Existence, characterization and number of ground states for coupled equations

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.

Keywords

• coupled equations
• ground states