Symmetry and monotonicity properties of singular solutions to some cooperative semilinear elliptic systems involving critical nonlinearity

We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearity in $\mathbb{R}^n$.


introduction
The aim of this paper is to investigate symmetry and monotonicity properties of singular solutions to some semilinear elliptic systems. In the first part of the paper we start by considering the following semilinear elliptic system where Ω is a bounded smooth domain of R n with n ≥ 2 and i = 1, ..., m (m ≥ 2). The technique which is mostly used in this paper is the well known moving plane method which goes back to the seminal works of Alexandrov [1] and Serrin [39]. See also the celebrated papers of Berestycki-Nirenberg [5] and Gidas-Ni-Nirenberg [23]. Such a technique can be performed in general domains providing partial monotonicity results near the boundary and symmetry when the domain is convex and symmetric. For simplicity of exposition we assume directly in all the paper that Ω is a convex domain which is symmetric with respect to the hyperplane {x 1 = 0}. The solution has a possible singularity on the critical set Γ ⊂ Ω. When u 1 = · · · = u m system (1.2) reduces to a scalar equations that was already studied in [19,37]. The moving plane procedure for semilinear elliptic system has been adapted first by Troy in [45] where he considered the cooperative system (1.1) with Γ = ∅ (see also [16,17,36]). This technique was also adapted in the case of cooperative semilinear systems in half spaces by Dancer in [15] and in the whole space by Busca and Sirakov in [9]. For the case of quasilinear elliptic system in bounded domains we suggest [34].
Moreover, motivated by [28], through all the paper, we assume that the following hypotheses (denoted by (h f i ) in the sequel) hold: (h f i ) (i) f i : R m + → R are assumed to be C 1 functions for every i = 1, ..., m. (ii) The functions f i (1 ≤ i ≤ m) are assumed to satisfy the monotonicity (also known as cooperative) conditions ∂f i ∂t j (t 1 , ..., t j , ..., t m ) ≥ 0 for i = j, 1 ≤ i, j ≤ m.
In this paper the case of singular nonlinearities for systems is not included, while it was considered in the case of scalar equations, see [19]; about these problems we have also to mention the pioneering work of Crandall, Rabinowitz and Tartar [14] and also [8,12,21,27,43] for the scalar case. It would be interesting to consider in future projects a more general class of nonlinearities. In particular it would be interesting to study problems involving singular nonlinearities as in the scalar case, using some techniques developed in [12,21].
Since we want to consider singular solutions, the natural assumption in our paper is .., m and thus the system is understood in the following sense: for every i = 1, ..., m.
Remark 1.1. Note that, by the assumption (h f i ), the right hand side in the system (1.2) is locally bounded. Therefore, by standard elliptic regularity theory, it follows that u i ∈ C 1,α loc (Ω \ Γ), where 0 < α < 1. We just remark that, in 1968, E. De Giorgi provided a counterexample showing that the scalar case is special and the regularity theory does not work in general for elliptic systems, see [18], but in the case of equations involving Laplace operator, Schauder theory is still applicable.
Under the previous assumptions we can prove the following result: Let Ω be a convex domain which is symmetric with respect to the hyperplane {x 1 = 0} and let (u 1 , ..., u m ) be a solution to (1.1), where u i ∈ H 1 loc (Ω \ Γ) ∩ C(Ω \ Γ) for every i = 1, ..., m. Assume that each f i fulfills (h f i ). Assume also that Γ is a point if n = 2 while Γ is closed and such that The technique developed in the first part of the paper and in [19,20,37] (see also [33] for the nonlocal setting) is very powerful and can be adapted to some cooperative systems in R n involving critical nonlinearity. Papers on existence or qualitative properties of solutions to systems with critical growth in R n are very few, due to the lack of compactness given by the Talenti bubbles and the difficulties arising for the lack of good variational methods. We refer the reader for this kind of systems [9,13,24,25,26,35]. The starting point of the second part of the paper is the study of qualitative properties of singular solutions to the following m × m system of equations where i = 1, ..., m, m ≥ 2, n ≥ 3 and the matrix A := (a ij ) i,j=1,...,m is symmetric and such that a ij = 1 for every i = 1, ..., m. This kind of system, with Γ = ∅, was studied by Mitidieri in [31,32] considering the case m = 2, A = 0 1 1 0 and it is known in the literature as nonlinearity belonging to the critical hyperbola.
