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. For a class of quasilinear Schrödinger equations with harmonic potential of the form iφt = −△φ + |x| φ − |φ|φ − 2(△|φ|)φ, t ≥ 0, x ∈ R , we prove firstly the existence of stable standing waves for 1 < p < 3 + 4 N and then study the instability of standing waves for 3 + 4 N ≤ p < 3N+2 N−2 . Our results indicate that the quasilinear term (△|φ|)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.

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