Second-Order Elliptic Equation with Singularities

where Δgu � − ∇∇iu is the Laplacian–Beltrami operator and N � 2n/(n − 2) is the critical Sobolev exponent. Equation (1) is one of the nonlinear second-order equations involving the singular term a and with critical Sobolev growth. Such problem arises from various fields of geometry and physics. (ere are many results for second-order elliptic equations, but most of them are focused on bounded domains Ω of R or on compact Riemannianmanifold (M, g), see [1–16] for a survey. A variety of techniques have been used to solve second-order equations, and variational methods are the most suitable. Certainly, if the singular term a is replaced by (n − 2)/4(n − 1)Sg, where Sg is the scalar curvature and f � 1, then equation (1) becomes the famous prescribed constant scalar curvature equation which is very known in the literature as the Yamabe problem. To solve this problem, Yamabe has used the variational method, and the main difficulty of this problem is the lack of compactness for Sobolev embedding theorem. (e problem is now solved, but it took a very long time to find the good approach. If f is not a constant, the problem is known as the prescribed scalar curvature problem. For more details, we refer the reader to [12, 13] and the references therein. A famous result concerning the equation of type (1) has been obtained in [17], and it consists of the classification of positive solutions of the equation


Introduction
Let (M, g) be an (n ≥ 3)-dimensional compact Riemannian manifold, and let a ∈ L p (M), where p > n/2, and f be a positive C ∞ (M) function on M. In this paper, we are interested in studying on, (M, g), the following nonlinear singular elliptic equation: where Δ g u � − ∇ i ∇ i u is the Laplacian-Beltrami operator and N � 2n/(n − 2) is the critical Sobolev exponent. Equation (1) is one of the nonlinear second-order equations involving the singular term a and with critical Sobolev growth. Such problem arises from various fields of geometry and physics. ere are many results for second-order elliptic equations, but most of them are focused on bounded domains Ω of R n or on compact Riemannian manifold (M, g), see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] for a survey. A variety of techniques have been used to solve second-order equations, and variational methods are the most suitable. Certainly, if the singular term a is replaced by (n − 2)/4(n − 1)S g , where S g is the scalar curvature and f � 1, then equation (1) becomes the famous prescribed constant scalar curvature equation which is very known in the literature as the Yamabe problem. To solve this problem, Yamabe has used the variational method, and the main difficulty of this problem is the lack of compactness for Sobolev embedding theorem. e problem is now solved, but it took a very long time to find the good approach. If f is not a constant, the problem is known as the prescribed scalar curvature problem. For more details, we refer the reader to [12,13] and the references therein.
A famous result concerning the equation of type (1) has been obtained in [17], and it consists of the classification of positive solutions of the equation where 0 < λ < (n − 4) 2 /4, into the family of functions where c λ � �������������� � 1 − (4λ/(n − 2) 2 ). e singular term a was introduced as follows: in [11], Madani studied equation (1) with f is a constant, a � (n − 2)/4(n − 1)S g , and such that the metric g admits a finite number of points with singularities and is smooth outside these points.
is problem can be seen as the Yamabe problem with singularities. More precisely, let (M, g) be a compact Riemannian manifold of dimension n ≥ 3; we denote by T * M the cotangent space of M. e metric g ∈ H g u| p + |∇ g u| p + |u| p dv g . By Sobolev's embedding, we get that, for all p > (n/2), H where [n/p] denotes the entire part of n/p; then, the Christoffel symbols belong to H p 1 (M), and the components of the Riemannian curvature tensor Rm g , Ricci tensor Ric g , and the scalar curvature S g are in L p (M). Solving the singular Yamabe problem is equivalent to finding a positive solution u ∈ H p 2 (M) of the equation where k is a real constant, and in this case, the latter equation is the singular Yamabe equation. Under these assumptions on the metric g, the author in [11] proved the existence of a metric g � u N− 2 g conformal to g such that u ∈ H p 2 (M), u > 0, and the scalar curvature S g of g is constant if (M, g) is not conformal to the round sphere. Moreover, we define the Yamabe invariant as follows: If μ(M, g) > 0 and let a � ((n − 2)/4(n − 1))S g , the singular Yamabe operator P g � Δ g + a is weakly conformally invariant, coercive, and invertible.
In [1] Azaiz et al. studied some singular second-order elliptic equations. ey focused on the following equation: where 1 < q < 2 and λ is a positive real parameter. In particular, in eorem 1.1, under additional assumptions, they proved the existence of λ * > 0 such that, for any λ ∈ (0, λ * ), the equation has a nontrivial weak solution. For details, see [1] and the references therein.

