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.
Citation
J. B. McLeod. C. A. Stuart. W. C. Troy. "Stability of standing waves for some nonlinear Schrödinger equations." Differential Integral Equations 16 (9) 1025 - 1038, 2003. https://doi.org/10.57262/die/1356060555
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