Semilinear elliptic system with boundary singularity

In this paper, we investigate the asymptotic behavior of local solutions for the semilinear elliptic system \begin{document}$ -\Delta \mathbf{u} = |\mathbf{u}|^{p-1}\mathbf{u} $\end{document} with boundary isolated singularity, where \begin{document}$ 1 , \begin{document}$ n\geq 2 $\end{document} and \begin{document}$ \mathbf{u} $\end{document} is a \begin{document}$ C^2 $\end{document} nonnegative vector-valued function defined on the half space. This work generalizes the correspondence results of Bidaut-Veron-Ponce-Veron on the scalar case, and Ghergu-Kim-Shahgholian on the internal singularity case.

is the Laplace operator with n ≥ 2. u := (u 1 , u 2 , · · · , u m ) is a nonnegative vector-valued function defined on a domain in R n , m ≥ 1, and p > 1. Now coupled systems of nonlinear Schrödinger equations like (1.1) are parts of several important branches of mathematical physics. They appear in the Hartree-Fock theory for Bose-Einstein double condensates, the fiber-optic theory, the langmuir waves theory for plasma physics, and in studying the behavior of deep water waves and freak waves in the ocean. A general reference in book form on such systems and their role in physics is by Ablowitz-Prinari-Trubatch [1]. Our interests is to obtain the asymptotic behavior of local solutions near the boundary singularity for the semilinear elliptic systems (1.1).
The corresponding internal isolated singularity for the systems had been very well understood. It is worth mentioning that the classification of the entire solutions plays an important part in the study of the internal isolated singularity. Using the method of moving spheres, Druet-Hebey-Vetóis [13] had proved that any nonnegative C 2 solutions of the strongly coupled critical elliptic system −∆u = |u| 4 n−2 u in R n , is of the form u = uc, where c ∈ S + := {θ = (θ 1 , θ 2 , · · · , θ m ) ∈ S m−1 : θ i ≥ 0, i = 1, 2, · · · , m}, S m−1 is the unit sphere in R m , and u is the positive solution of the Caju-doÓ-Silva Santos [9] obtained that if u ∈ C 2 (R n \{0}) is the positive solution of −∆u = |u| Recently, Ghergu-Kim-Shahgholian [14] obtained that u = 0 is the only nonnegative C 2 solution of −∆u = |u| p−1 u in R n for 1 < p < n+2 n−2 . Furthermore, they also classified the solutions in the punctured space, and proved that if u ∈ C 2 (R n \{0}) is the positive solutions of −∆u = |u| p−1 u in R n \{0}, with lim sup x→0 |u(x)| = +∞, then u is radially symmetric.
In the same paper [14], they derived the priori estimates that there exists a constant C depending only on n, m such that for the local positive solutions in C 2 (B 1 \{0}) of −∆u = |u| p−1 u in B 1 \{0}, where B 1 := {x ∈ R n : |x| < 1}, 1 < p ≤ n+2 n−2 . And they got the asymptotic radial symmetry, u(x) = (1 + O(|x|))u(|x|) near x = 0, where u(r) is the average of u over ∂B r . Utilizing the classification of solutions in the punctured space and the above asymptotic radial symmetry, they further studied in [14] the exact asymptotic behavior of local solutions around the singularity. In precise, either u can be continuously extended at the origin, or there exists a lower bound around the origin.
Especially, for the internal isolated singularity of the scalar case, see [3,8,16,20,22]. See also Li [21] and Han-Li-Teixeira [19] for conformally invariant fully nonlinear elliptic equations. The Sobolev critical exponent case p = n+2 n−2 is of particular interest, because the equation connects to the Yamabe problem and the conformal invariance, which leads to a richer isolated singularity structure.
