Classification for positive solutions of degenerate elliptic system

In this paper, by using the Alexandrov-Serrin method of moving plane combined with integral inequalities, we obtained the complete classification of positive solution for a class of degenerate elliptic system.

1. Introduction. We consider the following degenerate elliptic system: where a > 1 is a constant, u = u(x, y), v = v(x, y) ∈ C 2 (R N +1 + ) with (x, y) ∈ R N × R + := R N +1 + . The main purpose of the present paper is the following: On the one hand, we show how the method of moving plane combined with integral inequalities, as developed in [4] (see also [9], [10], [6]) can be applied to produce simple proofs of Liouville type theorems for a class of degenerate elliptic systems with general nonlinearities, and no any boundary condition is imposed on the boundary y = 0. On the other hand, we give the complete classifications of positive solution for system (1). More precisely, we prove that system (1) has no positive solution when the nonlinearities are subcritical unless f (u, v) = mu p1 v q1 , g(u, v) = lu p2 v q2 with m, l being constants and p 1 +q 1 = N +2a+2 N +2a−2 , p 2 +q 2 = N +2a+2 N +2a−2 . Moreover, in the later case, all the positive solutions of (1) take the form u = a 1 U, v = b 1 U, where a 1 = ma p1 1 b q1 1 , b 1 = la p2 1 b q2 1 , and U > 0 is the unique positive solution (up to translation and scaling) of the following equation (see [7]): N +2a−2 = 0, U (x, y) ≥ 0, U (x, y) ∈ C 2 (R N +1 + ). (2)

YUXIA GUO AND JIANJUN NIE
As far as the system (1.1) is concerned, beside the difficulty caused by the degeneracy of the equations, the nonlinearities are assumed to be continuous, the method for classical solution can not be applied directly. Moreover, since the nonlinearities are coupled each other, we are not clear where we can start the method of the moving plane. Furthermore, in the degenerate case, the system is not rotation invariant and new maximal principles are needed. Fortunately, motivated by the work of Huang [7] and Huang and Li [8], based on the Kelvin's transform, we may use integral inequality in a narrow domain-as was used in Terracini in [9], [10] and Damascelli in [1] , [4] -which helps us to start the method of moving plane.
Then either (u, v) ≡ (C 1 , C 2 ) for some constants C 1 , C 2 with f (C 2 ) = g(C 1 ) = 0 or there exist positive constants m, l such that h(t) , where s is a positive parameter and x 0 is a point in R N , and a 1 , b 1 are constants satisfying . We remark that Theorem 1.1 can be extended to more general cases like (1) (see Section 3). For applications convenience, we have the following corollary.
Suppose that p ji , q ji ≥ 0, p ji + q ji ≤ N +2a+2 N +2a−2 , j = 1, 2, i = 1, 2, ..., n. Then (1.5) has no solutions unless p ji + q ji = N +2a+2 N +2a−2 , j = 1, 2, i = 1, 2, ..., n. Before the end of this introduction, we would like to mention that the nonexistence results obtained in our paper will lead to a priori estimates for positive solutions of some semi-linear degenerate elliptic system in bounded domains, which arising from the study of geometry problem. For the proof of priori bounds, one can use the blow up method, we refer to [5] for the corresponding analysis for Laplacian equation with Dirichlet boundary conditions and to [3] for a blow-up analysis for some mixed boundary problems in bounded domains. However, when the blow up point approach to the boundary, the problem becomes more complicated when the equation is degenerate and without any boundary conditions on the boundary, we refer to Huang [7] for the discuss for a single degenerate equation. For a general degenerate elliptic system, the problem will become more difficult, we will discuss this problem in forthcoming paper.
The paper is organized as follows. In Section 2, we discuss a simpler system (4) and give a detailed proof of Theorem 1.1. We consider the extension system (1) (Theorem 3.1) and (27) (Theorem 3.2) in Section 3. We will show that how the techniques used in section 2 allow us to deal with other more general problems. We pay our attention on the proof of the analogous of those key lemmas in Section 2 and the uniqueness form of positive the solutions. We also state and prove a non-existence result (Theorem 3.9) under stronger smoothness assumptions on f, g but with simpler other conditions.

