Symmetry, quantitative Liouville theorems and analysis of large solutions of conformally invariant fully nonlinear elliptic equations

Abstract

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem. Mathematics Subject Classification

A new method of proving a priori bounds for superlinear elliptic PDE

“…This is why when the scaling method is applicable, it leads to a priori bounds for a larger range of growth of the nonlinearities. However, the nonexistence result of [9] hinges both on the conformal invariance of the Laplacian and the specific form of the nonlinearity u p (for developments in this direction we refer to [14] and the references there). Since our method is based solely on properties shared by all uniformly elliptic operators, and is independent of the form of the nonlinearity, it does not appear to be amenable to obtain results for nonlinearities with power growth above (n + 1)/(n − 1).…”

Towards a Liouville Theorem for Continuous Viscosity Solutions to Fully Nonlinear Elliptic Equations in Conformal Geometry

“…As mentioned earlier, a combination of the above strong comparison principle and Hopf Lemma and the proof of [32,Theorem 1.1] give the following Liouville theorem.…”

“…Liouville theorems for (1) have been studied extensively. We mention here earlier results of Gidas, Ni and Nirenberg [15], Caffarelli, Gidas and Spruck [10] in the semi-linear case, of Viaclovsky [39,40] for the σ k -equations for C 2 solutions which are regular at infinity, of Chang, Gursky and Yang [11] for the σ 2 -equation in four dimensions, of Li and Li [26,27] for C 2 solutions, and of Li and Nguyen [32] for continuous viscosity solutions which are approximable by C 2 solutions on larger and larger compact domains.…”

“…where y * = x * (ε, y) and x * is defined in (32). Let us assume (42) for now and go on with the proof.…”

“…The key use of the C 2 regularity in the proof of the Liouville theorem in [32] is the strong comparison principle and Hopf Lemma for (1). In fact, if the strong comparison principle and Hopf Lemma can be established for C 1,α solutions (0 ≤ α ≤ 1), a Liouville theorem is then proved in C 1,α regularity by the same arguments.…”