Classification of solutions to the critical order elliptic system with general nonlinearity

Consider the following elliptic system (0.1) \(\begin{cases}(-\Delta)u=f(u,v),\\(-\Delta)v=g(u,v),\end{cases}\) $$\{\begin{array}{l}(-\mathrm{\Delta })u=f(u,v),\\ (-\mathrm{\Delta })v=g(u,v),\end{array}$$ where f and g are continuous functions and satisfy the finite total curvature conditions. Recently, Guo and Liu (2008) derived Liouville-type results for the positive solution of the semilinear elliptic system (0.1) in the whole space ℝ^N (N ⩾ 3). However, the case N = 2 is different and difficult because u and v may change signs. Using the method of moving spheres in the integral form combined with integral inequalities, we give a complete classification of the classical solutions to the above system in ℝ². This seems the first result for the classification of solutions (u, v) (here u and v are not required to be positive solutions) to the critical order system (0.1) in the two-dimensional case.