Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space

In this paper we mainly classify the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space \begin{document}$\mathbb{R}_+^{n}$\end{document} , and also present some remarks on the extremal functions of logarithmic Hardy-Littlewood-Sobolev inequality on the whole space \begin{document}$\mathbb{R}^{n}$\end{document} . Our main techniques are Kelvin transformation and the method of moving spheres in integral forms.


1.
Introduction. The logarithmic Hardy-Littlewood-Sobolev(HLS) inequality is an endpoint case of the classic HLS inequality. It plays an important role in analysis, statistical mechanics, conformal geometry and spectral theory.
The dual form of the logarithmic HLS inequality on S n , also called Onofri inequality, was first derived on S 2 by Onofri [14] in 1982. Later, the sharp logarithmic HLS inequality was established on the whole space R n by Carlen and Loss [3] and Beckner [1] in the early 1990s. Morpurgo [11] showed that the logarithmic HLS inequality on S n was the analytic expression of an extremal problem for the regularized zeta function of the Paneitz operators. Later, Branson, Fontana and Morpurgo [2] proved the sharp logarithmic HLS inequality and the corresponding Onofri inequality on the CR sphere (Heisenberg group). On Euclidean spaces and Riemannian manifolds, the best constants of inequalities are often the critical elements to show the existence of solutions to PDEs, to identify extremal geometries, to solve curvature problems and etc.. Hence, classification of extremal functions is crucial in the study of elliptic equations with critical exponents.
Recently, Dou and Zhu [6] established HLS inequality with boundary terms on the upper half space for f ∈ L p (∂R n + ), g ∈ L t (R n + ), where 1 < α < n, 1 < t, p < ∞ satisfy n−1 n · 1 p + 1 t + n−α+1 n = 2, and C e (n, α, p) > 0 is the best constant. Inequality (1) is an extension of the classic HLS inequality. They discussed the extremal functions of inequality (1) and determined the best constant. Moreover, they also considered the limiting case of inequality (1). That is, as α → n − , the following logarithmic HLS inequality holds.
Then we obtain the following system where m 1 and m 2 are constants. The extremal functions are classified by the following result.
where a 1 , a 2 > 0, d > 0, ξ 0 ∈ ∂R n + . According to Theorem 1.2, we get the extremal functions of inequality (3), i.e., for x, x 0 ∈ ∂R n + , We note that the classic logarithmic HLS inequality studied by Beckner in [1] is of the form: where f (x) and g(x) are nonnegative LlnL functions on R n , with R n f (x)dx = R n g(x)dx = 1 and K n is the best constant. Based on the rearrangement technique and stereographic projection, the corresponding extremal functions are classified (up to a conformal automorphism) by Carlen and Loss [3] also classified the extremal functions of logarithmic HLS inequality (6) by competing symmetries technique.
Noticing the similarity between (3) and (6), we can also write the Euler-Lagrange equations of the extremal functions of inequality (6) as follows, Similar to Theorem 1.2, we have the following result. Theorem 1.3. Let (u, v) be a pair of C 1 solutions to system (8). If u, v satisfy R n e nu(x)−1 dx = R n e nv(y)−1 dy = 1, then, u, v must be of the following forms on R n : where x, x 0 ∈ R n , a 1 , a 2 are some constants. Furthermore, based on the proof similar to that in [16], it can be verified that under some suitable assumptions, the integral system (8) is equivalent to That is, Lemma 1.4. Let (u, v) be a pair of C 1 strong solutions to (9) with u = o(|x| 2 ), v = o(|x| 2 ) at infinity and R n e nu(x) dx < ∞, R n e nv(y) dy < ∞. Then (u, v) satisfies (8) and vice versa.
Combining Lemma 1.4 with Theorem 1.3, we can obtain Theorem 1.5. Let (u, v) be a pair of C 1 solutions to system (9) Then, u, v must be of the following forms on R n : where x, x 0 ∈ R n , a 1 , a 2 are some constants.
If u = v in (9) and n = 2, the above theorem was first proved by Chen and Li [4] more than 20 years ago. As an extension to (9), the higher-order equation has been received a great deal of attention. For some specific values of n, via the method of moving planes, equation (10) is classified by many authors, see e.g. Ni [13], Chou and Wan [5], Lin [10], Wei and Xu [15,16] and the reference therein. Xu [17] considered the following conformal invariant integral equation For K(y) = (n − 1)!, Xu proved that the integral equation (11) is equivalent to equation (10), and classified the C 1 solutions of equation (11) via the method of moving spheres introduced by Li and Zhu in [9].
To classify solutions to elliptic equations on the upper half space R 2 Li and Zhu [9] introduced a method of moving spheres, which directly yields the form of solutions: In this project, we employ the method of moving spheres to prove Theorem 1.2. As Li and Nirenberg improved the proof of their key calculus lemmas in [7] by assuming that the function f is only continuous, we show the three calculus lemmas for exponential functions via applying the improved proof of Li and Nirenberg.
Since the constant C e (n, α, p) in inequality (1) could not be immediately applied to derive the logarithmic HLS inequality which is different from that on the whole space, we first need to give an estimate to C e (n, α, p), then deduce the logarithmic HLS inequality by the differentiation argument used in [1]. Theorem 1.3 tells us that we can compute the explicit form of the best constant K n of inequality (6) from the explicit form of u and v. However, according to the conclusion of Theorem 1.2, we could not show the explicit form of the best constant C n of inequality (3) (or (2)) since we don't know the form of the extremal function v on R n + . The paper is organized as follows. Kelvin transformation and some key lemmas are presented in Section 2. In Section 3, by employing Li and Nirenberg's standard procedure in [7], we prove Theorem 1.2 via the method of moving spheres in integral form, which is different from the proof provided by Xu in [17].
In the end, we provide the proof of Theorem 1.1 in appendix.
2. Some lemmas. In this section, we show Kelvin transformation to system (4), and present some key calculus lemmas which ensure that the moving sphere procedure could be carried out.
For R > 0, we give some notations as follows.
We define the following transformation: Moreover, For any x ∈ ∂R n + and λ > 0, let The n space forms in y variable and η variable are related by

