# Second-Order Elliptic Equation with Singularities

Authors: Hichem Boughazi
Journal: International Journal of Differential Equations
 (2020)

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## Abstract

On the compact Riemannian manifold of dimensionn≥5, we study the existence and regularity of nontrivial solutions for nonlinear second-order elliptic equation with singularities. At the end, we give a geometric application of the above singular equation.


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## Body

## 1. Introduction

LetM,g be an n≥3-dimensional compact Riemannian manifold, and let a∈LpM, 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:(1)Δgu+au=fuN−2u,where Δgu=−∇i∇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. There are many results for second-order elliptic equations, but most of them are focused on bounded domains Ω of ℝn or on compact Riemannian manifold 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−2/4n−1Sg, 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. The 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(2)Δu−λx2u=un+2/n−2,inℝn∖0,where 0<λ<n−42/4, into the family of functions(3)uλx=cλxa−11+x2an/2−1,where cλ=1−4λ/n−22.The singular terma was introduced as follows: in [11], Madani studied equation (1) with f is a constant, a=n−2/4n−1Sg, and such that the metric g admits a finite number of points with singularities and is smooth outside these points. This 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. The metric g∈H2pM,T∗M⊗T∗M is the space of all sections g (2-covariant tensors) such that, in normal coordinates, the components gij of g are in H2pM, where H2pM is the completion of the space C0∞M with respect to the norm uH2pM2=∫M∇g2up+∇gup+updvg. By Sobolev’s embedding, we get that, for all p>n/2, H2pM,T∗M⊗T∗M⊂C1−n/pM,T∗M⊗T∗M, where n/p denotes the entire part of n/p; then, the Christoffel symbols belong to H1pM, and the components of the Riemannian curvature tensor Rmg, Ricci tensor Ricg, and the scalar curvature Sg are in LpM. Solving the singular Yamabe problem is equivalent to finding a positive solution u∈H2pM of the equation(4)Δgu+n−24n−1Sgu=kuN−2u,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¯=uN−2g conformal to g such that u∈H2pM, u>0, and the scalar curvature Sg¯ of g¯ is constant if M,g is not conformal to the round sphere. Moreover, we define the Yamabe invariant as follows:(5)μM,g=infu∈H12M,u≠0Eu,whereEu=∫MuPgudvg∫MuNdvg2/N.IfμM,g>0 and let a=n−2/4n−1Sg, the singular Yamabe operator Pg=Δg+a is weakly conformally invariant, coercive, and invertible.In [1] Azaiz et al. studied some singular second-order elliptic equations. They focused on the following equation:(6)Δgu+au=λuq−2u+fuN−2u,where 1<q<2 and λ is a positive real parameter. In particular, in Theorem 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.


