Existence results for critical elliptic systems in \(\mathbb R^N\). (English) Zbl 1259.35093

In this paper, by variational methods and critical point theory, the authors study the existence of positive solutions of the following critical elliptic system: \[ \begin{cases} -\Delta u+\lambda\Delta v+a(x)u=u^{2^{\ast}-1} &x\in \mathbb{R}^{N},\\ -\Delta v+\lambda\Delta u+b(x)v=v^{2^{\ast}-1} &x\in \mathbb{R}^{N},\\ u\geq0,\,\,v\geq0;\end{cases} \tag{1} \] where \(\lambda\in(0,1),\) \(N\geq3\), and \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent. Firstly, the authors consider the limiting problems: \[ \begin{cases} -\Delta u+\lambda\Delta v=u^{2^{\ast}-1} &x\in \mathbb{R}^{N},\\ -\Delta v+\lambda\Delta u=v^{2^{\ast}-1} &x\in \mathbb{R}^{N},\\ u\geq0,\,\,v\geq0;\end{cases} \tag{2} \] they show the existence of ground state solutions of problem (2); moreover, they study properties of the ground state solutions. In the last section, the authors show that equations (1) have no ground state solutions, and present results on the existence of positive solutions with higher energy. Under some standard conditions on the potentials, the existence theorems are proved using variational methods and critical point theory.


35J50 Variational methods for elliptic systems
35B33 Critical exponents in context of PDEs
35B09 Positive solutions to PDEs