JINGBO DOU AND YE LI
To simplify the calculations, we define That is, u can be written as follows: By a direct calculation we have Combining the above, we have Therefore, On the other hand, In view of R n + e nv(y)−1 dy = 1, we find that Therefore, Using the same method, we obtain Finally, we discuss the sign of P (x, λ; ξ, η). For ξ ∈ Σ n−1 x,λ , η ∈ Σ n x,λ and λ > 0, it is easy to verify that Hence, we conclude that P (x, λ; ξ, η) = ln |ξ−x||η−ξ x,λ | λ|ξ−η| > 0.
To prove the calculus key lemmas for exponential functions, we need to recall the following key lemmas introduced by Li and Nirenberg in [7]. The earlier visions with stronger assumptions were first proved by Li and Zhu [9], and Li and Zhang [8].
) Let n ≥ 1 and µ ∈ R, and f ∈ C 0 (R n ). Suppose that for every x ∈ R n , there exists λ > 0 such that Then there are a ≥ 0, d > 0 andx ∈ R n , such that The following three calculus key lemmas are essential to carry out the moving sphere procedure.
Lemma 2.5. Let n ≥ 1 and µ ∈ R, and f ∈ C 0 (R n ). Suppose that for every x ∈ R n , there exists λ > 0 such that Then there are a ≥ 0, d > 0 andx ∈ R n , such that

By Lemma 2.3, we have
The lemma is proved.
Lemma 2.6. For n ≥ 1, if f is a function defined on R n + and valued in (−∞, +∞) satisfying Proof. The proof of this lemma is similar to that of Lemma 3.7 in [6] which extended Lemma 5.7 in [7] to the upper half space. For any z = (z , z n ) ∈ R n + , choose y i = (y , y i n ) with y = z , y i n > z n and y i n → z n as i → ∞. Choose b i > 1, so that and by the assumption of Lemma 2.6, we obtain f (z , z n ) ≤ f (y , z n ). Since y and z are arbitrary, we prove the lemma.
3. Proof of Theorem 1.2. We devote this section to completing Theorem 1.2 by the method of moving spheres in integral form.
Appendix. In appendix, we present the proof of Theorem 1.1. To achieve this aim, we first need to project HLS inequality (1) to B 1 by Kelvin transform as follows (see Corollary 5.1 in [6]): where p = 2(n−1) n+α−2 , t = 2n n+α and It is not easy to calculate the exact value of C e directly. Since we are interested in the limiting case α → n − , we only need to obtain an estimate of C e as α → n − . The following proposition provides an estimate of C e as α → n − . A similar proof of the estimate of C e can be found in [12]. Proposition 1. For n ≥ 2 and θ = n−α 2 , the following holds: C e (n, α) = C e (θ) = 1 + θ n − 1 ln(nω n ) + θ n ln B1 e In(ξ) dξ + o(θ), as θ → 0, where I n (ξ) = −2ω −1 n ∂B1 ln |ξ − η|dS η . Proof. For any ξ ∈ B 1 , there exists a positive constant C independent of ξ such that ∂B1 | ln |ξ − η||dS η ≤ C.
Hence, using the Taylor's expansion of C e (θ) with respect to θ at 0, we conclude (22).