Another motivation stems from that the scalar case of system (1.1) with a boundary singularity had been considered by a series of seminal papers. The authors proved in [10,11,18] respectively that the nonexistence of positive bounded solu- where R n + stands for the half space and p ≥ 1. Xiong lately removed in [25] the condition on boundness of the solutions. Based on the classification of the solution on the half space, the asymptotic behavior of the positive singular solutions in has been established by many works, where B + 1 := B 1 ∩ R n + and ∂ B + 1 := B + 1 ∩ ∂R n + . See Bidaut-Véron-Vivier [6] for 1 < p < n+1 n−1 , Bidaut-Véron-Ponce-Véron [4,5] for n+1 n−1 ≤ p < n+2 n−2 and Xiong [25] for p = n+2 n−2 . Under a blow up rate assumption: |x| 2 p−1 u(x) is bounded in B + 1 , then Bidaut-Véron-Ponce-Véron [4,5] obtained refined asymptotic behaviors for the supercritical case n+2 n−2 < p < n+1 n−3 . We refer to [4] and references therein for related results on boundary singularity. The fact that the exponent n+1 n−1 corresponds to n n−2 for the internal singularity was discovered by Brézis-Turner [7].
In the present paper, our primary interest is to analyze the behavior of the singular positive solutions in C 2 (B + 1 )∩C(B + 1 \{0}) for the semilinear elliptic system with Dirichlet boundary value conditions where u := (u 1 , u 2 , · · · , u m ), m ≥ 1 and p > 1. Via the method of moving spheres, we first classify the solutions on the half space, which will be used in the blow up analysis and is consistent with the work of [10,11,18].
The description of the boundary behavior of positive solutions of (1.2) is greatly helped by using a specific separable solutions of the same equation. This was early performed by Gmira-Véron [17] in 1991, and recently Porretta-Véron [24] also use the method for quasilinear Lane-Emden equations. Motivated by these, our next work is to look for the special positive solutions in By a direct calculation, w ∈ C 2 (S n−1 where ∆ s denotes the Laplace-Beltrami operator in the unit sphere S n−1 , S n−1 + := S n−1 ∩ R n + , and 2192 YIMEI LI AND JIGUANG BAO l n,p = 2(n − p(n − 2)) (p − 1) 2 .
Indeed, we need to obtain the existence of solutions for the Dirichlet problem (1.5) on semisphere. On the technical level, we shall transform (1.5) to a similar problem in an Euclidean space by stereographic projection. Inspired by the result of [14], we shall obtain a similar description Theorem 1.2 in this case, which is consistent with the work of [4].
(ii) Let n+1 n−1 < p ≤ n+3 n−1 , then (1.5) admits positive solution of the form w := wc, where c ∈ S + , and w is the positive solution of Remark that for the case m = 1, (1.5) admits a unique positive solution for n+1 n−1 < p < n+1 n−3 , and no positive solution for 1 < p ≤ n+1 n−1 or p ≥ n+1 n−3 . So far, we have no idea whether this conclusion holds or not for m ≥ 2, p > n+3 n−1 . We next establish a universal upper estimate near the singularity for (1.2) using doubling property (see [23,Lemma 5.1]). It is consistent with the work of [4] for the equation. In the upcoming sections of this paper, we focus on the exact asymptotic behavior of the local solutions of (1.2) for the subcritical case 1 < p < n+2 n−2 , which generalizes the work of [4] for the equation and [14] for internal isolated singularity of the same systems. In [14], the upper bound and the classification of solutions in R n \{0} play a key role in the asymptotic analysis. Next, we devote to studying the lower critical exponent case p = n+1 n−1 . Due to the multiplicity of components |u| p−1 u, the lower critical exponent case p = n n−2 is very different from the situation n n−2 < p < n+2 n−2 for the internal isolated singularity. To overcome this problem, the authors in [14] first give a more precise upper estimates and then obtain a asymptotic near the singularity. The similar problem also happens for the scalar case of system (1.1) with a boundary singularity. Motivated by the precise work, we also use a similar method to obtain the following theorem. where e := (1, 1, · · · , 1) ∈ R m and k is a positive constant depending only on n.
As for the case 1 < p < n+1 n−1 , we study a more general case involving boundary measures in a subsequent work by a very different method.