If u 1 = u 2 = · · · = u m then (1.3) reduces to the classical critical Sobolev equation that can be found in [19,37]. If Γ reduces to a single point we find the result contained in [44], while if Γ = ∅ then system (1.5) reduces to the classical Sobolev equation (see [11]). For existence results of radial and nonradial solutions for (1.3), we refer to some interesting papers [24,25]. We want to remark that in [24,25] the authors treat the general case of a matrix A in which its entries a ij are not necessarily positive and this fact implies that it is not possible to apply the maximum principle. As remarked above the natural assumption is u i ∈ H 1 loc (R n \ Γ) ∀i = 1, ..., m and thus the system is understood in the following sense: for every i = 1, ..., m.
What we are going to show is the following result: Assume that the matrix A = (a ij ) i,j=1,...,m , defined above, is symmetric, a ij ≥ 0 for every i, j = 1, ..., m and it satisfies (1.4). Moreover at least one of u i has a non-removable 1 singularity in the singular set Γ, where Γ is a closed and proper subset of {x 1 = 0} such that Cap 2 R n (Γ) = 0. 1 Here we mean that the solution (u 1 , ..., u m ) does not admit a smooth extension all over the whole space. Namely it is not possible to findũ i ∈ H 1 loc (R n ) with u i ≡ũ i in R n \ Γ, for some i = 1, ..., m.
Then, all u i are symmetric with respect to the hyperplane {x 1 = 0}. The same conclusion is true if {x 1 = 0} is replaced by any affine hyperplane. If at least one of u i has only a nonremovable singularity at the origin for every i = 1, ..., m, then each u i is radially symmetric about the origin and radially decreasing.
Another interesting elliptic system involving Sobolev critical exponents is the following one: where α, β > 1, α + β = 2 * := 2n n−2 (n ≥ 3) The solutions to (1.7) are solitary waves for a system of coupled Gross-Pitaevskii equations. This type of systems arises, e.g., in the Hartree-Fock theory for double condensates, that is, Bose-Einstein condensates of two different hyperfine states which overlap in space. Existence results for this kind of system are very complicated and the existence of nontrivial solutions is deep related to the parameter α, β and n. This kind of system (1.7) with Γ = ∅ was studied in [2,3,4,35,38,41]. In particular in [35] the authors show a uniqueness result, for least energy solutions, under suitable assumptions on the parameters α, β and n, while in [13] the authors study also the competitive setting, showing that the system admits infinitely many fully nontrivial solutions, which are not conformally equivalent. Motivated by their physical applications, weakly coupled elliptic systems have received much attention in recent years, and there are many results for the cubic case where Γ = ∅, α = β = 2 and 2 * is replaced by 4 in low dimensions n = 3, 4, see e.g. [2,3,4,29,30,40,41]. Since our technique does not work when 1 < α < 2 or 1 < β < 2, here we study the case α, β ≥ 2 and n = 3 or n = 4, since we are assuming that α + β = 2 * . Theorem 1.4. Let n = 3 or n = 4 and let (u, v) ∈ H 1 loc (R n \ Γ) × H 1 loc (R n \ Γ) be a solution to (1.7). Assume that the solution (u, v) has a non-removable 2 singularity in the singular set Γ, where Γ is a closed and proper subset of {x 1 = 0} such that Moreover let us assume that α, β ≥ 2 and that holds α + β = 2 * . Then, u and v are symmetric with respect to the hyperplane {x 1 = 0}. The same conclusion is true if {x 1 = 0} is replaced by any affine hyperplane. If at least one between u and v have only a nonremovable singularity at the origin, then (u, v) is radially symmetric about the origin and radially decreasing.
