Standing waves for a generalized Davey–Stewartson system: Revisited

Abstract

The existence of standing waves for a generalized Davey–Stewartson (GDS) system was shown in Eden and Erbay [A. Eden, S. Erbay, Standing waves for a generalized Davey–Stewartson system, J. Phys. A 39 (2006) 13435–13444] using an unconstrained minimization problem. Here, we consider the same problem but relax the condition on the parameters to χ + b < 0 or χ + b m 1 < 0 . Our approach, in the spirit of Berestycki, Gallouet and Kavian [H. Berestycki, T. Gallouet, O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983) 307–310] and Cipolatti [R. Cipolatti, On the existence of standing waves for a Davey–Stewartson system, Comm. Partial Differential Equations 17 (1992) 967–988], is to use a constrained minimization problem and utilize Lions’ concentration–compactness theorem [P.L. Lions, The concentration–compactness principle in the calculus of variations. The locally compact case. Part 1, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984) 109–145]. 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, S. Erbay, Standing waves for a generalized Davey–Stewartson system, J. Phys. A 39 (2006) 13435–13444], this leads to a global existence result for small-mass solutions. Moreover, following an argument in Eden, Erbay and Muslu [A. Eden, H.A. Erbay, G.M. Muslu, Two remarks on a generalized Davey–Stewartson system, Nonlinear Anal. TMA 64 (2006) 979–986] 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 → ∞ .