On the Neumann Problem of Hardy-Sobolev critical equations with the multiple singularities

Let $N \geq 3$ and $\Omega \subset \mathbb{R}^N$ be $C^2$ bounded domain. We study the existence of positive solution $u \in H^1(\Omega)$ of \begin{align*} \left\{ \begin{array}{l} -\Delta u + \lambda u = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} + \frac{|u|^{2^*(s)-2}u}{|x-x_2|^s}\text{ in }\Omega\\ \frac{\partial u}{\partial \nu} = 0 \text{ on }\partial\Omega, \end{array}\right. \end{align*} where $0<s<2$, $2^*(s) = \frac{2(N-s)}{N-2}$ and $x_1, x_2 \in \overline{\Omega}$ with $x_1 \neq x_2$. First, we show the existence of positive solutions to the equation provided the positive $\lambda$ is small enough. In case that one of the singularities locates on the boundary and the mean curvature of the boundary at this singularity is positive, the existence of positive solutions is always obtained for any $\lambda>0$. Furthermore, we extend the existence theory of solutions to the equations for the case of the multiple singularities with different exponents.


Introduction
The Hardy-Sobolev inequality asserts that for all u ∈ H 1 0 (R N ), there exists a positive constant C = C(N, s) such that where N ≥ 3, 0 < s < 2 and 2 * (s) = 2(N −s) N −2 . Suppose Ω ⊂ R N , then the Hardy-Sobolev inequality holds for u ∈ H 1 0 (Ω). The best constant of the Hardy-Sobolev inequality is defined as When Ω = R N , S s (R N ) is attained by for some a > 0 (see [6,10]). Moreover, g a (x) are the only positive solutions to (2). Hence, in case 0 ∈ Ω, by a standard scaling invariance argument, it is easy to see S s (Ω) = S s (R N ) and S s (Ω) cannot be attained unless Ω = R N . However, if 0 ∈ ∂Ω, the existence of the minimizer for S s (Ω) is established under the assumption that the mean curvature of ∂Ω at 0, H(0) is negative (see [5]).
Hence, there does not exist a positive solution to (3). So, only the case where λ > 0 are adderessed in literature. In this case, Ghossoub-Kang [4] showed that (3) has a positive solution if the mean curvature of ∂Ω at 0, H(0) is positive. Furthermore, Chabrowski [3] investigated the solvability of the nonlinear Neumann problem with indefinite weight functions −∆u + λu = Q(x)|u| 2 * (s)−2 u |x| s , u > 0 in Ω and gives some sufficient condition on Q(x) provided the mean curvature of ∂Ω at 0, H(0) > 0. Recently, concerning the equation (3) the first author investigated the case when H(0) ≤ 0 in [7]. He showed the existence of λ * such that for λ ∈ (0, λ * ), a least energy solution of (3) exists, and when λ > λ * a least energy solution does not exist. We remark that the sufficient conditions for Dirichlet and Neumann problems are completely different.
In this paper, we consider the Neumann problem with the multiple singularities where Ω is C 2 -bounded domain and x 1 , The main results of this article are as follows  Note that Theorem 1.2 asserts the singularity at boundary prevails the singularity in the interior. To establish the existence theory, we study the functional defined on H 1 (Ω) where u + = max(u, 0). It is not hard to see that J λ is a C 1 functional and for φ ∈ H 1 (Ω). Moreover, by Sobolev embedding theorem, we obtain Hence there exists α > 0 and ρ > 0 such that The scenario for the proof of the theorems is to apply the mountain pass lemma to attack the existence theory. However, the crux is to decide the threshold of the energy level so that the Palais-Smale condition would hold. We use concentration compactness principle to find this energy level.
In section 2, we investigate the threshold of the Palais-Smale condition for J λ . In section 3 and 4, we prove the existence of solutions as described in Theorem 1.1 and Theorem 1.2, respectively. In section 5, the positivity of solutions is established. In section 6, regularity of solution is considered. Lastly in section 7, we give brief accounts for the Neumann problem with the multiple singularities. Namely, the existence of solutions to

Palais-Smale Condition
In this section, we investigate the threshold of the Palais-Smale condition for J λ . In what follows, S s denotes S s (R N ). First we recall the Hardy-Sobolev inequality for functions supported on neighborhood of boundary. For the Sobolev inequality, see Lemma 2.1 in [15]. The following lemma is obtained by applying the technique of [15]. (2) For any ε > 0, there exists δ > 0 such that if |∇h| ≤ δ, then The proof is based on P. L. Lions' concentration-compactness principle [11,13]. Suppose {u m } be a (P S) c sequence. That is Taking off one-half of (9) from (7), we obtain Hence, we derive from (7) that Hence {u m } is a bounded sequence in H 1 (Ω). So, up to a subsequence, we have the following weak convergence : Here Then the concentration-compactness principle gives in the sense of measure where δ x is the Dirac-mass of mass 1 concentrated at x ∈ R N . Here, I is at most countable index set and the numbers Letting l → ∞, we see that where ν 1 := ν 1 and ν 2 := ν 2 . Now we shall show some relation between ν k and µ k for k = 1, 2. We consider v m = u m − u and In case of x k ∈ Ω, we have In case of x k ∈ ∂Ω, applying Lemma 2.1, we see that for k = 1, 2.
To complete the proof, we need to show that µ i = 0 for i = 1, 2 or i ∈ I.
One can readily check that We claim that Let Ω l i := Ω ∩ supp(∇φ l (· − x i )). First we consider the case where x i is not a limit point of {x k : k ∈ I}. In this case, we see that In the case of x i is a limit point of {x k : k ∈ I}, there is additional term which also goes to 0 as l → ∞. So we get Using the same argument, we have for i = 1, 2, If we assume µ i > 0 for i = 1 or 2, then But from (11) and (12), we have which is a contradiction. This prove Proposition 1.