When the paper was completed we learned that the case of bounded domains was also considered in [7] (see [6]), obtaining similar results.

Notations and preliminary results
We need to fix some notations. For a real number λ we set which is the reflection through the hyperplane T λ := {x 1 = λ}. Also let Since Γ is compact and of zero capacity, u i is defined a.e. on Ω and Lebesgue measurable on Ω for every i = 1, ..., m. Therefore the functions are Lebesgue measurable on R λ (Ω). Similarly, ∇u i and ∇u i,λ are Lebesgue measurable on Ω and R λ (Ω) respectively.
In the same spirit of [19] we recall some useful properties of the 2-capacity. It is easy to see that, if Cap 2 R n (Γ) = 0, then Cap 2 R n (R λ (Γ)) = 0. Another consequence of our assumptions is To this end we consider the following Lipschitz continuous function and we set where we have extended ϕ ε by zero outside B λ ε . Clearly ψ ε ∈ C 0,1 (R n ), 0 ≤ ψ ε ≤ 1 and Now we set γ λ := ∂Ω ∩ T λ . Recalling that Ω is convex, it is easy to deduce that γ λ is made of two points in dimension two. If else n ≥ 3 then it follows that γ λ is a smooth manifold of dimension n − 2. Note in fact that locally ∂Ω is the zero level set of a smooth function g(·) whose gradient is not parallel to the x 1 -direction since Ω is convex. Then it is sufficient to observe that locally ∂Ω ∩ T λ ≡ {g(λ, x ′ ) = 0} and use the implicit function theorem exploiting the fact that ∇ x ′ g(λ, x ′ ) = 0. This implies that Cap 2 R n (γ λ ) = 0, see e.g. [22]. So, as before, Cap 2 I λ τ (γ λ ) = 0 for any open neighborhood of γ λ and then there exists ϕ τ ∈ C ∞ c (I λ τ ) such that ϕ τ ≥ 1 in a neighborhood I λ σ with γ λ ⊂ I λ σ ⊂ I λ τ . As above, we set (2.13) .., m. We will prove the result by showing that, actually, it holds w + i,λ ≡ 0 for i = 1, ..., m. To prove this, we have to perform the moving plane method.
Now we are ready to prove an essential tool that we will use to start the moving plane procedure.
Lemma 3.2. Under the assumptions of Theorem 1.2, let a < λ < 0. Then w + i,λ ∈ H 1 0 (Ω λ ) for every i = 1, ..., m and where |Ω| denotes the n−dimensional Lebesgue measure of Ω and C i is a positive constant depending only by f i .
Proof. For ψ ε as in (2.12) and φ τ as in (2.13), we consider the functions ϕ i defined in Lemma 3.1. In view of the properties of ϕ i , stated in Lemma 3.1, and a standard density argument, we can use ϕ i as test function in (1.2) and (3.14) so that, subtracting, we get Exploiting Young's inequality in the right hand side of (3.18), we get that The last term of the right hand side of (3.19) can be rewritten as follows Using the fact that f i are C 1 functions (h f i ) − (i) and they satisfy ( Now compiling all the previous estimates and exploiting Young's inequality in the right hand side of 3.21 we obtain Adding all the equations of (1.3) we get Taking into account the properties of ψ ε and φ τ , we see that By Fatou Lemma, as ε and τ tend to zero, we have (3.17). To conclude we note that , as ε and τ tend to zero, by definition of ϕ i for every i = 1, ..., m. Also, ∇ϕ → ∇w + i,λ in L 2 (Ω λ ), by (3.16). Therefore, w + i,λ in H 1 0 (Ω λ ), since ϕ i ∈ H 1 0 (Ω λ ) again by Lemma 3.1, for every i = 1, ..., m, which concludes the proof.
