Standing wave solutions of a quasilinear Schrödinger equation. Part I: The low frequency limit
Abstract
Abstract This article is concerned with the quasilinear Schrödinger equation $$ \Delta u-\omega u+|u|^{p-1}u+\delta \Delta (|u|^2)u=0, $$ Δ u - ω u + | u | p - 1 u + δ Δ ( | u | 2 ) u = 0 , where $$\delta >0$$ δ > 0 , $$N=2$$ N = 2 and $$p>1$$ p > 1 or $$N\geqslant 3$$ N ⩾ 3 and $$1 1 < p < 3 N + 2 N - 2 . After proving uniqueness and non-degeneracy of the positive solution $$u_\omega $$ u ω for all $$\omega >0$$ ω > 0 , our main results establish the asymptotic behavior of $$u_\omega $$ u ω in the limit $$\omega \rightarrow 0^+$$ ω → 0 + . Three different regimes arise, termed ‘subcritical’, ‘critical’ and ‘supercritical’, corresponding respectively (when $$N\geqslant 3$$ N ⩾ 3 ) to $$1 1 < p < N
Fields of Study & Subjects
Advanced Mathematical Physics Problems
Nonlinear Partial Differential Equations
Nonlinear Photonic Systems
Publication details
- Journal / source
- Nonlinear Differential Equations and Applications NoDEA
- Publisher
- Springer Science and Business Media LLC
- Volume
- 33
- Issue
- 2
- Publication date
- 10 January 2026
- Language
- EN