Classification of Solutions to Conformally Invariant Systems with Mixed Order and Exponentially Increasing or Nonlocal Nonlinearity

Abstract

In this paper, without any assumption on and under the extremely mild assumption at for some arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed order and exponentially increasing nonlinearity in where , and that satisfies the finite total curvature condition . In order to show the integral representation formula and the crucial asymptotic property for , we derive and use an inequality, which is itself of independent interest. When , the system is closely related to single conformally invariant equations and on , which have been quite extensively studied (cf. [H. Brezis and F. Merle, Comm. Partial Differential Equations, 16 (1991), pp. 1223–1253; D. Cao, Comm. Partial Differential Equations, 17 (1992), pp. 407–435; S.-Y. A. Chang and P. C. Yang, Math. Res. Lett., 4 (1997), pp. 91–102; W. Chen and C. Li, Duke Math. J., 63 (1991), pp. 615–622; W. Chen, C. Li, and Y. Li, Adv. Math., 308 (2017), pp. 404–437; W. Chen, Y. Li, and R. Zhang, J. Funct. Anal., 272 (2017), pp. 4131–4157], etc.). We also derive classification results for nonnegative solutions to a conformally invariant system with mixed order and Hartree type nonlocal nonlinearity in . Extensions to mixed order conformally invariant systems in with general dimensions are also included.