Proof of Theorem 1.2. We define for all t ∈ (a, λ] and for every i=1,...,m.} and to start with the moving plane procedure, we have to prove that Step 1 : Λ 0 = ∅. Fix λ 0 ∈ (a, 0) such that R λ 0 (Γ) ⊂ Ω c , then for every a < λ < λ 0 , we also have that R λ (Γ) ⊂ Ω c . For any λ in this set we consider, on the domain Ω, the function where φ τ is as in (2.13) and we proceed as in the proof of Lemma 3.2. That is, by Lemma 3.1 and a density argument, we can use ϕ i as test function in (1.2) and (3.14) so that, subtracting, we get Exploiting Young's inequality and the assumption (h f i ), then we get that Taking into account the properties of φ τ , we see that We therefore deduce that By Fatou Lemma, as τ tends to zero, we have where C i,p (·) is the Poincaré constant (in the Poincaré inequality in H 1 0 (Ω λ )). Since C i,p (Ω λ ) → 0 as λ → a, we can find λ 1 ∈ (a, λ 0 ), such that ∀λ ∈ (a, λ 1 ) and for every i = 1, ..., m, so that by (3.28), we deduce that for λ close to a, which implies the desired conclusion Λ 0 = ∅.
Step 2: here we show that λ 0 = 0. To this end we assume that λ 0 < 0 and we reach a contradiction by proving that u i ≤ u i,λ 0 +ν in Ω λ 0 +ν \ R λ 0 +ν (Γ) for any 0 < ν <ν for some smallν > 0 and for every i = 1, ..., m. By continuity we know that Since Ω is convex in the x 1 −direction and the set R λ 0 (Γ) lies in the hyperplane of equation Therefore, by the strong maximum principle we deduce that u i < u i,λ 0 in Ω λ 0 \ R λ 0 (Γ) and for every i = 1, ..., m.
Step 3: conclusion. Since the moving plane procedure can be performed in the same way but in the opposite direction, then this proves the desired symmetry result. The fact that the solution is increasing in the x 1 -direction in {x 1 < 0} is implicit in the moving plane procedure. Since u has C 1 regularity, the fact that ∂ x 1 u i is positive for x 1 < 0 follows by the maximum principle, the Höpf lemma and the assumption (h f i ).

Proof of Theorem 1.3
Proof of Theorem 1.3. We first note that, thanks to a well-known result of Brezis and Kato [10] and standard elliptic estimates (see also [42]), the solution (u 1 , ..., u m ) to (1.3) is smooth in R n \ Γ. Furthermore we observe that it is enough to prove the theorem for the special case in which the origin does not belong to Γ. Indeed, if the result is true in this special case, then we can apply it to the functions u where Γ * = K(Γ) and i = 1, ..., m. It follows that (û 1 , ...,û m ) weakly satisfies (1.3) in R n \ {Γ * ∪ {0}} and that Γ * ⊂ {x 1 = 0} since, by assumption, Γ ⊂ {x 1 = 0}. Furthermore, we also have that Γ * is bounded (not necessarily closed) since we assumed that 0 / ∈ Γ.
To proceed further we recall some useful lemma whose proofs are contained in [19].  Let us now fix some notations. We set (4.41) Σ λ = {x ∈ R n : x 1 < λ} .
where C i,S are the best constants in Sobolev embeddings.