We note that u ∈ C 2 (R n + ) ∩ C(R n + ) if u i ∈ C 2 (R n + ) ∩ C(R n + ) for any i ∈ {1, 2, · · · , m}, m ≥ 1, and u is nonnegative if u i is nonnegative for any i ∈ {1, 2, · · · , m}. In a word, we say a vector u has some properties means that every component of u has the same properties. Moreover, if for a fixed i ∈ {1, 2, · · · , m}, u i vanishes somewhere, then since u i is super-harmonic and nonnegative, we know that u i vanishes everywhere on R n + . As a result, if u ≡ 0, then there is k ∈ {1, 2, · · · , m} such that after suitable rearrangement in the components of u, u i > 0 if and only if i = 1, 2, · · · , k. Without loss of generality, in our paper, u is nonnegative either u i is positive for any i ∈ {1, 2, · · · , m} or u = 0.
Our paper is organized as follows. Section 2 includes one proposition to prove Theorem 1.1. Section 3 is devoted to obtain the existence of solutions for (1.5). The upper bound for solutions of (1.2) will be provided in Section 4. In Section 5, we shall show the removability under some blow up assumption. Finally, we obtain the asymptotic symmetry in Section 6, including Theorem 1.4.

2.
Nonexistence of entire solutions. We now give the following proposition, which is about the monotonicity of positive solutions, that is, the positive solutions is monotone increasing in x n direction. Proof of Theorem 1.1. Proposition 2.1 implies that the monotonicity of u with respect to the variable x n . If we also have proved that u is bounded in R n + , then Moreover, the positive vector-valued function u ∞ ∈ C 2 (R n−1 ) satisfies Together with 1 < p ≤ n+2 n−2 < n+1 n−3 and the Liouville Theorem [14], we derive that u ∞ = 0. It is a contradiction. Then we complete the proof of Theorem 1.1. Now, we shall prove that u is bounded in R n + . If not, then there exist x k ∈ R n + , k = 1, 2, · · · , such that |u(x k )| → +∞ as k → +∞.
By Proposition 2.1 the monotonicity of u in the x n direction, we may assume that and denote Then By the definition of v k , we have Thus, we have 2 Using the equations satisfied by u, a direct calculation gives that 3), (2.4) and standard elliptic estimates, after extracting a subsequence, we have and |w ∞ (0)| = 1. By the Liouville Theorem [14], we have w ∞ = 0 for 1 < p < n+2 n−2 , and for the critical case p = n+2 n−2 , we obtain that for some z ∈ R n , r ≥ 0, and a unit nonnegative vector e ∈ R m . But we know from the monotonicity of w k that w ∞ must be non-decreasing in x n direction. This is a contradiction and the claim is proved.
Proof of Proposition 2.1. Suppose that for any λ > R, we have Then for any y ∈ R n + , and every a > 0, it follows that u(y) ≤ u x R ,R+yn+a/2 (y). Let R → +∞, we obtain by the above inequality that u(y) ≤ lim R→+∞ u x R ,R+yn+a/2 (y) = u(y 1 , y 2 , · · · , y n−1 , y n + a), It is a straightforward computation to show that Applying the Strong Maximum principle, we conclude that ∂u ∂xn is always zero or strictly positive in R n + . If ∂u ∂xn ≡ 0, together with the boundary condition, we conclude that u ≡ 0. It is a contradiction with the positive of the solution. Then we finish the proof.
In order to prove Proposition 2.1, it suffices to obtain (2.5). For the purpose, we introduceλ . First, we need the following lemma to guarantee that the set over which we are taking the supremum is non-empty such thatλ(R) is well defined.
Next we shall prove Equivalent to (2.5), the following job gives the proof of Lemma 2.2 and Lemma 2.3.

Proof of Lemma 2.2. A direct calculation gives that
We will make use of the narrow domain technique to conclude (2.6). Denote

Multiplying both sides of the equation by w − λ and integrating by parts in
(2.7)

YIMEI LI AND JIGUANG BAO
For any λ ∈ (R, 2R) and y ∈ B + λ (x R ), we have With the help of Hölder inequality and Sobolev inequality, we obtain that where S(n) is a constant depending only on n. By the Mean Value Theorem, there exists a θ ∈ (0, 1) such that Using the Hölder inequality and Sobolev inequality again, From the above argument, it follows that (2.7) implies It follows that . Hence, we complete the proof.
Proof of Lemma 2.3. We establish Lemma 2.3 by contradiction. Ifλ(R) < +∞ for some R, we shall prove that there exists a positive constant ε such that for all which contradicts with the definition ofλ(R).
