Stability of standing waves for some nonlinear Schrödinger equations

Abstract

The paper concerns the monotonicity with respect to $\lambda$ of the $L^2$-norm of the branch of positive solutions of the nonlinear eigenvalue problem $$ u''(x)+ g(x, u(x)^2) \ u(x) + \lambda u(x) = 0, \ x\in {\bf R}, \ \lim\limits_{|x|\rightarrow \infty} \ u(x) = 0.$$ For the particular case $ g(x) = p(x) + s^{\sigma},$ with $\sigma > 0$ and $p(x)$ an even function, decreasing for $x > 0$ and with $p(\infty) = 0$, the main theorem implies that the $L^2$-norm decreases as we increase $\lambda$ 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.