Existence and asymptotic behavior of ground states for linearly coupled systems involving exponential growth

Abstract

In this paper we study the following class of linearly coupled systems in the plane:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + u = f_1(u) + \lambda v,\quad \text{ in }\quad \mathbb {R}^2, \\ -\Delta v + v = f_2(v) + \lambda u,\quad \text{ in }\quad \mathbb {R}^2, \\ \end{array}\right. } \end{aligned}$$

where \(f_{1}, f_{2}\) are continuous functions with critical exponential growth in the sense of Trudinger-Moser inequality and \(0<\lambda <1\) is a parameter. First, for any \(\lambda \in (0,1)\), by using minimization arguments and minimax estimates we prove the existence of a positive ground state solution. Moreover, we study the asymptotic behavior of these solutions when \(\lambda \rightarrow 0^{+}\). This class of systems can model phenomena in nonlinear optics and in plasma physics.