Fractional elliptic systems with critical nonlinearities

Abstract

This paper deals with existence, uniqueness and multiplicity of positive solutions to the following nonlocal system of equations:

\begin{equation*}\left\{\begin{aligned}& {(-{\Delta})}^{s}u=\frac{\alpha }{{2}_{s}^{{\ast}}}\vert u{\vert }^{\alpha -2}u\vert v{\vert }^{\beta }+f(x)\quad \text{in}\enspace {\mathbb{R}}^{N},\\ & {(-{\Delta})}^{s}v=\frac{\beta }{{2}_{s}^{{\ast}}}\vert v{\vert }^{\beta -2}v\vert u{\vert }^{\alpha }+g(x)\quad \text{in}\enspace {\mathbb{R}}^{N},\\ & u,\enspace v{ >}0\quad \text{in}\hspace{2pt}{\mathbb{R}}^{N},\end{aligned}\right.\qquad \qquad \qquad \qquad (\mathcal{S})\end{equation*}

where 0 < s < 1, N > 2s, α, β > 1, α + β = 2N/(N − 2s), and f, g are nonnegative functionals in the dual space of ${\dot {H}}^{s}({\mathbb{R}}^{N})$, i.e., ${}_{{({\dot {H}}^{s})}^{\prime }}\langle f\hspace{-1pt},u{\rangle }_{{\dot {H}}^{s}}{\geqslant}0$, whenever u is a nonnegative function in ${\dot {H}}^{s}({\mathbb{R}}^{N})$. When f = 0 = g, we show that the ground state solution of $(\mathcal{S})$ is 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 $(\mathcal{S})$ 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.