Improved Beckner's inequality for axially symmetric functions on $\mathbb{S}^4$

We show that axially symmetric solutions on $\mathbb{S}^4$ to a constant $Q$-curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter $\alpha$ in front of the Paneitz operator belongs to $[\frac{473 + \sqrt{209329}}{1800}\approx0.517, 1)$. This is in contrast to the case $\alpha=1$, where a family of solutions exist, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on $ \mathbb{S}^2$. As a consequence, we prove an improved Beckner's inequality on $\mathbb{S}^4$ for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when $\alpha=\frac15$ by exploiting Pohozaev-type identities, and prove existence of a non-constant axially symmetric solution for $\alpha \in (\frac15, \frac12)$ via a bifurcation method.

Here S 4 is the 4-dimensional sphere, is the Paneitz operator on S 4 and α is a positive constant.The volume form dw is normalized so that The corresponding energy functional is defined in H 2 (S 4 ) as e 4u dw.
In what follows, we shall consider axially symmetric functions that are only dependent on ξ 1 .We shall show that (1.1) under axially symmetric setting admits only constant solutions when α ∈ [ 473+ or equivalently, One can refer to Section 2 for the detailed derivation of (1.6).By direct computations, we see that the corresponding functional I α (u) can be expressed as follows Here the function space is H 2 (−1, 1), which is the restriction of H 2 (S 4 ) in the set of functions axially symmetric about ξ 1 -axis and ξ 1 = x.
The set L is replaced by (1.7) L r = u ∈ H 2 (S 4 ) : u = u(x) and We conjecture that Theorem 1.1 holds for 1 2 < α < 1.Indeed, the lower bound 473+ can be improved slightly to 0.5145 (see discussions in Section 6).We believe that J 1/2 (u) ≥ 0 for u ∈ L, given the similar inequality for S 2 as shown in [13].
Now we define the following first momentum functionals on H 2 (S 4 ) e 4u ξ i dw) 2 .
As a consequence of Theorem 1.1, we have the following form of first Szegö limit theorem on S 4 for axially symmetric functions.
Concerning the classification of axially symmetric solutions at another critical parameter α = 1 5 , we have the following theorem.
5 and u is an axially symmetric solution to (1.1), then u must be constant.
Using a bifurcation approach and Theorem 1.1-1.3,we can also show the existence of nonconstant axially symmetric solution for α ∈ ( ) and a sequence of non constant solutions We also establish the following proposition concerning the centers of mass and first order momentums of solutions to (1.1).Proposition 1.5.If u solves (1.1), then S 4 e 4u ξ i dw = 0 and The paper is organized as follows.First, we list some preliminaries and integral identities in Section 2 which will be substantially used in the later context.Section 3 is devoted to the proof of Theorems 1.1-1.2.In Section 4, we derive various Pohozaev-type identities and employ them to validate Theorem 1.3 together with Proposition 1.5.In Section 5, we carry out a bifurcation analysis of (1.6) and its equivalent form, and prove Theorem 1.4 based on Theorems 1.1 and 1.3.The last section is devoted to some discussion of the improvement of the best constant for α.

Preliminaries and Integral Identities
In this section, we state several important preliminaries and integral identities which will be needed in the proof of Theorem 1.1.We begin by stating some basic facts on spherical geometry of S 4 .
Let θ i , i = 1, 2, 3, 4 denote the usual angular coordinates on the sphere with and define x = ξ 1 = cos(θ 1 ).Then the metric tensor can be given as follows: For axially symmetric functions, we have One further has that Thus, the Paneitz operator on S 4 can be expressed as for u = u(x).Then, we transform the original equation (1.1) on S 4 into an ODE (1.6).
Note that the eigenfunctions associated with Paneitz operator coincide with those associated with the Laplacian.It is natural to introduce Gegenbauer polynomials (see [23,Chapter 2.4]), which can be considered as a family of generalized Legendre polynomials.Let Note here F k is a sphere harmonic of degree k.Then it is readily checked that for x ∈ (−1, 1), [23,12]), .
