Pattern formation from homogeneous states for an elliptic system with mixed interactions

Abstract

In this paper, we are interested in the asymptotic behavior of a ground state vector solution for the following coupled nonlinear Schr & ouml;dinger system { Delta u(1)-lambda(1)u(1)+beta(11)(u(1))(3)+alpha beta(12)u(1)(u(2))(2)-beta beta(13)u(1)(u(3))(2)=0 Delta u(2)-lambda(2)u(2)+alpha beta(21)(u(1))(2)u(2)+beta(22)(u(2))(3)-beta beta(23)u(2)(u(3))(2)=0 Delta u(3)-lambda(3)u(3)-beta beta(31)(u(1))(2)u(3)-beta beta(32)(u(2))(2)u(3)+beta(33)(u(3))(3)=0 in Omega and partial derivative u i /partial derivative n = 0 on partial derivative Omega when alpha , beta > 0 are very large. The existence of a ground state vector solution for (1) was proved in Byeon et al. [Pattern formation via mixed interactions for coupled Schr & ouml;dinger equations under Neumann boundary condition. J Fixed Point Theory Appl. 2017;19:559-583] when alpha , beta , alpha/beta 2 - n/2 are large. We prove that if lambda 3 is small, as alpha , beta , alpha/beta 2 - n /2 -> infinity , u(3) converges to a constant, u1 and u2 develop a small peak on partial derivative . Under an additional condition alpha/beta (2 - 2 n + delta -> infinity) for some delta>1, we show that the peak point converges to a maximum point of the mean curvature of partial derivative Omega .