2.
Preliminaries and the proof of Theorem 1.1. In this section, we first give some general facts for degenerate system (4). We begin with some notations and comments. Let u, v ∈ C 2 (R N +1 + ) be the solution of (4), set x N +1 = 2 √ y, we definē .
Combining Hölder inequality with the previous estimates, we have (21) Now we estimate the term Next we estimate the term I 2 . In view of the asymptotic behavior of w(x) at ∞ (see (10)), we have By using dominated convergence and Sobolev's inequality, letting → 0 in (21), we get The desired result follows immediately.
It follows that there exist λ 0 > 0, such that To proceed, we will need the following invariant of Maximum principle (see [7]). Consider the following elliptic operator: Then either u is a constant or u can not attain its maximum in B 1 , where B 1 is the unit ball in R N +1 . Lemma 2.5. Suppose that all the coefficients a ij , b i , a are the same as in Lemma If u attains its maximum at x 0 ∈ ∂B 1 , then either u is a constant or − ∂u ∂n | x=x0 < 0, where n is the outward normal to ∂B 1 at x 0 . Now we define Proof. By continuity, we see that Similarly, Combine with the previous arguments, we can deduce that W λ (x) ≤ 0 and Z λ (x) ≤ 0 in Σ λ for λ < Λ, which contradicts with the definition of Λ.
Before the proof of Theorem 1.1, we first have: Proposition 2.7. Let u, v and f, g be as in Theorem 1.1 and suppose that (u, v) is positive in R N +1 + . Let (w, z) be the Kelvin's transform of (u, v) centered at a point p. Then (w, z) is radially symmetric with respect to some point q in R N +1 . Moreover if h, k are not constants in (0, sup x∈R N +1 v(x)) and in (0, sup x∈R N +1 u(x)), respectively, then (w, z) is radially symmetric around the pole of Kelvin's transform, i.e q = p.
Proof. To prove that w and z are radially symmetric, we use the method of moving plane and prove the symmetry in every direction. Without loss of generality, for simplicity, we choose x 1 direction and prove that w, z are symmetric with respect to the x 1 direction.
If Λ = 0, we conclude by continuity that w(x) ≤ w 0 (x) and z(x) ≤ z 0 (x) for all x ∈ Σ 0 . We perform the moving plane procedure from the left and find a corresponding Λ l = 0 (again if Λ l < 0, an analogue to Lemma 2.6 shows that w and z are symmetric with respect to T l Λ ), from which we get w 0 (x) ≤ w(x) and z 0 (x) ≤ z(x) for all x ∈ Σ 0 . So w and z are symmetric with respect to the plane x 1 = 0. Therefore, if Λ = 0 for all directions, then w, z and hence u, v are radially symmetric around the point of the Kevin's transform. Now we suppose that Λ > 0, then we have w ≡ w Λ and z ≡ z Λ . This implies that w, z are regular at the origin, and hence u, v are regular at infinity. Since Note that for any x ∈ Σ Λ , |x| > |x Λ |, and h is nonincreasing, it follows from (25) Similarly h is constant in any right neighborhood of t = v( x |x| 2 ), x 1 < Λ, in particular it is true for t is close to 0 since t = v( x |x| 2 ) convergence to 0 at infinity. Therefore we conclude that if Λ > 0, h is constant in the range of v. By the same arguments, we have that if Λ > 0 then k is constant in the range of u.
Proof of Theorem 1.1. We first note that under the assumptions of the theorem, we have either By Proposition 2.7, we know that the Kelvin's transform w, z of u, v centered at any point p are radially symmetric around some point q. Moreover, if h, k are not constants on the value of u, v respectively, then p = q, this implies that u, v are also radially symmetric around p. It follows from the arbitrary choice of p that u, v are constants.
If h is constant, for example h = m, then f (t) = mt N +2a+2 N +2a−2 for t ∈ v(R N +1 + ). By using the similar arguments as in the proof of Lemma 2.6, one has either for any pole p, Λ = 0, which implies that v is constant, or there exists some pole p with Λ > 0. In this case, we have w ≡ w Λ in Σ Λ , and hence 0 is not a singular point of w and u is regular at infinity and decay at infinity as 1 |x| N +2a−2 . Repeating the arguments as we did in the proof of Proposition 2.7, one has h, k are constants and u, v are radially symmetric, moreover one has In fact, this can be easily proved. For simplicity we assume l = m = 1, then Noting that y ∈ R + , we obtain u = v. By the known result for a single equation 3. Extensions. In this section, by using the similar techniques in Section 2, we consider a more general system: and its extension version of We will use the same notations as the previous section.
Then either (u, v) ≡ (C 1 , C 2 ) for some constants C 1 , C 2 such that f (C 1 , C 2 ) = g(C 1 , C 2 ) = 0 or there exist m, l > 0 such that f (s, t) = ms p1 t q1 , g(s, t) = ls p2 t q2 , and u, v are radially symmetric and regular at infinity. Moreover u = a 1 U, v = b 1 U, where a 1 = ma p1 1 b q1 1 , b 1 = la p2 1 b q2 1 and U > 0 satisfies yU yy +aU y +∆ x U +U is nonincreasing in (s, t) , and f i (s, t) ≥ 0 but not equal to zero at the same time; (iv) g(s, t) is nondecreasing in s; is nonincreasing in (s, t); (vi) for i ≥ 1, there exist p 2i > 0, q 2i ≥ 0, p 2i + q 2i = N +2a+2 N +2a−2 such that g i (s, t) s p2i t q2i is nonincreasing in (s, t) and g i (s, t) ≥ 0 but not equal to zero at the same time.
