Electronic Journal of Differential Equations, 
Vol. 2019 (2019), No. 127, pp. 1-18.

Title:  Existence, characterization and number of ground states for coupled  equations

Authors:  Qihan He (Guangxi Univ., Nanning, China)
               Shuangjie Peng (Central China Normal Univ., Wuhan, China}

Abstract:
 This article concerns the existence, characterization and number of ground
 states for the system consisting  of m coupled semilinear equations
 $$\displaylines{
 -\Delta u_i +\lambda u_i =\sum_{j=1}^m k_{ij}
 \frac{q_{ij}}{p+1}|u_j|^{p_{ij}}|u_i|^{q_{ij}-2}u_i, \quad x\in \Omega,\cr
 u_i \in H^1_0(\Omega), \quad i=1,2,\ldots,m.
 }$$
 We extend the characterization results obtained by Correia [5,6]
 to the above problem. Also we give a new characterization of
 the ground states, which  provides a more convenient way for finding or checking
 ground states.  This study may be the first result not only positive ground states
 but also  for semi-trivial ground states, and it shows that the positive ground
 state is unique for some special cases.
 
Submitted July 18, 2018. Published November 26, 2019.
Math Subject Classifications:  35B99, 35J47, 35J60.
Key Words: Coupled equations; ground states.