It is clearly to see by the definition ofλ(R) that In order to obtain (2.8), we divided the region B + λ (x R ) into two parts, where δ is a small positive constant will be fixed later.
Since K 1 is compact, From the fact that the uniform continuity of u on compact sets, we can choose ε < δ sufficient small such that for any λ ∈ (λ(R),λ(R) + ε), Consequently, in view of the above argument, we obtain that for any λ ∈ (λ(R),λ(R) + ε), Now let us focus on the region K 2 . Using the narrow domain technique as that in Lemma 2.2, we can fix the value of δ small such that for any λ ∈ (λ(R),λ(R) + ε), Together with the above argument, we can see that the moving spheres procedure may continue beyondλ(R) where we reach a contradiction. And we complete the proof of Lemma 2.3.

3.
Existence of solutions of the PDES in S n−1 + . First, by applying stereographic projection, the upper semisphere S n−1 + is mapped into the unit ball of R n−1 . Then as for (1.5), the Laplace-Beltrami operator in S n−1 + can be reduced to the Euclidean Laplace operator, which is convenient to study. Then, we shall show that w = wc, where w is a positive solution of (1.6), c ∈ S + . Then we can complete the proof. For the purpose, let us summarize some well-known properties of this transformation: For any point ξ ∈ S n−1 \{S}, S is the south pole. Let g ξ be the straight line through the points ξ and S, and let g ξ ∩ {X ∈ R n |X n = 0} := (x, 0), x ∈ R n−1 . The mapping ξ → x is conformal and satisfies:

YIMEI LI AND JIGUANG BAO
Proof of Theorem 1.2. By applying stereographic projection ξ → x, S n−1 + is transformed into B 1 ⊂ R n−1 , and ∆ s is transformed into where ∆ now is the Euclidean Laplace operator in R n−1 . Suppose w is the solution of (1.5) and set (3.1) Then we can obtain that W(x) = W(|x|), and W r (x) ≤ 0 using the method of moving plane as [15], which turned out to be a very powerful technique in proving symmetry results for positive solutions of semilinear elliptic problems in symmetric domains. Then the symmetry proofs in this work depend on a number of technical steps. In order to guarantee the method of moving plane is effective, we need that the right-hand side of (3.1) is non-increasing in r. It is sufficient that That is, 1 < p ≤ n+3 n−1 for n ≥ 2; and p = n+1 n−3 for n ≥ 4. Define t := − ln |x| = − ln r and U(t) := |x| In particular, for any i ∈ {1, 2, · · · , m}, we have Hence, we obtain that for any i, j ∈ {1, 2, · · · , m}, where c is a constant. The following we shall show that c = 0. Suppose that c = 0. Without loss of generality, we can assume that c > 0. A direct calculation gives that Since there exist a positive constant M such that |U| ≤ M , integrating from 0 to t, we have It follows that if t sufficiently large. It is a contradiction. Hence, from the above argument, we have Therefore, we conclude that there exist constants c ji such that Fixed any i ∈ {1, 2, · · · , m}, define c i := (c 1i , c 2i , · · · , c mi ), we have Back to the definition of U, we have Since |c i |w i is a solution of (1.6), we know by the work of [4] that if 1 < p ≤ n+1 n−1 for n ≥ 2, and p = n+1 n−3 for n ≥ 4, then (1.6) admits no positive solution; if n ≥ 2, n+1 n−1 < p ≤ n+3 n−1 , then (1.6) admits a unique positive solution. Therefore, we complete the proof of this theorem.
then there exists a positive constant C which is independent of the solution such that
Proof. Assume by contradiction that (4.2) is false. Then, for every integer k ≥ 1, there exist 0 < r k < 1 2 , a solution u k of (4.1) with r = r k , and y k ∈ B + 2r k \B + r k such that and for any z ∈ D k and |z − x k | ≤ k/M k (x k ), It follows from (4.3) that for any k ∈ N + , and Combining (4.5), we obtain that for any y ∈ B k , that is, Therefore, w k is well defined in B k and a calculation gives that w k satisfies and |w k (0)| = 1. Moreover, from (4.4), we find that for all y ∈ B k , Since (4.7) Hence, the sequence w k is uniformly bounded, it follows that −∆w k is also uniformly bounded. Passing to a subsequence if necessary, we may assume that either If −M k (x k )(x k ) n → −∞, from interior elliptic estimates, we have for every q ∈ (1, ∞), w k W 2,q loc (R n ) ≤ C q . Then up to a subsequences w k converges locally uniformly in R n to some smooth function w ∞ such that w ∞ satisfies −∆w ∞ = |w| p−1 w ∞ in R n , and |w ∞ (0)| = 1, from the Liouville Theorem [14] that w ∞ = 0, it is a contradiction.