Existence of solution to (4) for small λ
In this section, we show the existence theory of Theorem 1.1. Plugging constant function c into the functional J λ , we have provided the positive solution parameter λ is small enough.

Existence of solution to (4) with boundary singularity
In this section, we prove the existence of a solution in Theorem 1.2. We shall follow the strategy of [15,4] to prove Theorem 1.2. We may assume x 1 = (0, · · · , 0) ∈ ∂Ω and the mean curvature H(0) is positive. Then, up to rotation, the boundary near the origin can be represented by Here α 1 , α 2 , · · · , α N −1 are the principal curvature of ∂Ω at 0 and the mean for small parameter ε > 0. Then, it follows that Choose δ such that x 2 / ∈ B 3δ (0). Set a cut-off function η such that for sufficiently large T . We define P = p(t) p(t) : [0, 1] → H 1 (Ω) is continous map with p(0) = 0 ∈ H 1 (Ω) and p(1) = T ηU ε (x)| Ω . Let Then, thanks to Proposition 1, it suffices to show In the following discussion, we denote First we deal with K ε 0 . By using Leibniz rule, one has Similarly, it follows that The last term is more delicate. We consider the case of N = 3 and the case of N ≥ 4 separately. When N = 3, we have Observe that So we have which leads to The curvature assumption (H(0) > 0) implies Moreover, , for any σ > 0, there exists C(σ) > 0 such that So we have which implies Therefore we obtain On the other hand, we have Observe that So, we have Similarly, we can get Thus, we obtain Moreover, direct calculation gives Actually when N = 3, we have When N = 4, we see that When N ≥ 5, we have Lastly, we are concerned about K ε 2 . Since x 2 / ∈ B 3δ (0) and supp(η) ⊂ B 2δ (0), we see that Let t ε be a constant satisfying In case N = 3, we see that K 3 (ε) = O(ε 1 2−s ). Hence, So to prove (14), it suffices to show that Taking (13), (16) and (18) into account, (19) is equivalent to which is true for small ε > 0, because C > 0 and So to prove (14), it suffices to show that Taking (13), (16) and (18) into account, (20) is equivalent to (21) Hence to verify (21), we have to prove By (15), (17) and L'Hôpital's rule, we obtain lim ε→0 II(ε)

Integration by parts gives for
So, plugging β = N + 2 − s into (22), we obtain lim ε→0 II(ε) Therefore we obtain II(ε) for sufficiently small ε and complete the proof.

Positivity of solution
In this section, we establish the positivity of solutions. One first observes that where u − = min(u, 0). Since λ > 0, we have u ≥ 0. Then the interior positivity of u follows from the maximum principle Proof. We employ the argument in [14]. If u vanishes somewhere in Ω \ {x 1 , x 2 }, then there exists y 0 ∈ Ω \{x 1 , x 2 } and a ball B = B R (y 1 ) satisfying u(y 0 ) = 0, B ⊂ Ω \ {x 1 , x 2 }, y 0 ∈ ∂B and 0 < u < a in B. We observe that u > 0 on Then u(x) = 0 ≤ u(x) on ∂B R (y 1 ) and u(x) = c ≤ u(x) on ∂B R 2 (y 1 ). Moreover, on A, we have for sufficiently large k 1 , k 2 . We claim that u ≥ u on A. Suppose not, there exists Ω 1 ⊂ A such that u > u on Ω 1 . And we have So, by multiplying u − u and integrating over Ω 1 , we obtain which is a contradiction.
Since u(y 0 ) = u(y 0 ) = 0, u ≥ u on A and v ′ > 0, u ′ (y 1 ) should be positive which contradicts to y 0 is minimum point. 6. Regularity of solution to (4) In this section, we verify the regularity of solution. Recall the following lemma.

Neumann Problem with the multiple singularities
In this section, we deal with the existence theory for the equation where 0 < s i < 2, Ω is C 2 bounded domain with x i ∈ ∂Ω for 1 ≤ i ≤ I ′ − 1 and x i ∈ Ω for I ′ ≤ i ≤ I. In addition, we assume that The energy functional is given by We see that J λ is C 1 and In the same fashion as the proof of Proposition 1, we obtain the following proposition : Actually we may assume x 1 = (0, · · · , 0) ∈ ∂Ω and the mean curvature H(0) is positive. Then, up to rotation, the boundary near the origin can be represented by where x ′ = (x 1 , x 2 , · · · , x N −1 ) ∈ D δ (0) = B δ (0) ∩ {x N = 0} for some δ > 0.