Stability of standing waves for a class of quasilinear Schrödinger equations

Abstract

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.