Existence of Positive Solutions to a Fractional-Kirchhoff System

Let Ω be a bounded smooth domain in ℝ^N (N ≥ 3). Assuming that 0 < s < 1, \(1 < p,q \le {{N + 2s} \over {N - 2s}}\) $$1<p,q\le \frac{N+2s}{N-2s}$$ with \((p,q) \ne ({{N + 2s} \over {N - 2s}},{{N + 2s} \over {N - 2s}})\) $$(p,q)\ne (\frac{N+2s}{N-2s},\frac{N+2s}{N-2s})$$ , and a, b > 0 are constants, we consider the existence results for positive solutions of a class of fractional elliptic system below,

\(\left\{{\matrix{{(a + b[u]_s^2){{(- \Delta)}^s}u = {v^p} + {h_1}(x,u,v,\nabla u,\nabla v),} \hfill & {x \in \Omega,} \hfill \cr {{{(- \Delta)}^s}v = {u^q} + {h_2}(x,u,v,\nabla u,\nabla v),} \hfill & {x \in \Omega,} \hfill \cr {u,v > 0,} \hfill & {x \in \Omega,} \hfill \cr {u = v = 0,} \hfill & {x \in {\mathbb{R}^N}\backslash \Omega.} \hfill \cr}}\right.\) $$\{\begin{array}{ll}(a+b{[u]}_s^2){(-\Delta )}^{s}u=v^p+h_1(x,u,v,\nabla u,\nabla v),\hfill & x\in \Omega ,\hfill \\ {(-\Delta )}^{s}v=u^q+h_2(x,u,v,\nabla u,\nabla v),\hfill & x\in \Omega ,\hfill \\ u,v>0,\hfill & x\in \Omega ,\hfill \\ u=v=0,\hfill & x\in {ℝ}^N\backslash \Omega .\hfill \end{array}$$

Under some assumptions of h_i(x, u, v, ∇u, ∇v)(i = 1, 2), we get a priori bounds of the positive solutions to the problem (1.1) by the blow-up methods and rescaling argument. Based on these estimates and degree theory, we establish the existence of positive solutions to problem (1.1).