We will focus on the gradient of u on the sphere throughout the rest of the paper.Define where u = u(x) is a solution to (1.1).Then we have the following decomposition using the orthogonal polynomials F k 's: We first derive a lemma concerning the constant term a 0 in (2.5).Lemma 2.1.If u is a critical point of I α whenever α = 1 , then the function G(x) belongs to H 2 (−1, 1) and satisfies that 1 −1 (1 − x 2 )G = 0.In other words, a 0 = 0. Proof.In view of equation (1.6), we have By differentiating (2.6), we further have Multiplying (2.8) by (1 − x 2 ) 2 and employing (2.6), we have We use integration by parts for the first term of (2.10): Similarly, for the last term in (2.10), one has We conclude from the (2.10)-(2.12) that 24 − 24 α 1 −1 Next, we state some important integral identities which will be used frequently in the proof of Theorem 1.1.Lemma 2.2.We establish the following equalities for G(x) = (1 − x 2 )u ′ where u is a solution of (1.6) and α > 0. (2.13) (2.16) recalling that β is defined in (2.5) and γ is given in (2.7).

Proof of Theorem 1.1
Inspired by [10] and [14], our basic strategy is to assume β = 0, and show that it leads to a contradiction with the range of α.It is fairly easy to see from (2.16) that if β = 0, then ∇u = 0, which shows that u is a constant.One important new ingredient is the surprising a priori estimate in Lemma 3.1 regarding the derivative of the gradient of u.
We now give the key estimate on the derivative of G, which is defined in (2.4).Note that the lemma is true for general α > 0.
), then we may assume without loss of generality that sup It is well known that ū(r) can be extended evenly and ū(r) then, one has A direct calculation shows that It is easy to see that lim x→1 G ′ (x) = 2ū rr (0).Note that lim r→0 ūr (r) = ūr (0) = 0 and lim r→0 ūr r = ūrr (0).
We can write The last inequality is insured by the fact that (3.3) implies Furthermore, by (3.4), By similar arguments, we obtain that near r = 0, Therefore, By similar calculations again, near r = 0, This ensures that Using (3.2) together with (3.5)-(3.7),we have Remark 3.1.When α = 1, there is a family of solutions u = − ln(1 − ax) to (1.6) for any a ∈ (0, 1).Straightforward computations show that the estimate in Lemma 3.1 is indeed optimal in general.
However, given some extra information, the estimate may be improved slightly (see the discussion in Section 6 below for details).
Proof.Using the facts that the first eigenvalue of Laplacian on S 4 is λ 1 = 4 as in (2.1) and the first eigenvalue of P 4 is λ 1 (λ 1 + 2) = 24, we obtain from Lemma 3.2 immediately that when α > 2/3, G must be constant 0 and hence u must be constant.
Proof of Theorem 1.1.We shall use higher order eigenfunctions in (2.1) to gain better estimate for α and prove the main theorem.We first define the following quantity From (2.16), Lemma 3.2 and the definition of G 2 , it is easy to see that (3.20) In what follows, we assume that β = 0. From (2.13) and (2.14), one has Hence one has In particular, we obtain It follows from α > Then one has After some straightforward computations, we obtain (3.25) Next, we fix an integer n ≥ 3.After some computations, we get A straightforward calculation shows .
Therefore, (3.28) is equivalent to We use the same technique as in (3.25) to obtain When n = 3, we derive from (3.27) that A direct calculation suggests that It is readily checked that We now claim that there exists some d n > 0 for n = 3 or 4, such that for α ∈ [α n+1 , α n ), We see from (3.29) and (3.32) that There is a contradiction.We are ready to prove the assertion (3.32).First, we study more accurately for the bound in (3.27) where Thus, One The previous arguments show that α < α 5 .This proves Theorem 1.1 with α ∈ ( 473+ √ 209329 1800 ≈ 0.51695, 1).The range of α for Theorem 1.1 to hold can be slightly improved to 0.5145 ≤ α < 1, see Section 6 for discussions.
Remark 3.2.The approach used in the case n = 3 or 4 for (3.32) does not work for n ≥ 5.The main obstacle is that cn contains a term involving −λ 2 n+1 , so we can not guarantee that the value of ω n is positive for n ≥ 5. Let us take n = 5 as an example.Some computations indicate that γ 5 ≤ 0.2994 and then ω 5 = A 5 − γ 5 B 5 ≈ −632, which shows that there does not exist such a d 5 that the assertion (3.32) holds for n = 5.Therefore, it seems impossible to get a contradiction similar to (3.39).
