Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients

Abstract

The Paneitz operator discovered in [11] is the fourth order operator defined on a 4-dimensional Riemannian manifold (M, g) bywhere ∆ g u = −div g ∇u is the Laplacian of u with respect to g, S g is the scalar curvature of g, and Rc g is the Ricci curvature of g. An extension to manifolds of dimension n ≥ 5, due to Branson [2], is the fourth order operator defined byBoth P 4 g and P n g have conformal properties: for all u ∈ C ∞ (M ), P 4 g u = e −4ϕ P 4 g u when n = 4 andg = e 2ϕ g, while P n g (uϕ) = ϕ (n+4)/(n−4) P ñ g u when n ≥ 5 and g = ϕ 4/(n−4) g. With respect to these relations, P 4 g in dimension 4 is a natural analogue of ∆ g in dimension 2, while P n g in dimension n ≥ 5 is a natural analogue of the conformal Laplacian ∆ g + n−2 4(n−1) S g in dimension n ≥ 3. Possible references on the subject are the survey articles [3] by Chang, and [4] by Chang and Yang.We let here (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 5, and say that a fourth order operator P g is a Paneitz type operator with constant coefficients if P g u = ∆ 2 g u + α∆ g u + au (0.1)

Existence and nonexistence results for critical growth biharmonic elliptic equations

“…Related generalisations of the Struwe compactness lemma to higher order problems can be found in [1,3,11,19]. However, none of these applies directly to our situation.…”

Conformal invariants and partial differential equations

“…In dimensions higher than four, the conformal Paneitz operator is being investigated by a number of authors. In particular, Djadli-Hebey-Ledoux ( [40]) studied the question of coercivity of the operators P as well as the positivity of the solution functions; Djadli-Malchiodi-Ahmedou ( [41]) and Hebey-Robert ( [67]) have studied the blow-up analysis of the Paneitz equation. A serious difficulty in dealing with higher order operators is the lack of a good maximum principle.…”

Multiple solutions to a nonlinear elliptic equation involving Paneitz type operators

“…(1.5) This is a special case of what one usually refers to as a Paneitz-Branson operator with constant coefficients, namely, an operator which is expressed as 6) where α and β are real numbers. In this direction, we can refer the reader to Djadli et al [11], Esposito and Robert [14], Felli et al [15], Hebey [17,18], Hebey and Robert [19], and finally to Robert [25].…”

“…For the Paneitz-Branson operator, we mention the references described above [11,14,15,17,18,19,25]. Hereafter, the space H 2 2 (M) will be endowed with the norm · , 13) which is equivalent to norm · H 2 2 (M) .…”