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A Liouville theorem of degenerate elliptic equation and its application

In this paper, we apply the moving plane method to some degenerate elliptic equations to get a Liouville type theorem. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic equations.

(1. 6) For the case p = 2, Lin and Xu got the similar results respectively in [16] and [23]. Wei-Xu extended the results to the case 2 ≤ 2p ≤ n, p ∈ Z in [22]. Chen-Li-Ou and Li proved the results for the most general case 0 < p < n 2 by the integral form of the moving plane method(moving sphere method) respectively in [3] and [19]. Applying Chen-Li-Ou's method to systems as in [4] and [18], one can also get the similar conclusions. Chang and Yang in [6] also extended this results to manifolds.
The main method used in solving problem (1.4) is the moving plane method which was first proposed by Alexandrov [1] and developed by Serrin [21], Gidas, Ni and Nirenberg [9,10]. Now moving plane method has been widely used in study of the symmetry of the positive solutions of many elliptic partial differential equations and systems. The key point of using the moving plane method in (1.4) is the conformal invariant property and the rotation invariant property of (1.4).
In our case, we also use the moving plane method and the conformal invariant property. To do so, we must establish some new maximal principles and overcome the difficult that (1.2) is not rotation invariant.
In this paper, we obtain the following results for (1.1).
It is easy to see that for a = k 2 , k ∈ N + , Theorem 1.1 is exactly the result of (1.4) in R n+k . For general a > 1, we may consider Theorem 1.1 as the extension of the results of (1.4) to R n+2a with real dimension n + 2a.
As an application of Theorem 1.1, we also derive a priori bounds for positive solutions of some semi-linear degenerate elliptic equations which arising from the study of geometry, (1.7) Let φ ∈ C 2 (N (∂Ω)) be the defining function of ∂Ω, namely, and that near ∂Ω for the eigenvalues of (a ij ) λ 1 and λ 2 , there hold, for some constant c 0 , Then it follows that |u| L ∞ ≤ C. (1.13) The invariance of g(x) is proved in [13]. The numerator of g(x) is the well-know Fichera number. The concept of Fichera number is very important when we deal with degenerate elliptic problems with boundary characteristic degenerate. It indicates whether we should impose boundary condition in such case. This fact was first observed by M.V.Keldyš in [15] and developed by Fichera in [7,8]. The Fichera number also affects the regularities of the solutions up to the boundary, see [13]. For more details of Fichera number, refer to [20].
Remark 1.2. It might be hard to understand that the nonlinearity of f (x, u) should be related to g(x). We can take equation (1.1) for instance to explain why this happens. In this situation, φ = y, f = u α . It is easy to see g(x, 0) = a − 1 by a direct computation. Theorem 1.1 tells us that the existence of non-trivial positive solution depends on the nonlinear power α which is involved in a. When we use blow up method to get a priori estimates of (1.7), one of the limit cases is (1.1) as the blow up point approaching the boundary. It is nature that the the nonlinearity of f (x, u) should be related to g(x) if we want to get the a priori bounds.
The proof of Theorem 1.2 mainly follows the blow up method used in [12]. The mainly difficulty we encounter is the case when the blow-up point approach to the boundary. This case becomes complicated with the degeneracy of the equation on the boundary and without boundary condition. We should take a suitable transformation of coordinates to make the limit equation exists and establish some regularities estimates up to the boundary to guarantee the point-wise convergence.
The present paper is organized as follows. In Section 2, we establish some lemmas which are similar to Lemma 2.3 and Lemma 2.4 in [2], and necessary for utility of the moving plane method. In Section 3, we shall use the moving plane method to prove Theorem 1.1. In Section 4, as an application of Theorem 1.1, we derive a priori bounds for positive solutions of some semi-linear degenerate elliptic equations.