If −M k (x k )(x k ) n → −c ≤ 0, it follows from interior-boundary elliptic estimates that w k W 2,q loc ({y∈R n :yn≥−c}) ≤ C q . Then up to a subsequence w k converges locally uniformly in {y ∈ R n : y n ≥ −c} to some smooth function w ∞ such that w ∞ satisfies and |w ∞ (0)| = 1, from the Theorem 1.1 that w ∞ = 0, it is impossible. Hence, we finish the proof.
This establishes the result.

5.1.
The case for p > n+1 n−1 . then u ∈ C α (B + 1/2 ) for any α ∈ (0, 1) and u(0) = 0. Proof. In terms of spherical coordinates, we can write (1.2) as 3) it follows by assumption (5.1) that there exists T 0 > 0 such that v is bounded on [T 0 , +∞) × S n−1 + . Then by the Agmon-Douglis-Nirenberg estimates ( see [2] ) we have for any q ∈ (1, +∞), v W 2,q ((t−1,t+1)×S n−1 where t ∈ [T 0 + 3, +∞), and C depends on q but not on t. Together with v is bounded on [T 0 , +∞) × S n−1 + , we have v W 2,q ((t−1,t+1)×S n−1 for t ∈ [T 0 + 3, +∞), q is any number in (1, +∞). Multiplying both sides of the i-th components of the system (5.3) by v i and integrating by parts in S n−1 + , it follows that Then in order to obtain the theorem, for any t > 0, we define A direct calculation gives that for every t ∈ (0, +∞), Using Hölder's inequality we have Computing the derivative with respect to t on both sides of identity (5.5), we get From this identity and estimate (5.6), we deduce that On the other hand, since the first eigenvalue of the Laplace-Beltrami operator −∆ s in W 1,2 0 (S n−1 + ) is n − 1, The Hölder inequality gives that Combining with the above estimates, we have From the condition we have lim t→+∞ |v(t, ·)| = 0 uniformly in S n−1 + .
Hence, using the Maximum principle for X i and Z, we obtain that for any t ∈ (t 0 , +∞), Applying estimates (5.4) with q > n 2 , we have v W 2,q ((t−1,t+1)×S n−1 The Morrey's inequality implies that v L ∞ ((t−1,t+1)×S n−1 The estimate above implies that for |x| small, Together with ε > 0 is sufficient small, we conclude that u is Hölder continuous up to x = 0 and u(0) = 0.

5.2.
The case for p = n+1 n−1 . Proof. Since p = n+1 n−1 , then (5.3) can be written as where v defined as (5.2). For any t ∈ (0, +∞), i ∈ {1, 2, · · · , m}, let Multiplying both sides of the i-th components of the system (5.10) by t n−1 v i and integrating by parts in S n−1 + and by the fact that then together with (5.9), we conclude that for any ε > 0 there exists t 0 sufficiently large such that for every t ∈ (t 0 , +∞), On the other hand, applying Lemma A.2 in [4] we know that one solution of the equation satisfies the following asymptotic behaviors as t → +∞, (1)).
Let ε > 0 small enough and T 0 ≥ t 0 large enough such that Choose a positive constant C 0 ∈ R such that Then from X i (t) → 0 as t → +∞. Using the Maximum principle for X i and C 0 Z, we deduce that for t ∈ (T 0 , +∞), In particular, for t large, By the same argument as Theorem 5.1, we conclude that u is bounded and u can be continuously extended to 0. 6. Asymptotic.