Notations and Preliminaries
In this section, we introduce some notations and materials necessary in our study. Let (M, g) be a smooth compact Riemannian manifold of dimension n ≥ 3; we will work on the Sobolev space H 2 1 (M) which is the space of functions u such that u, |∇ g u| ∈ L 2 (M) and equipped with the usual norm By Sobolev's embedding (see [12]), H 2 1 (M) ⊂ L q (M), where 1 < q ≤ N, and this embedding is compact when q < N. e number N � 2n/(n − 2) is known as the critical exponent of the Sobolev embedding.
Let K 0 (n, 1) denote the best constant in Sobolev inequality that asserts that there exists a constant B > 0 such that, for any u ∈ H 2 1 (M), Notice that where ω n is the volume of S n . Denote by P g the operator defined in the weak sense on H 2 1 (M) by P g is an elliptic operator self-adjoint and is called coercive if there exits C > 0 such that, for any u ∈ H 2 1 (M), Let F be the functional defined on H 2 1 (M) by and let E be the Sobolev quotient. en, for any Traditionally, to obtain solutions of equation (1), we will use, when necessary, ideas developed in [9,11,18], and we will use classical variational techniques by minimizing the functional F. However, serious difficulties appear compared with the smooth case. In order, we define the quantity where Clearly, the functional E is well defined in H 2 1 (M) and is of class C 1 , and the identity zE(u) � 0 being the equation [1], where zE(u) is the differential of the functional E at u. So, for all v ∈ H 2 1 (M), we have Note that if f � 1 and a � ((n − 2)/4(n − 1))S g , it is easy to see that λ(M, g) is not the Yamabe invariant. Indeed, λ(M, g) is the infimum over the set A, and as above, the Yamabe invariant μ(M, g) satisfies 2 International Journal of Differential Equations roughout the paper, we will denote by B(P, δ) a geodesic ball of center P and of radius δ with 0 < δ < (r g (M)/2), where r g (M) is the injectivity radius, and let η be a smooth function on M such that We also let S g (P) be the scalar curvature of M at P. Now, we state our main results.
is satisfied, and (1) has a nontrivial positive weak solution Notice that this theorem is regarded as combined results between eorem 3-eorem 5.
en, there exists u ∈ H 2 1 (M) nontrivial solution to the following equation: where r, λ 2 (M, g), and K 2 (n, 2, − 2) are given in Section 5. Our paper is organized as follows: in Sections 1 and 2, we introduce some notations and preliminaries. In Section 3, we establish the existence and regularity result to equation (1). Section 4 is devoted to test functions which verify geometric assumptions and by the same way complete the proofs of our main theorems ( eorem 1 and eorem 2). Section 5 deals with applications to particular equations which could arise from conformal geometry, and in Section 6, we consider the critical case α � 2. e classical reference for conformal geometry is a survey by Lee and Parker [13].

Existence and Regularity of the Solution
In this section, we establish the existence and regularity result to equation (1). An elementary result we wish to briefly discuss here is the following.

Proposition 1. If P g is coercive, the following norm is equivalent to the usual norm on
Proof. If P g is coercive, one finds a constant c > 0 such that, for any u ∈ H 2 1 (M), Again with Brezis-Lieb lemma applying to (u m ) m , we get that en, and inequality (44) will be written as and multiplying this inequality by 1 + ‖a‖ p , then we get Using Sobolev's inequality, we still get us, Now, if we assume we find Hence, φ m converges strongly to 0 in H 2 1 (M), and then u m converges strongly to u in H 2 1 (M) and in L N (M). It follows that which leads to and we can conclude that u ∈ A, and u is a nontrivial positive weak solution of (1). Concerning the regularity of solutions of equations (1), Madani in [11] proved through the classical techniques a regularity result with f a constant function and a � ((n − 2)/4(n − 1)S g . By following the same procedure, though the presence of the nonconstant function f adds further technical difficulties, we can prove the regularity of solutions of equations (1). is result is formulated in the following theorem.
Proof. e proof of this theorem is reduced to show that u ∈ L N+ϵ for some ϵ > 0. Indeed, u verifies the equation and if u ∈ L N+ϵ , it follows that a − fu N− 2 ∈ L r (M), where r � min(p, (n/2) + ϵ) > (n/2) (see [10] for some details); hence, one has Δ g u ∈ L p (M), and by the regularity theorem, we deduce that u ∈ H p 2 (M). Let l > 0 be a real number and H, F be two continuous functions on R + given by International Journal of Differential Equations where c � 2q − 1 and 1 < q < (n(p − 1)/n(p − 2)). Since u ≥ 0 and u ∈ H 2 1 (M), then it follows that H°u and F°u are both in H 2 Let u be a weak solution of (55); then, for all v ∈ H 2 1 (M), one has Now, as in Section 2, we define a cutoff function Chosen v � η 2 H°u and plugging this function in (59), we get We put h � F ∘ u. Now, let us evaluate each of the above integrals by using h and (58). We have ∇ g h � F ′ ∘ u∇ g u; thus, by applying the second relationship of (58), this implies (62) We deduce that the first integral of (61) is bounded; then, e first relationship of (58) and the Cauchy-Schwarz inequality imply that the second integral of (61) is bounded; hence, By using the latter relationship of (58), we obtain uH°u ≤ h 2 . In the same vain, the two integrals of the righthand side member in (61) are bounded; thus, where ‖u‖ N N,r � B(P,r) u N dv g . If we group these estimates together, equality (61) becomes Now, let a 1 , b 1 , c 1 , and d 1 be four real numbers; if a 2 1 − 2a 1 b ≤ c 2 1 + d 2 1 , we easily obtain that a 1 ≤ 2b 1 + c 1 + d 1 . en, (66) becomes By Sobolev's embedding, we then get that there exists a constant c > 0 depending only on n such that Since q < N and after using (67), we obtain For δ sufficiently small, one has When l goes to +∞, we then get that there exists a constant C > 0 depending only on n, δ, ‖η‖ ∞ , ‖∇ g η‖ ∞ ‖a‖ p , and f such that Now, from the boundedness of u in L N (M) and as (2p/(p − 1))q < N, we still get Since M is compact, it can be covered by a finite number of balls B(P i , δ) i∈I , and let (η i ) i∈I be a partition of unity subordinated to the covering; then, It follows that u ∈ L qN (M) with qN > N.