Preliminary Results
In this section we collect some preliminary results which will be needed for our later analysis.
Noting that u ∈ C 2 (R n+1 + ), we must have This allows us to extendū to the lower half-space bȳ Consider the following elliptic operator All the coefficients a ij (x), b i (x), a(x) ∈ C(R n+1 ), a(x) ≥ 0 and (a ij ) is a positive definite matrix. Then we shall have the following two lemmas.
Then either u is a constant or u can not attain its minimum in B 1 .
Proof. It suffices to prove the second case. Without loss of generality, we may assume that u attains its minimum at the origin. Denote B r 2 (P ) has the same center as B r (P ) but half radius. Set Σ = B r (P )\B r 2 (P ). We consider Then we have for some positive constant c 0 > 0 if we take β large enough. Now let v = u + ǫh, then Lv < 0. This implies that v must attain its minimum on the boundary of Σ. Consider v on the boundary of Σ, (i) on ∂B r (P ), noting that h| ∂Br (P ) = 0, u| ∂Br(P ) ≥ u(0), then we have u + ǫh| ∂Br (P ) ≥ u(0).
If u attains its minimum at x 0 ∈ ∂B 1 , then either u ≡ const or Proof. Assume that u is not a constant. By Lemma 2.1, u(x) can not attain its minimum in B 1 . If u attains the minimum at x 0 ∈ ∂B 1 \{x n+1 = 0}, it is the immediate consequence of the standard Hopf' lemma. Without loss of any generality, we assume that v attains its minimum at if we choose β large enough. Choosing ǫ > 0 small enough, one can get This completes the proof of the present lemma.
Turn back to (2.1) and consider From the definition of v, we will have the following asymptotic behavior at ∞ Next we generalize the important lemmas which are essential for the application of moving plane method to (1.4) in [2] to the equation (1.1) studied in the present paper. Denote Then there hold the following lemmas Lemma 2.3. Let v be a function in a neighborhood of infinity satisfying the asymptotic expansion (2.5). Then there exist two positive constants R, Hence Then there exist two constants ǫ and S > 0 such that (2.8) We claim: there exists δ > 0 so small that Here e i means the vector whose i-th coordinate is 1 and others are 0. By Lemma 2.2, we see that It is easy to see that Therefore (2.9) is proved by the maximum principle. In particular, Combining this with the asymptotic expansion yields as |h| < δ 4C and |x| is large. This proves the first part of the present lemma. As for the second part, from the asymptotic expansion and the result of the first part, it follows that and |λ − λ 0 | is sufficiently small compared to ǫ. This completes the proof of the present lemma.
In all the above arguments, the discussion is always carried out outside a neighborhood of the origin. Now let us investigate the behavior of v in this neighborhood. The following idea mainly comes from [17] and [5].
) is a positive solution to the following problem with n + 2a > 2, , with 0 < s < 1 and a suitable constant l(s), From the definition of h s (x), there holds By the maximum principle we have for any fixed It is easy to see l(s) = − (m 1 + 1)s n+2a−2 1 − s n+2a−2 → 0 as s → 0. Thus passing to the limit s → 0, we have proved v(x) ≥ m 1 . This finishes the proof of the present lemma.
3 The proof for Theorem 1.1 Now we can prove Theorem 1.1. Proof of Theorem 1.1: Set By Lemma 2.3 and Lemma 2.5, we have |λ 0 | < ∞. Also we can claim that Without loss of generality, we may assume λ 0 > 0. Since for the case λ 0 = 0, we can start the moving plane from −∞ and stop at x 1 = λ 1 . If λ 1 < 0, we can prove v(x) = v(x λ1 ) by the same arguments as we do in the case λ 0 > 0. Otherwise λ 1 = 0, the claim (3.1) holds immediately. Now we turn to prove the claim (3.1) for Also from the definition of λ 0 and Lemma 2.4, one can choose λ k ↑ λ 0 as k → ∞ such that Noting Lemma 2.5 and the continuity of v, we can see that if we choose r small enough and k large enough. This implies that It is easy to see that ∃x k ∈ σ k such that w k (x k ) = inf w k (x). Next we consider x ∞ = lim k→∞ x k in two cases This proves the assertion (3.1). If α < n+2a+2 n+2a−2 , we have τ > 0. To prove the radial symmetry of v, one should take a transfor- ). It follows that, There is a singularity at 0, and hence λ 0 must be 0. Notice that (3.2) is rotation invariant about This implies thatū Repeating the above arguments, similarly we havē . In fact, b ′ can be chosen arbitrarily, thusū must be a constant. This means thatū ≡ 0. Now we consider the case α = n+2a+2 n+2a−2 or τ = 0. By the same arguments as we did in the case τ > 0, there exists λ = (λ 1 , ..., λ n+1 ) such that In fact, λ n+1 must be 0. Otherwise, it follows that It shows that for the fixed x ′ , v is periodic with respect to x n+1 with period 2λ n+1 . This means that v must vanish which is impossible. For λ ′ = (λ 1 , ..., λ n ), we have two cases. ( is radially symmetric with respect to the origin. (2) λ ′ = 0: This means that 0 is not the symmetric center of v, v must be C 2 at 0. In other words,ū(x) has the similar asymptotic behavior at ∞ as v(x). This allows us to apply the moving plane method toū(x) directly to obtain thatū(x) is radially symmetric with respect to some point b ∈ R n+1 , b n+1 = 0.
The above arguments show thatū(x) is radially symmetric with respect to a point b ∈ {b n+1 = 0}. Now we can follow the arguments of Section 3 in [3], then we can complete the proof of Theorem 1.1. Comparing Theorem 1.1 with (1.5), we can regard (1.1) as an equation defined in dimension n + 2a. Therefore, we can consider the following more general equation for some x 0 ∈ R n and t ≥ 0.
The proof of Theorem 3.1 is just the same as the proof of Theorem 1.1, as we can easily establish the similar lemmas as in Section 2 for (3.4). Thus we omit the details here.

