Existence of stable standing waves and instability of standing waves to a class of quasilinear Schrödinger equations with potential

Abstract

Abstract. For a class of quasilinear Schrödinger equations with harmonic potential of the formwe prove firstly the existence of stable standing waves for 1 < p < 3 + 4 N and then study the instability of standing waves for 3 +. Our results indicate that the quasilinear term (△|ϕ| 2 )ϕ makes the standing waves more stable than their counterpart in the semilinear case, which is consistent with the physical phenomena and is in striking contrast with the classical semilinear Schrödinger equations with potential.

Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schrödinger equations

“…By their results, the standing wave solution of (1.6) is stable if 2 < q < 4α + 4 N and unstable if 4α + 4 N ≤ q < 2α • 2 * . Chen and Rocha in [10] studied the equation with a harmonic potential iϕ t + ∆ϕ + 2(∆|ϕ| 2 )ϕ − |x| 2 ϕ + |ϕ| q−2 ϕ = 0 for x ∈ R N , t > 0 ϕ(x, 0) = ū0 (x), x ∈ R N .…”

Instability of ground states for a quasilinear Schrӧdinger equation

“…Note that in [9], a discussion on the values of m κ (c) with respect to p and c is given. See also Definition 3.1 in [23] and Definition 4.1 in [5] for (1.9).…”