Proof. We immediately see that w + i,λ ∈ L 2 * (Σ λ ), since 0 ≤ w + i,λ ≤û i ∈ L 2 * (Σ λ ) for every i = 1, ..., m. The rest of the proof follows the lines of the one of Lemma 3.2. Arguing as in section 2, for every ε > 0, we can find a function ψ ε ∈ C 0,1 (R N , [0, 1]) such that Fix R 0 > 0 such that R λ ({Γ * ∪ {0}) ⊂ B R 0 and, for every R > R 0 , let ϕ R be a standard cut off function such that 0 ≤ ϕ R ≤ 1 on R n , ϕ R = 1 in B R , ϕ R = 0 outside B 2R with |∇ϕ R | ≤ 2/R, and consider for every i = 1, ..., m. Now, as in Lemma 3.1 we see that Therefore, by a standard density argument, we can use ϕ i as test functions respectively in (4.43) and in (4.44) so that, subtracting we get (4.48) Exploiting also Young's inequality and recalling that 0 ≤ w + i,λ ≤û i , we get that (4.49) Furthermore we have that where c(n) is a positive constant depending only on the dimension n. Let us now estimate I 3 . Sinceû i (x),û i,λ (x) > 0, by the convexity of t → t 2 * −1 , for t > 0, we obtain for every x ∈ Σ λ and i = 1, ..., m. Thus, by making use of the monotonicity of t → t 2 * −2 , for t > 0 and the definition of w + i,λ we get for every i = 1, ..., m. Therefore where we also used that 0 ≤ w + i,λ ≤û i for every i = 1, ..., m and Hölder inequality. Taking into account the estimates on I 1 , I 2 and I 3 , by (4.48) we deduce that (4.53) By Fatou Lemma, as ε tends to zero and R tends to infinity, we deduce that ∇w + i,λ ∈ L 2 (Σ λ ) for every i = 1, ..., m. We also note that ϕ i → w + i,λ in L 2 * (Σ λ ), by definition of ϕ i , and that ∇ϕ i → ∇w i,λ in L 2 (Σ λ ), by (5.68) and the fact that w + i,λ ∈ L 2 * (Σ λ ) for every i = 1, ..., m. Therefore by (4.53) we have where C i,S are the best constants in Sobolev embeddings. Thus, passing to the limit in (4.55) and using the above convergence results, we get the desired conclusion (4.46).
We can now complete the proof of Theorem 1.3. As for the proof of Theorem 1.2, we split the proof into three steps and we start with Step 1: there exists M > 1 such thatû i ≤û i,λ in Σ λ \ R λ (Γ * ∪ {0}), for all λ < −M and i = 1, ..., m.
Arguing as in the proof of Lemma 4.3 and using the same notations and the same construction for ψ ε , ϕ R and ϕ i , we get (4.56) where I 1 , I 2 and I 3 can be estimated exactly as in (4.49), (4.50) and (4.51). The latter yield (4.57) Taking the limit in the latter, as ε tends to zero and R tends to infinity, leads to Recalling thatû i ,û j ∈ L 2 * (Σ λ ) for every i, j = 1, ..., m, we deduce the existence of M > 1 such that for every λ < −M and i = 1, ...., m. The latter and (5.87) lead to This implies that for every i = 1, ..., m we have w + i,λ = 0 by Lemma 4.3 and the claim is proved.
Step 3: conclusion. The symmetry of the Kelvin transform (û 1 , ...,û m ) follows now performing the moving plane method in the opposite direction. The fact that everyû i is symmetric w.r.t. the hyperplane {x 1 = 0} implies the symmetry of the solution (u 1 , ..., u m ) w.r.t. the hyperplane {x 1 = 0}. The last claim then follows by the invariance of the considered problem with respect to isometries (translations and rotations).

Proof of Theorem 1.4
Proof of Theorem 1.4. As we observed in the proof of Theorem 1.3, thanks to a well-known result of Brezis and Kato [10] and standard elliptic estimates (see also [42]), the solution (u, v) is smooth in R n \ Γ. Furthermore we recall that it is enough to prove the theorem for the special case in which the origin does not belong to Γ.
We can now complete the proof of Theorem 1.4. As for the proof of Theorem 1.2 and Theorem 1.3, we split the proof into three steps and we start with Step 1: there exists M > 1 such thatû ≤û λ andv ≤v λ in Σ λ \ R λ (Γ * ∪ {0}), for all λ < −M.
Step 3: conclusion. The symmetry of the Kelvin transform v follows now performing the moving plane method in the opposite direction. The fact thatû andv are symmetric w.r.t. the hyperplane {x 1 = 0} implies the symmetry of the solution (u, v) w.r.t. the hyperplane {x 1 = 0}. The last claim then follows by the invariance of the considered problem with respect to isometries (translations and rotations).