An application to a priori estimates of semi-linear degenerate elliptic equations
The proof for Theorem 1.2: our proof is by contradiction and uses a scaling argument reminiscent to that used in the theory of Minimal Surfaces, also refer to [12]. If (1.13) is false, we can get a sequence u k ∈ C 2 (Ω) ∩ L ∞ (Ω) such that (4.1) Hence, we can find x k ∈ Ω →x ∈Ω as k → ∞ such that u k (x k ) ≥ M k 2 . Next we shall distinguish two cases to investigate.
Dividing both sides of (4.10) by A 22 , one can get in H k We must take care of the limit of , then (4.14) changes to It is important to show that J k can be chosen arbitrarily large as k → ∞. Since p ∈ H k , it follows that From y k 2 → 0 and Since Also, we have v k (0) ≥ 1 2 . As for any R, we can choose k large enough such that B R (0) ⊂ J k . Thus (4.15) is uniformly elliptic in B R (0) with uniformly bounded coefficients. This allows us to follow the same steps in Case 1. Namely, passing to limit k → ∞, we have if we notice that for q ∈ B R (0), Therefor, (4.19) gives rise a contradiction.
Then for r suitable small, we have (4.23) with |D j η ǫ | ≤ C j ǫ −j for p 2 ∈ (ǫ, 2ǫ). Denote ψ r,ǫ = ψ r η ǫ . Multiplying both sides of (4.21) by ψ r,ǫ u and integrating by parts, we can get Now we estimate the terms on the right side of (4.24). The first term, ∂ψ r,ǫ ∂p 2 u 2 ≤ |∂ p2 ψ r |η ǫ u 2 + |∂ p2 η ǫ |ψ r u 2 ≤ C 1 + C 2 2ǫ ǫ ǫ −1 dp 2 ≤ C, (4.25) where C is a constant only depending on the quantities in (4.22). Also The second term, The last term, Then by Lemma 4.1, one can getb 2 (p 1 , 0) ≥ b > 2. All the coefficients of (4.14) are By a linear change of coordinates and a stretching of coordinates, we have that From the assumption of Theorem 1.2, it follows that 2 <b ≤ a and By Theorem 5.1, we see that v ∈ C 2 (R 2 + ) and have v ≡ 0 which follows from Theorem 1.1. This is a contradiction to v(0, c) > 0. This ends the proof of Theorem 1.2.

Appendix
In the present Appendix, we shall give a result about the regularity of solutions to some degenerate elliptic equation in [14]. For the convenience of readers, we shall give a brief proof for it. We shall use the notations in [14]. Define I q (v) and I β (v) by: where Λ 1 is a singular integral operator with the symbol σ(Λ 1 ) = |ξ|. Also we say a function v(x, y) inĊ α (R n+1 Let ψ ∈ C ∞ c (R n+1 ) be a cutoff function with ψ(x, y) = 1 as |x| ≤ 1/2, y ≤ 1/2 and ψ = 0 as |x| ≥ 1 or |y| ≥ 1. Set ψ r (x, y) = ψ( x r , y r ).