## Standing waves for a generalized Davey-Stewartson system: Revisited.(English)Zbl 1165.35458

Summary: The existence of standing waves for a generalized Davey-Stewartson (GDS) system has been shown by A. Eden and S. Erbay [J. Phys. A, Math. Gen. 39, No. 43, 13435–13444 (2006; Zbl 1156.35360)] using an unconstrained minimization problem. Here, we consider the same problem but relax the condition on the parameters to $$\chi +b<0$$ or $$\chi + \frac{b}{m_1}< 0$$. Our approach, in the spirit of H. Berestycki, T. Gallouet and O. Kavian [C. R. Acad. Sci., Paris, Sér. I 297, 307–310 (1983; Zbl 0544.35042)] and R. Cipolatti [Commun. Partial Differ. Equations 17, No. 5–6, 967–988 (1992; Zbl 0762.35109)], is to use a constrained minimization problem and utilize Lions’ concentration-compactness theorem [P. L. Lions, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109–145 (1984; Zbl 0541.49009)]. When both methods apply, we show that they give the same minimizer and obtain a sharp bound for a Gagliardo-Nirenberg type inequality. As in [A. Eden and S. Erbay, loc. cit.], this leads to a global existence result for small-mass solutions. Moreover, following an argument [A. Eden, H. A. Erbay and G. M. Muslu, Nonlinear Anal., Theory Methods Appl. 64, No. 5 (A), 979–986 (2006; Zbl 1091.35088)], we show that when $$p>2$$, the $$L^p$$-norms of solutions to the Cauchy problem for a GDS system converge to zero as $$t\rightarrow \infty$$.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B40 Asymptotic behavior of solutions to PDEs 35J20 Variational methods for second-order elliptic equations
Full Text:

### References:

 [1] Babaoglu, C.; Eden, A.; Erbay, S., Global existence and nonexistence results for a generalized davey – stewartson system, J. phys. A, 39, 11531-11546, (2004) · Zbl 1064.35176 [2] Babaoglu, C.; Erbay, S., Two-dimensional wave packets in an elastic solid with couple stresses, Internat. J. non-linear mech., 39, 941-949, (2004) · Zbl 1141.74307 [3] Berestycki, H.; Gallouet, T.; Kavian, O., Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. acad. sci. Paris Sér. I math., 297, 307-310, (1983) · Zbl 0544.35042 [4] 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 [5] T. Cazenave, An introduction to nonlinear Schrödinger equations, in: Textos de Métodos Matemáticos, vol. 22. Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 1989 [6] Cipolatti, R., On the existence of standing waves for a davey – stewartson system, Comm. partial differential equations, 17, 967-988, (1992) · Zbl 0762.35109 [7] Eden, A.; Erbay, S., On travelling wave solutions of a generalized davey – stewartson system, IMA J. appl. math., 70, 15-24, (2005) · Zbl 1079.35093 [8] Eden, A.; Erbay, S., Standing waves for a generalized davey – stewartson system, J. phys. A, 39, 13435-13444, (2006) · Zbl 1156.35360 [9] Eden, A.; Erbay, H.A.; Muslu, G.M., Two remarks on a generalized davey – stewartson system, Nonlinear anal. TMA, 64, 979-986, (2006) · Zbl 1091.35088 [10] A. Eden, E. Kuz, Davey-Stewartson system generalized: Existence, uniqueness and scattering, preprint · Zbl 1180.35480 [11] Lions, P.L., The concentration – compactness principle in the calculus of variations. the locally compact case. part 1, Ann. inst. H. Poincaré, anal. non linéaire, 1, 109-145, (1984) · Zbl 0541.49009 [12] Papanicolaou, G.C.; Sulem, C.; Sulem, P.L.; Wang, X.P., The focussing singularity of the davey – stewartson equations for gravity – capillarity surface waves, Physica D, 72, 61-86, (1994) · Zbl 0815.35104 [13] Weinstein, M.I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. math. phys., 87, 567-576, (1983) · Zbl 0527.35023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.