Stability of standing waves for a class of quasilinear Schrödinger equations

European Journal of Applied Mathematics - Tập 23 Số 5 - Trang 611-633 - 2012
Jianqing Chen1, Yongqing Li1, Zhi-Qiang Wang2
1School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, P.R. China
2Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA; School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, P. R. China

Tóm tắt

This paper is concerned with the stability and instability of standing waves for the quasilinear Schrödinger equation of the form which has been derived in many models from mathematical physics. We find the exact threshold depending upon the interplay of quasilinear and nonlinear terms that separates stability and instability. More precisely, we prove that for α ∈ and odd p ∈ , when $1 < p < 4\alpha -1 +{4\over N}$, the standing wave is stable, and when $4\alpha -1 +{4\over N} \leq p < 2\alpha\cdot 2^\ast -1$ (where $2\alpha\cdot 2^\ast = \frac{4N\alpha}{N-2}$ for N ≥ 3 and 2 α ċ 2* = +∞ for N = 2), the standing wave is strongly unstable. Our results show that the quasilinear term 2 α(△|φ|2α)|φ|2α−2φ makes the standing waves more stable, which is consistent with the physical phenomena.

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European Journal of Applied Mathematics

Tập 23 Số 5

611-633

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