Remark 3.3. Unlike Theorem 1.1, in Theorems 3.1 ( also for Theorem 3.2), we consider only positive solutions, that is u > 0, v > 0. In fact, since f (s,t) Let t → 0, one has f (s, 0) ≥ 0. Note that f (s, t) is nondecreasing in t, we get f (s, t) ≥ 0 for any s, t. Thus −yu yy − au y − ∆ x u ≥ 0, it follows that u ≡ 0 or u > 0. The same results are true for v. But we can not exclude the case of u ≡ 0, v > 0 or v ≡ 0, u > 0. And we consider only the positive solutions of (26) and (27).
But in Theorem 1.1, there are only two cases occur, namely, either u ≡ C 1 , v ≡ C 2 or u > 0, v > 0. As far as system (26) is concerned, the situation is different. Suppose that (u, v) is a nonnegative solution of (26). Then either u ≡ 0 or u > 0.
If z ≤ z λ , we have for some constant C λ .
, for w ≥ w λ . Therefore, by the similar arguments as in the proof of Lemma 2.2, we get On the other hand, by Sobolev inequality, one has Let → 0 in (34), we get the desired result. Proof. By Lemma 3.4, we proceed as the same as the proof of Lemma 2.3.
Proof. We prove that if w = w Λ at some point x 0 ∈Σ λ , then in a neighborhood of x 0 , w = w Λ , and hence w ≡ w Λ inΣ Λ , by continuity. Indeed, by continuity, we see that a neighborhood of x 0 . By using the same arguments as in the proof of Lemma 3.4 and the fact that if s > s , t > t , then Where C > 0 is a constant depending on x 0 , r. Hence By the Maximal principle, W Λ ≡ 0 in B r (x 0 ). Next, we claim that W Λ ≡ 0 implies Z Λ ≡ 0. In fact, by the equations (28) and (29), ). Since f (s, t) is nondecreasing in t, we deduce from the above inequality that On the other hand, It follows from (36) that By (38) and assumption (i) of Theorem 3.1, As a consequence of (37) and (39), we have z q1 = z q1 Λ , and hence z = z Λ since q 1 > 0.
Proof of Theorem 3.1. Suppose that (u, v) is a positive solution of (3.1). Make the Kelvin's transform around a point p ∈ R N +1 and define Λ = Λ(p, ν) as (24) for a direction ν ∈ R N +1 . If Λ(p, ν) = 0 for all p and ν, then (u, v) is radially symmetric with respect to all p ∈ R N +1 , and therefore must be constant. If Λ(p, ν) > 0 for some p and ν, then the corresponding Kelvin's transform (w, z) is radially symmetric with respect to a point q other than p and regular at the pole p, hence (u, v) is regular at infinity, that is u( |x| N +2a−2 as |x| → ∞. Without loss of generality, we assume that p is the origin. If a point x is not on the line pq, then we can find two points x − , x + such that The same is true for the function g. So the problem (3.1) reduces to where m, l are positive constants. Moreover (u, v) is regular at infinity. We can apply the method of moving plane to (u, v) and conclude that (u, v) is radially symmetric. It remains to determine the form of u, v. Suppose that (u, v) is radially symmetric with respect to the origin: Let (w, z) be the Kelvin's transform of (u, v) with the pole p = 0, (w, z) is radially symmetric with respect to some point q. Notice that for x = λp + e, λ ∈ R, (e, p) = 0, we have |x − p| 2 = (λ − 1) 2 |p| 2 + |e| 2 , | x − p |x − p| 2 + p| 2 = 1 + 2(λ − 1)|p| 2 (λ − 1) 2 |p| 2 + |e| 2 + |p| 2 , and w(x) = 1 is a function of λ and |e|, which implies that w is axisymmetric around the line op, and q must be on this line. If q = p, then (u, v) is radially symmetric with respect to p too, hence constant. If q = p, it follows from [3, lemma 7] that for some A, B, s = s(p, q) > 0, satisfies yU yy + aU y + N +2a−2 = 0 in R N +1 + and a 1 , b 1 satisfies Proof of Theorem 3.2. We proceed the similar technique arguments as in the proof of Theorem 3.1. For reader's convenience, in the following, we only give the proof of how w ≡ w Λ implies that z ≡ z Λ . In fact, by the equations (28) and (29), we have Since f (s, t) is increasing in t, we deduce that Moreover, By (44) and (45), for each i, it holds that That is combining with (42), we see that z = z Λ provided f i > 0 for some i.
We observe that in Theorem 3.1 and 3.2, we only assume that f (s, t), g(s, t) are continuous functions. If f (s, t), g(s, t) are Lipschitz continuous functions with respect to t and s respectively, then the same results of theorem 3.1 hold but with simpler assumptions. (ii)f (s, t) is increasing and locally Lipschitz continuous in t, that is provided that m ≥ t ≥ t > 0, m ≥ s ≥ 0. And g(s, t) is increasing and locally Lipschitz continuous in s in the sense that 0 ≤ g(s, t) − g(s , t) ≤ L 1 (m)(s − s ),

or
(ii)' f (s, t) is increasing in t and nondecreasing in s ; g(s, t) is increasing in s and nondecreasing in t.
If z ≥ z λ , . Proceeding the same arguments as in Lemma 3.4, we show that the integral inequality we need is true. The rest of the proof are the same as Theorem 3.1 with necessary modifications.