Test Functions
e purpose of this section is to find conditions such that (29) will be true. Consider a normal geodesic coordinate system centered at a point P. Denote by S(r) the geodesic sphere centered at P and of radius r with r < r g (M), where r g (M) is the injectivity radius. Let dΩ be the volume 6 International Journal of Differential Equations element of the n − 1-dimensional Euclidean unit sphere S n− 1 , and put where ω n− 1 is the volume of S n− 1 and |g| is the determinant of the Riemannian metric g. e formula of Taylor's expansion of G(r) in a neighborhood of P is given by 6n where S g (P) is the scalar curvature of M at P. As in Section 2, let η be a smooth function on M such that For ϵ > 0, we define the radial function u ϵ as follows: where r � d(P, x) is the distance from P to x and f(P) � max x∈M f(x). For further computations, we need the following integrals; then, for any real positive numbers p, q such that p − q > 1, we put Furthermore, it can be easily seen that (79) then, (29) is true.
Proof. To proof this theorem, it suffices to show that e aim of the following is to compute expansions of these integrals: on the geodesic ball B (P, δ). To compute the first term, we need the following limited development of f at P: We have where 6n 6n .
(86) en, Now, we set By changing the variable as above, it follows that International Journal of Differential Equations we get that (91) erefore, 6n Let us compute the third integral. First, we have en, in a similar way, we get 6n at is, Now, compute the second integral J 2 . By using H€ older's inequality, we get en, a direct computation shows that 8 International Journal of Differential Equations 6n r n+1 3n where It follows that Independently, we can easily show that, for φ � |∇ g u ε | 2 or φ � f|u ε | N or φ � au 2 ε , we get Let us derive estimates for E(u ε ). Using expansions of J 1 , J 2 , and J 3 , we get that which yields 6n Next, International Journal of Differential Equations 9 E u ε ≤ 1 2 6n Let ϵ be sufficiently small such that 1 2 where A � (n − 2)n((ω n− 1 I (n/2)− 1 n )/2) 2/n (f(P)) − (2/N) , and as 2 − (n/p) > 0, en, we get that 6n erefore, 6n Put i.e., Direct calculation gives Knowing that (n − 2)n ω n− 1 I (n/2)− 1 we obtain 10 International Journal of Differential Equations E u ε ≤ K − 2 0 (n, 1)(f(P)) − (2/N) ‖a‖ p + 1 To ensure assumption (29), we must take en, we get that It follows that (114) means that

Application
Let P ∈ M; we define a function on M by where δ(M) is the injectivity radius of M. For brevity, we denote this function by r. We define the weighted L p (M, r c ) space as the set of measurable functions u on M such that r c |u| p are integrable, where p ≥ 1. We endow L p (M, r c ) with the norm In this section, we need the Hardy-Sobolev inequality and the Rellich-Kondrakov embedding whose proofs are given in [11].
is type of inequality in one dimension was introduced by Hardy and generalized for all dimensions. For more details, see the book of V. G. Maz'ja, where we can find the proof of this theorem. In our case, we are interested when β � 0 and l � 1.
e following theorems were proved by Madani in [11].
Theorem 7. Let (M, g) be a Riemannian compact manifold of dimension n and p, q and c be real numbers such that ((c + n)/p) � − 1 + (n/q) > 0 and 1 ≤ q ≤ p ≤ (qn/(n − q)). For any ϵ > 0, there exists A(ϵ, q, c) such that, for any u ∈ H q 1 (M), where the number K(n, q, c) � c is the best constant in Hardy inequality.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.