Next we shall show Theorem 1.2 as an immediate consequence of Theorem 1.1 and invariance of J 4 5 under a family of conformal transformations φ P,t , P ∈ S 4 , t > 0 of S 4 .
Given u ∈ H 2 (S 4 ) and t > 0, let We have the following invariance property of J Indeed, after a proper rotation, we may assume that P = P 0 .Letting a = 1−t 2 1+t 2 , we have e 4u ξ1 dw) 2 and for i = 2, 3, 4, 5 e 4u ξi dw.
When P = P 0 is chosen to coincide with the direction of the center of mass of e 4u , we also observe from the above proof that if we also choose a = − S 4 e 4u ξ1 dw S 4 e 4u dw .Then, for any u ∈ H 2 (S 4 ), there is a φ P,t such that v(ξ) = u(φ P,t (ξ)) + 5 4 2 ln |det(dφ P,t )|, ξ ∈ S 4 belongs to L.Moreover, we have that J α (u) = J α (v) for v ∈ L.
Then Theorem 1.2 follows immediately from Theorem 1.1 and Proposition 3.4.We note that a similar but more general Szegö limit theorem for u ∈ H 1 (S 2 ) is proven in [6] using a variational method with a mass center constraint, in combination with the improved Moser-Trudinger inequality in [13].In general, similar Szegö limit theorem should be true for S n , n ≥ 5 with α = 4   5   replaced by α = n n+1 , provided that an improved Beckner's inequality could be proven for α ≤ n n+1 .Note that a counter part of Proposition 3.4 always holds for general S n .

Pohozaev-type Identities and Classification Result
Pohozaev-type identities are very powerful tools in studying the symmetry of solutions to semilinear elliptic equations.They play a vital role in proving classification results (see, e.g., [11], [25]).Recently Shi et.al. [26] obtain several Pohozaev-type identities and apply them to prove the uniqueness of axially symmetric solution of mean field equation on S 2 for α.In this section, we first list several useful Pohozaev-type identities corresponding to solutions of (1.1), then we prove Theorem 1.3 based on these identities.
On the other hand, let Then (4.1) can be written as (4.2) By the Kazdan-Warner condition (1.5), one obtains Therefore, ( for all n ≥ 2 by the same method.Here for n even; ), for n odd.
Theorem 1.3 has been proven.
Note that critical points of I α (u), satisfy To apply Theorem 5.1, we define a nonlinear operator T : R × V → W as S 4 e 4u dw , Obviously, the operator T is well defined.After direct computations, one has S 4 e 4u dw .
Let S denote the closure of the set of nontrivial solutions of (5.2) It is clear that (5.2) and (5.1) are equivalent and a solution of (5.2).
Moreover, the range of the operator ∂ u T (ρ k , 0)) is given by and it has co-dimension 1.In addition, we have Proof.We can choose It is easy to compute that Then (5.3) follows from (2.1).From the orthogonal property (2.2), we deduce that R(∂ u T (ρ k , 0)) coincides with the orthogonal of ker(∂ u T (ρ k , 0)).
Note ker(∂ u F) = ker(∂ u T ).Differentiating ∂ u T with respect to ρ at the point (ρ k , 0), we get which, combined with the relation ).The lemma is proven.
For k ∈ N + , the following local bifurcation result is an immediate consequence of Theorem 5.1 and Lemma 5.3.
Remark 5.1.When k = 1, the bifurcation leads to the family of solutions u = − ln(1−ax), a ∈ (−1, 1) and ρ = 6.It is clear that (ρ k , 0) is not a transcritical bifurcation point for k odd since F k is an odd function and ρ ′ (0) = 0 in this case.It should be true that (ρ k , 0) is a transcritical bifurcation point for k even, we only need to check if k = 0 in this case, which can be confirmed for small k numerically.However, in this paper we only need to use the transcriticality of (ρ 2 , 0).
