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Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities. (English) Zbl 1098.74614

Summary: We show that the ground-state solitary waves of the critical nonlinear Schrödinger equation \(i\psi_t(t,r)+\Delta\psi +V(\epsilon r)\psi^{4/d}\psi=0\) in dimension \(d\geq2\) are orbitally stable as \(\epsilon\rightarrow0\) if \(V(0)V^{(4)}(0)<G_d[ V^{\prime\prime}(0)]^2\), where \(G_d\) is a constant that depends only on \(d\).

MSC:

74J35 Solitary waves in solid mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
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[1] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345 (1983) · Zbl 0533.35029
[2] Fibich, G.; Merle, F., Self-focusing on bounded domains, Physica D, 155, 132-158 (2001) · Zbl 0980.35154
[3] D. Gilbert, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983.; D. Gilbert, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1983.
[4] Gill, T. S., Optical guiding of laser beam in nonuniform plasma, Pramana J. Phys., 55, 842-845 (2000)
[5] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry, J. Funct. Anal., 74, 160-197 (1987) · Zbl 0656.35122
[6] Kwong, M. K., Uniqueness of positive solutions of Δ \(u\)−\(u+u^p=0\) in \(R^n\), Arch. Ration. Mech. Anal., 105, 243-266 (1989) · Zbl 0676.35032
[7] S. Levandosky, Stability and instability of fourth-order solitary waves, J. Dynam. Differential Equations 10 (1998) 151-188.; S. Levandosky, Stability and instability of fourth-order solitary waves, J. Dynam. Differential Equations 10 (1998) 151-188.
[8] Liu, C. S.; Tripathi, V. K., Laser guiding in an axially nonuniform plasma channel, Phys. Plasmas, 1, 3100-3103 (1994)
[9] Liu, Y.; Wang, X. P., Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation, Commun. Math. Phys., 183, 253-266 (1997) · Zbl 0879.35136
[10] Merle, F., Nonexistence of minimal blow-up solutions of equations i \(u_t\)=−Δ \(u\)−\(k(x)|u|^{4/N}u\) in \(R^N\), Ann. Inst. Henry Poincare Phys. Theorique., 64, 33-85 (1996) · Zbl 0846.35129
[11] Ni, W. M.; Takagi, I., Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70, 247-281 (1993) · Zbl 0796.35056
[12] Shatah, J., Stable Klein-Gordon equations, Commun. Math. Phys., 91, 313-327 (1983) · Zbl 0539.35067
[13] Shatah, J.; Strauss, W. A., Instability of nonlinear bound states, Commun. Math Phys., 100, 173-190 (1985) · Zbl 0603.35007
[14] Strauss, W. A., Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55, 149-162 (1977) · Zbl 0356.35028
[15] Wang, X. F.; Zeng, B., On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal., 28, 633-655 (1997) · Zbl 0879.35053
[16] K. Wang, On the strong instability of standing wave solutions of the inhomogeneous nonlinear Schrödinger equations, M.Phil. Thesis, HKUST, 2001.; K. Wang, On the strong instability of standing wave solutions of the inhomogeneous nonlinear Schrödinger equations, M.Phil. Thesis, HKUST, 2001.
[17] Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87, 567-576 (1983) · Zbl 0527.35023
[18] Weinstein, M. I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math., 39, 51-68 (1986) · Zbl 0594.35005
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