Critical growth fractional systems with exponential nonlinearity

Abstract: We study the existence of positive solutions for the system of fractional elliptic equations of the type, \begin{equation*} \begin{array}{rl} (-\Delta)^{\frac{1}{2}} u &=\frac{p}{p+q}\lambda f(x)|u|^{p-2}u|v|^q + h_1(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1),\ (-\Delta)^{\frac{1}{2}} v &=\frac{q}{p+q}\lambda f(x)|u|^p|v|^{q-2}v + h_2(u,v) e^{u^2+v^2},\;\textrm{in}\; (-1, 1), u,v&>0 \;\textrm{in } \; (-1,1), u&=v=0 \; \text{in} \; \mathbb R\setminus (-1,1). \end{array} \end{equation*} where {$1<p+q\<2$}, $h_1(u,v)=(\alpha{+}2u^2)|u|^{\alpha-2}u|v|^\beta, h_2(u,v)=(\beta{+}2v^2) |u|^\alpha |v|^{\beta-2}v$ and ${\alpha+\beta\>2}$. Here $(-\Delta)^{\frac{1}{2}}$ is the fractional Laplacian operator. We show the existence of multiple solutions for suitable range of $\lambda$ by analyzing the fibering maps and the corresponding Nehari manifold. We also study the existence of positive solutions for a superlinear system with critical growth exponential nonlinearity.