Singular solutions of higher-order non-linear elliptic systems

The behaviour of solutions of non-linear elliptic systems is studied in the neighbourhood of a singular point, finite or infinite. It is shown that if the order of the singularity lies in a certain interval depending on the modulus of ellipticity of the system, then it is equal to the order of the singularity of some singular solution for the poly-Laplacian operator. Sharp two-sided energy estimates in the neighbourhood of the singular point are obtained for such solutions. Global solutions are considered, for which lower energy bounds are derived from upper estimates. Counterexamples constructed for second-order equations and systems demonstrate that the interval of regular behaviour of the order of the singularity is precisely described.