Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth

Abstract

Abstract. We are concerned in this paper with the bubbling phenomenon for nonlinear fourth-order four-dimensional PDE's. The operators in the equations are perturbations of the bi-Laplacian. The nonlinearity is of exponential growth. Such equations arise naturally in statistical physics and geometry. As a consequence of our theorem we get a priori bounds for solutions of our equations.We are concerned in this paper with understanding the bubbling phenomenon for fourth-order four-dimensional PDE's of exponential growth. Such equations arise naturally in statistical physics and in geometry (see [7] and [9]). In what follows, we let (M, g) be a smooth compact Riemannian 4-manifold without boundary. We also let (b ε ) ε>0 and (f ε ) ε>0 be sequences of smooth functions on M , and we let (A ε ) ε>0 be a sequence of smooth (2, 0)-symetric tensor fields. We assume that (b ε ), (f ε ) and (A ε ) converge as ε → 0 in the C k -topologies, k a positive integer, to limiting objects of the same nature, b 0 , f 0 and A 0 . Then we consider sequences (u ε ) ε>0 of solutions ofwhere) is the Laplace-Beltrami operator andFollowing standard terminology, we say that the u ε 's blow up if u ε (x ε ) → +∞ as ε → 0 for a sequence (x ε ) of points in M . We letbe the limit operator in (1). At last, we let G be the Green function of L 0 . The Green function is unique up to a constant when the kernel of L 0 consists only of constants. We write G as

Quantization effects for a fourth-order equation of exponential growth in dimension $4$

“…This result is optimal, as shown in the preceding example involving the function g k . Druet and Robert [6] studied the corresponding problem on four-dimensional Riemannian manifolds, where the bi-Laplacian is replaced by a fourth-order elliptic operator referred to as P : when the kernel of P is such that Ker P = {constants}, we get similar results to those in theorem 1.2 with the additional information that α i = 1 for all i ∈ {1, . .…”

“…, N}. The techniques used in [6] are different from the techniques used here; the main reason is that, for equation (E), the kernel of the bi-Laplacian contains more than the constant functions. Related references in the context of Riemannian manifolds are [10,11].…”

ConstantT-curvature conformal metrics on 4-manifolds with boundary

“…One can consider analogous questions in dimensions higher than four, where there are there are higher-order analogues of the Laplace-Beltrami operator and of the Paneitz operator and also of the associated curvatures (called again Qcurvatures); see [Fefferman and Graham 2002;1985;Graham et al 1992].…”

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“…Actually, several works on equations (4) and (5) have extended the result of Brezis and Merle, showing that, under the crucial assumption that the prescribed curvatures K k converge in C 0 , the amount of curvature concentrating at each point is a multiple of 4π, i.e., a multiple of the total Gaussian curvature of S 2 ; see, e.g., [Li and Shafrir 1994]. (Also, higher-dimensional extensions were studied under the same strong assumptions of convergence of K k in C 0 or even C 1 ; see, e.g., [Druet and Robert 2006;Malchiodi 2006;Martinazzi 2009b]. ) In [Brezis and Merle 1991] the functions K k can belong to L p ‫,)ޒ(‬ with 1 < p ≤ +∞.…”