameter. A critical point of $F(u)$ constrained to $S(c)$ gives rise to solutions of the Schr\"odinger-Poisson equation {equation*} -\Delta u - \lambda u + (|x|^{-1}\ast |u|^2) u - |u|^{p-2}u = 0 \ \{in}\ \ \R^3,{equation*} where $\lambda \in \R$ is the associated Lagrange parameter. In the range $p \in [3, 10/3]$ we explicit a threshold value for $c>0$ separating existence and non-existence of minimizers. We also derive a non-existence result of critical points of $F(u)$ restricted to $S(c)$ when $c>0$ is sufficiently small. As a byproduct of our approach we extend some results of \cite{CJS} where a constrained minimization problem linked to a quasilinear equation is considered.
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