Multi-bump bound states for a system of nonlinear Schrodinger equations

作者：Lucia, Marcello; Tang, Zhongwei^*

来源：Journal of Differential Equations, 2012, 252(5): 3630-3657.

DOI：10.1016/j.jde.2011.11.017

• Nonlinear Schrodinger systems
• Potential well
• Semiclassical limit
• Multi-bump solutions
• Variational methods

Motivated by problems arising in nonlinear optics and Bose-Einstein condensates, we consider in R-N, with N <= 3, the following system of coupled Schrodinger equations:
{-Delta u(i) + lambda V-i(x)u(i) = u(i)Sigma(d)(j=1)alpha(ij)u(j)(2), i=1, ..., d,
u(i)>= 0, lim(vertical bar x vertical bar ->infinity) u(i)(x) = 0,
where lambda > 0 is a parameter. alpha(ij) = alpha(ji) are positive constants, and V-i non-negative given potentials. We assume that the interior of boolean AND(d)(i=1) V-i(-1) (0) admits ni connected components Omega(1), ..., Omega(m) which are of class C-1, and isolated in each V-i(-1) (0). For each non-empty J subset of {1, ..., m}, we prove that the system admits for any lambda large a multi-bump solution u(lambda) :R-N -> R-d which is small in R-N\boolean OR(j is an element of J)Omega(j), and on each Omega(j) (j is an element of J) close in H-1-norm to a least energy solution of the limit problem:
-Delta u(i) = u(i)Sigma(d)(j=1)alpha(ij)u(j)(2), i=1, ..., d,
subjected to homogeneous Dirichlet boundary condition. An explicit condition on the matrix (alpha(ij)) is given to ensure our solutions have at least two positive components.

• 出版日期2012-3-1
• 单位北京师范大学