Estimate of Morse index of cooperative elliptic systems and its application to spatial vector solitons》.pdf

Journal of Computational and Applied Mathematics 281 (2015) 169–181 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: /locate/cam Estimate of Morse index of cooperative elliptic systems and its application to spatial vector solitons Xianjin Chen a,∗, Jianxin Zhou b a School of Mathematical Science, University of Science and Technology of China, Hefei 230026, PR China b Department of Mathematics, Texas AM University, College Station, TX 77843, United States a r t i c l e i n f o a b s t r a c t Article history: Local instability index of unstable solutions to single partial differential equations (PDEs) Received 26 June 2014 by a local minimax method (LMMM) was established in Zhou (2005). It is known that the local min-orthogonal method (LMOM) which was first proposed in Zhou (2004) and MSC: then further developed in Chen et al. (2008) can find more general unstable solutions to 35A40 both single PDEs and cooperative elliptic systems. This paper is to carry out instability 35A15 analysis of unstable solutions by LMOM, to which an infinite-dimensional functional space 58E05 58E30 can be decomposed as a direct sum of a finite-dimensional subspace and its orthogonal complement. A Morse index approach is developed to show that with LMOM, instability