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Article Contents

# Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation

• We establish a sharp instability theorem for the standing-wave solutions of the inhomogeneous nonlinear Schrödinger equation

$i u_t + \Delta u + V(\epsilon x ) |u|^{p-1} u = 0, \quad x \in \mathbf R^n$

with the critical power $p = 1 + 4/n, n \ge 2,$ under certain conditions on the inhomogeneous term $V$ with a small $\epsilon > 0.$ We also demonstrate that these localized standing-waves converge to standing waves of the nonlinear Schrödinger equation with the homogeneous nonlinearity.

Mathematics Subject Classification: Primary: 35B35, 35B60, 35Q40, 35Q55, 76B25.

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