In order to analyze the global bifurcation diagram, we employ a global bifurcation theorem via degree arguments (see [19,24]) and also exploit special properties of solutions to (5.1).
Proposition 5.6.In Theorem 5.4, the bifurcation at (ρ k , 0) is global and satisfies the Rabinowitz alternative, i.e., a global continuum of solutions to (5.1) either goes to infinity in R × W or meets the trivial solution curve at (ρ m , 0) for some m ≥ 1 and m = k.
Next we state and prove the following more specific global bifurcation result regarding (5.1).
2) Similarly, for k ≥ 2, there exists a global continuum of solutions B − k which coincides in a small neighborhood of (ρ k , 0) with {(ρ k (ε), k is contained in N 2 and satisfies the boundedness for ρ in any fixed finite interval [ρ m , ρ M ] ⊂ (12, ∞).Furthermore, the improved Rabinowitz alternative holds.
3) Moreover, The global continuum of solutions B − 2 of (5.1) must be contained in the set ), and there exists a sequence of (ρ As an immediate consequence, there is a nontrivial solution to (5.1) for any ρ ∈ (12, 30).
Proof.To prove 1) and 2), we only need to first apply the general global bifurcation theory and then use A general compactness result ([22, Theorem 1.1]) says that the solutions to (5.1) can only blow up in L ∞ ([−1, 1]) at ρ = 6k for an positive integer k when (5.1) is considered as an fourth order Q-curvature type equation on S 4 , and k is the number of blowup points.(See also [15,Theorem 4.3] from a view point of constrained inequalities.)Since an axially symmetric solution can blow up at most two points at a finite parameter ρ, we must have k = 1, 2. Therefore, this leads to the boundedness of B + k , k ≥ 2 and B − k , k ≥ 3 for ρ in any fixed finite interval [ρ m , ρ M ] ⊂ (30, ∞).To prove 3), we note that u(x) = v(−x) is a solution to (5.1) if so is v(x), and u(x) is not an even function for u ∈ {u : This completes the proof.
Remark 5.2.The above theorem implies that B − 2 does not coincide with other bifurcation branches.It would be interesting to see whether the solution branches bifurcating from different points (ρ k , 0) coincide with each other or not, i.e., whether Proof of Theorem 1.4: Theorem 1.4 follows immediately from Theorem 5.7.This leads to the existence of a nontrivial solution to (1.6) for α ∈ ( 1 5 , 1 2 ).

Discussion
In this Section, we shall discuss some ideas to close the gap α ∈ ( 1 2 , 473+ √ 209329 1800 ).Note that Gui and Wei [14] used an induction method to show with the sequence λ n → ∞.So it follows 1 α − β → 0 as n → ∞, which leads to a contradiction.Following the arguments in [14], we divide (3.29) by λ n+1 to get (6.1) A direct calculation shows that (6.2) LHS of (6.1) ≤ 100β λ n+1 , which is the basic ingredient for the induction procedure in [14].Next, the major task is to find an appropriate d so that ).However, there does not exist such a constant d for (6.6) to hold.Hence, this method does not seem to yield the optimal constant α = 1 2 for this problem.Remark 6.1.We also intend to replace denominator in (6.3) and (6.4) by λ t n and λ t n+1 , respectively, for some t > 0. For this purpose we only need to slightly modify the previous procedure.After some calculations, (6.5) becomes Therefore, we need to show that for α ∈ ( 1 2 , 473+ √ 209329 1800 ) and n large, which suggests that t = 1 is the best choice, since λ n → ∞ as n → ∞.
We next will use some notations in Section 3 and assume that (6.17) 0.5165 ≤ α < 0.51696.On the other hand, we need to modify some inequalities in Section 3 by exploiting (6.15) instead of (3.1).First, the inequality (3.16) becomes 24 α − 12 1 −1 Here, B denotes B(α, β).Similarly, we have  Hence, α < 0.5145.Unfortunately, it does not seem possible to improve the estimate of α significantly in this way and recursively, due to (6.22), let alone to obtain the possible optimal constant α = 1 2 .
uP 4 F 2 dw = αλ 2 (λ 2 + 2) However, we do not know what the initial value n 0 should be, which is dependent on the choice of d.