Standing waves for a generalized Davey-Stewartson system: Revisited

The existence of standing waves for a generalized Davey–Stewartson (GDS) system was shown in Eden and Erbay [8] using an unconstrainted minimization problem. Here, we consider the same problem but relax the condition on the parameters to χ+b < 0 or χ + b m1 < 0. Our approach, in the spirit of Berestycki, Gallouët and Kavian [3] and Cipolatti [6], is to use a constrained minimization problem and utilize Lions’ concentrationcompactness theorem [11]. 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 [8], this leads to a global existence result for small-mass solutions. Moreover, following an argument in Eden, Erbay and Muslu [9] we show that when p > 2, the L-norms of solutions to the Cauchy problem for a GDS system converge to zero as t→∞.