Ground States for K-Component Coupled Nonlinear Schrodinger Equations with Two Types of Strongly Indefinite Structure

作者：Chen, Peng^*; Chen, Huimao; Li, Yuanyuan

来源：JOURNAL OF GEOMETRIC ANALYSIS, 2023, 33(11): 362.

DOI：10.1007/s12220-023-01415-6

• Ground state
• Bose-Einstein condensates
• Strongly indefinite functional
• Non-Nehari manifold method
• Perturbation method

In this paper, we are concerned with a class of K-component coupled nonlinear Schrodinger equations: @@@ -Delta u(i) + V-i(x)u(i) = f(i)(x, u) in R-N, i = 1, 2, . . . K, @@@ with u = (u(1), u(2), . . . , u(K)) : R-N -> R-K, f(i)(x, u) = partial derivative(ui) F(x, u), which originates from Bose-Einstein condensates phenomenon. We mainly study the case that V-i(x) is asymptotically periodic or non-periodic, which is different from periodic case. We obtain ground state solutions of Nehari-Pankov type under mild conditions on the nonlinearity by further developing non-Nehari method with two types of strongly indefinite structure. If in addition the corresponding functional is even, we also obtain infinitely many geometrically distinct solutions by using some arguments about deformation type and Krasnoselskii genus. Furthermore, we fill the gaps about the existence of ground state solutions of K-component equations with spectrum point zero. Nevertheless, we need to overcome some difficulties: one is due to the absence of strict monotonicity condition, a key ingredient of seeking the ground state solution on suitable manifold, we need some new methods and techniques. The second is that the working space is only a Banach space, not a Hilbert space, due to 0 is a boundary point of the spectrum of operator. The third lies that some delicate analysis are needed for the dropping of classical super-quadratic assumption on the nonlinearity, periodic assumption on potential and in verifying the link geometry and showing the boundedness of Cerami sequences.

• 单位
  武汉工程大学