Stability of standing waves for some nonlinear Schrödinger equations. (Q1413613)

The authors deal with the nonlinear eigenvalue problems \[ u''(x)+g(x,u(x))u(x)+\lambda u(x)=0, \quad x\in \mathbb R,\;u(+\infty)=u(-\infty)=0. \] They are concerned with the monotonicity with respect to \(\lambda\) of the \(L^{2}\)-norm of the branch of positive solutions. In particular, when \(g(x,s)=p(x)+s^{\sigma}\) with \(\sigma\) positive, and p, an even function, decreasing for \(x>0\) and \(p(\infty)=0\), the main result implies that the \(L^{2}\)-norm decreases with increasing \(\lambda\) if \(\sigma\leq{2}\). It is also shown that this is no longer true if \(\sigma>2\). The result has implications for the orbital stability of standing waves of the nonlinear Schrödinger equation.