6.1. The case for p = n+1 n−1 . In this part, with a blow up rate assumption, we shall show some asymptotic symmetry. Proof. From our assumption (6.1), it follows that there exists where v is defined as (5.2) in Theorem 5.1. It follows from the estimates (5.4) and the Morrey's inequality that for any γ ∈ (0, 1), . Furthermore, we also have by elliptic estimates that v C 2,γ ((t−1,t+1)×S n−1 for any γ ∈ (0, 1), t ∈ [T 0 +3, +∞). With the above estimates, to prove the theorem, we first to show that v t (t, ·) → 0 uniformly in S n−1 Next we can prove that v(t, ·) → 0 uniformly in S n−1 + as t → +∞. (6.4) As the first step, we shall show (6.3). Multiplying the system (5.3) by v t and integrating over S n−1 Combining v t vanishes on the boundary [T 0 , +∞) × ∂S n−1 Since (6.2) gives that for some constant C > 0. Hence, integrating (6.5) on (T 0 + 3, +∞), we obtain On the other hand, since p = n+2 n−2 , n − 2(p+1) p−1 = 0, we conclude that (6.3) follows. Indeed, by (6.2) we obtain that v t , v tt is uniformly bounded in (T 0 +3, +∞)×S n−1 + . It follows that there exists a constant M > 0 such that for t ∈ (T 0 + 3, +∞). If (6.3) not true, for a given ε > 0 there exist a sequences {t l } → +∞ such that S n−1 + |v t (t l , ·)| 2 dσ > ε and choose η = ε 4M such that for any t ∈ (t l − η, t l + η), We can assume that t l < t l+1 − η < t l+1 < t l+1 + η < t l+2 , then It is a contradiction with (6.6). For (6.4), we study the limit set of the trajectories of v i , i ∈ {1, · · · , m} and for simplicity, we just consider i = 1, namely the set where the closure is computed with respect to the usual norm in C 0 (S n−1 + ). Since Γ is the intersection of a decreasing family of closed connected subsets of C 0 (S n−1 + ), Γ i is closed and connected. In addition, from (6.2) and the Arzelà-Ascoli theorem that Γ i is also compact and nonempty.
For any w 1 ∈ Γ, let t k be a sequence of nonnegative real numbers such that t k → +∞ and v 1 (t k , ·) → w 1 uniformly in S n−1 + Clearly, w 1 is nonnegative and w 1 = 0 on ∂S n−1 + . For each k ≥ 1, let V k : (s, σ) ∈ [0, 1] × S n−1 + → R m be the function defined by V k (s, σ) = v(t k + s, σ). For every φ ∈ C ∞ 0 (S n−1 + ) and for every ε ∈ (0, 1), from the equation satisfied by v we have Since the sequence V k is bounded in C 1 ([0, 1] × S n−1 + ), passing to a subsequence if necessary, we may assume that for some continuous functions W, V k → W uniformly in [0, 1] × S n−1 + . Furthermore, the fact that v t → 0 uniformly as t → +∞ gives ε 0 S n−1 Therefore, we conclude that for every ε ∈ (0, 1), Dividing both sides by ε, and letting ε → 0, we get Then (6.4) follows from the fact that W(0, ·) = 0 by Theorem 1.2, and we finish the proof.
For t ∈ (0, +∞), define Y i (t) := v i,2 (t, ·) L 2 (S n−1 + ) . By the orthogonality between φ 1 and v i,2 , we have From the first equality and the Hölder inequality gives that As in the proof of Theorem 5.1, we have Since the second eigenvalue of the Laplace-Beltrami operator −∆ s in W 1,2 0 (S n−1 + ) is 2n, Then multiplying the i-th of (5.10) by v i,2 and integrating over S n−1 + , together with lim t→+∞ |v(t, ·)| = 0 uniformly in S n−1 + , we obtain that as in the proof of Theorem 5.1, for every ε ∈ (0, 1) there exists t 0 > 0 such that for every t ∈ (t 0 , +∞), where C 0 a positive constant such that We also have lim t→+∞ Z(t) = 0.
Since Y i (t) → 0 as t → +∞ and −(n + 1 − ε) < 0, applying the Maximum principle one deduces that for any t ∈ (t 0 , +∞), This gives the estimate for v 2 and we complete the proof of (6.8).
If the above estimates holds, then u must can be continuously extended to 0 in view of Theorem 5.2.