A note on a class of higher order conformally covariant equations

By S. Chang, Wenxiong CHEN *. 2001

96 citations recorded by the source corpus.

In this paper, we study the higher order conformally covariant equation $(- \Delta )^{\frac{n}{2}} w = (n -1)! e^{n w} x \in R^n$ for all even dimensions n. Let $\alpha = \frac{1}{|S^n|} \int_{R^n} e^{n w} dx .$ We prove, for every $0 < \alpha < 1$, the existence of at least one solution. In particular, for $ n = 4$, we obtain the existence of radial solutions.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/61d76ca76c81d4efcb4405a3ac5445438a123d2b

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In Applied Mathematics

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Citations in the corpus

Concentration–compactness phenomena in the higher order Liouville's equation☆ (2008, background): “Adimurthi, Robert and Struwe [ARS] studied the case of dimension 4 (m = 2). As they showed, the situation is more subtle because the blow-up set (the set of points x such that uk(x) → ∞ as k → ∞) can have dimension up to 3 (in contrast to the finite blow-up set S in dimension 2). Moreover, as a cons”
The Paneitz equation in hyperbolic space (2008, unknown): “Infinitely many solutions have also been observed by Chang and Chen [4] in a different conformally covariant fourth order equation in R4 with exponential nonlinearity.”
CONCENTRATION PHENOMENA FOR LIOUVILLE'S EQUATION IN DIMENSION FOUR (2006, background): “In fact, by a result of Chang-Chen [4] for any α ∈]0, 16π2] there exists a solution uα of (7) of total volume ∫ R4 e 4uα dx = α which for α 16π2 fundamentally differs from the solution u(x) = log √ 96√ 96+|x|2 corresponding to the metric obtained by pull-back of the spherical metric on S4 under st”
Asymptotic method of moving planes for fractional parabolic equations (2020, background): “During the last decade, elliptic nonlocal equations, especially those involving fractional Laplacians, have been studied by more and more scholars and a series of results have been obtained, such as [9] for symmetry and nonexistence of solutions, [12, 16] for Liouville Theorems with indefinite nonli”
Classification of solutions to the higher order Liouville’s equation on $${\mathbb{R}^{2m}}$$ (2008, methodology): “Following an argument of [ CC ], we now see that solutions of the kind v + p actually exist, even among radially symmetric functions, with deg p = 2m − 2, and with deg p = 2. For simplicity, we only treat the case when m is even; if m is odd, the proof is similar.”
Fractional Adams–Moser–Trudinger type inequalities (2015, background): “For existence results for (26) in the case Ω = Rn, n odd, we refer to [20] and [25], when n is even we refer to [10], [11], [22], [35], [48].”
Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three (2013, background): “The proof will follow the ideas in [7], and will be a simple consequence of the following theorem about the prescribed Q-curvature on R3.”