Fractional elliptic systems with critical nonlinearities

Abstract

In this paper we study positive solutions to the following nonlocal system of equations: \begin{equation*} \left\{\begin{aligned} &(-Δ)^s u = \fracα{2_s^*}|u|^{α-2}u|v|^β+f(x)\;\;\text{in}\;\mathbb{R}^{N}, &(-Δ)^s v = \fracβ{2_s^*}|v|^{β-2}v|u|^α+g(x)\;\;\text{in}\;\mathbb{R}^{N}, & \qquad u, \, v >0\, \mbox{ in }\,\mathbb{R}^{N}, \end{aligned} \right. \end{equation*} where $N>2s$, $\alpha ,\beta >1$, $\alpha +\beta =2N/(N-2s)$, and $f,g$ are nonnegative functionals in the dual space of ${\dot{H}}^s({\mathbb{R}}^N)$. When $f=0=g$, we show that the ground state solution of the above system is {\it unique}. On the other hand, when $f$ and $g$ are nontrivial nonnegative functionals with ker$(f)$=ker$(g)$, then we establish the existence of at least two different positive solutions of the above system provided that $\Vert f{\Vert }_{({\dot{H}}^s{)}^{\prime }}$ and $\Vert g{\Vert }_{({\dot{H}}^s{)}^{\prime }}$ are small enough. Moreover, we also provide a global compactness result, which gives a complete description of the Palais-Smale sequences of the above system.

27 pages