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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
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