CLASSIFICATION OF SOLUTIONS TO CONFORMALLY INVARIANT SYSTEMS WITH MIXED ORDER AND EXPONENTIALLY INCREASING OR NONLOCAL NONLINEARITY

In this paper, without any assumption on v and under the extremely mild assumption u(x) = O(|x|^K) at \infty for some K \gg 1 arbitrarily large, we prove classification of solutions to the following conformally invariant system with mixed order and exponentially increasing nonlinearity in \BbbR²: ^{\Biggl\{ (}_-^-_\Delta^\Delta)_v(_x²¹₎^u₌^(x_u⁾₄⁼_(x^e₎^pv(x^),_x \in _\BbbR^x₂^\in\BbbR2, where p \in (0, +\infty), u \geq 0 and that satisfies the finite total curvature condition ^\int_\BbbR2 u⁴(x)dx < +\infty. In order to show the integral representation formula and the crucial asymptotic property for v, we derive and use an exp^L +Lln L inequality, which is itself of independent interest. When p = ₂³ , the system is closely related to single conformally invariant equations (-\Delta) 2¹ u = u³ and -\Deltav = e^2v on \BbbR², 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 \BbbR³. Extensions to mixed order conformally invariant systems in \BbbRⁿ with general dimensions n \geq 3 are also included.