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 -∆ui + λui = m∑j=1 kij qij / p+1 |uj|pij|ui|qij-2ui, x ∈ Ω, ui ∈ H10(Ω), i = 1, 2, ..., 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.