Concentrating standing waves for Davey-Stewartson systems

作者：He, Yi^*; Luo, Xiao

来源：PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 2022, 152(6): 1411-1450.

DOI：10.1017/prm.2021.59

• Concentration
• Davey-Stewartson system
• multi-peak solutions
• singular integral operator

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey-Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey-Stewartson systems. We emphasize that with respect to the classical Schrodinger equation, the presence of a singular integral operator in the Davey-Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

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