## 2. Notations and Preliminaries

In this section, we introduce some notations and materials necessary in our study. LetM,g be a smooth compact Riemannian manifold of dimension n≥3; we will work on the Sobolev space H12M which is the space of functions u such that u,∇gu∈L2M and equipped with the usual norm(7)uH12M=∫M∇gu2+u2dvg1/2.By Sobolev’s embedding (see [12]), H12M⊂LqM, where 1<q≤N, and this embedding is compact when q<N.The numberN=2n/n−2 is known as the critical exponent of the Sobolev embedding.LetK0n,1 denote the best constant in Sobolev inequality that asserts that there exists a constant B>0 such that, for any u∈H12M,(8)∫MuNdvg2/N≤K02n,1∫M∇gu2dvg+BuL2M2.Notice that(9)K02n,1=4nn−2ωn2/n,where ωn is the volume of Sn. Denote by Pg the operator defined in the weak sense on H12M by(10)Pgu=Δgu+au,Pg is an elliptic operator self-adjoint and is called coercive if there exits C>0 such that, for any u∈H12M,(11)∫MuPgudvg≥CuH12M2,where(12)∫MuPgudvg=∫M∇gu2+au2dvg.LetF be the functional defined on H12M by(13)Fu=∫M∇gu2+au2dvg,and let E be the Sobolev quotient. Then, for any u∈H12M−0,(14)Eu=Fu∫MfuNdvg2/N.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(15)λM,g=infu∈AFu,where(16)A=u∈H12Msuch that∫MfuNdvg=1+apN/2.Clearly, the functionalE is well defined in H12M and is of class C1, and the identity ∂Eu=0 being the equation [1], where ∂Eu is the differential of the functional E at u. So, for all v∈H12M, we have(17)∂Euv=ddtt=0Eu+tv.Note that iff=1 and a=n−2/4n−1Sg, it is easy to see that λM,g is not the Yamabe invariant. Indeed, λM,g is the infimum over the set A, and as above, the Yamabe invariant μM,g satisfies(18)μM,g=λM,g1+ap.Throughout the paper, we will denote byBP,δ a geodesic ball of center P and of radius δ with 0<δ<rgM/2, where rgM is the injectivity radius, and let η be a smooth function on M such that(19)ηx=1,onBP,δ,0,onM−BP,2δ.We also letSgP be the scalar curvature of M at P.Now, we state our main results.Theorem 1.
LetM,g be a Riemannian compact manifold of dimension n≥5. Assume that a∈LpM, where p>n/2, f is a positive C∞M function on M, and P∈M such that fP=supx∈Mfx. If(20)ΔfPfP<SgP3n+2n−4−ap−1ap+1−1,then(21)λM,g<1+apK02n,1supx∈Mfx2/N,is satisfied, and (1) has a nontrivial positive weak solution such that Fu=λM,g and u∈A. Moreover, u∈C1−n/pM and u>0.Notice that this theorem is regarded as combined results between Theorem3–Theorem 5.Theorem 2.
LetM,g be a Riemannian compact manifold of dimension n≥5, P∈M such that fP=supx∈Mfx, a be smooth function, and r denote the distance function. If(22)λ2M,gK02n,1fP2/N<1+ar2pK2n,2,−2,1+aPK2n,2,−2>0.Then, there existsu∈H12M nontrivial solution to the following equation:(23)Δgu+axr2u=fuN−2u,where r, λ2M,g, and K2n,2,−2 are given in Section 5.Our paper is organized as follows: in Sections1 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 (Theorem 1 and Theorem 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.The classical reference for conformal geometry is a survey by Lee and Parker [13].


## 3. 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.
IfPg is coercive, the following norm is equivalent to the usual norm on H12M:(24)u=∫M∇gu2+au2dvg1/2=∫MuPgudvg1/2.Proof.
IfPg is coercive, one finds a constant c>0 such that, for any u∈H12M,(25)∫MuPgudvg≥cuH12M2.
Sincea∈LpM, where p>n/2, and by the embedding LpM⊂Ln/2M, it follows that there exists a constant c0>0 such that an/2≤c0ap; then, from Ho¨lder’s inequality, we get that(26)∫MuPgudvg=∫M∇gu2+au2dvg≤∫M∇gu2dvg+∫Man/2dvg2/n∫Mu2n/n−2dvgn−2/n≤∫M∇gu2dvg+an/2uN2≤∫M∇gu2dvg+c0apuN2.
Now, from (8), we get that(27)uN2≤maxK02n,1,BuH12M2,so we obtain(28)∫MuPgudvg≤uH12M2+c0apmaxK02n,1,BuH12M2≤max1,c0apmaxK02n,1,BuH12M2≤c1uH12M2,where c1=max1,c0apmaxK02n,1,B>0.Theorem 3.
Assume that(29)λM,g<1+apK02n,1supx∈Mfx2/N.Then, (1) has a nontrivial positive weak solution such that Fu=λM,g and u∈A.In particular, iff=1 and a=n−2/4n−1Sg, condition (29) becomes the famous inequality(30)μM,g<1K02n,1.Proof.
First, we show thatλM,g is finite. For any u∈H12M and by Ho¨lder’s inequality, one has(31)∫Mau2dvg≤∫Man/2dvg2/n∫Mu2n/n−2dvgn−2/n≤an/2uN2≤c0apuN2,from(32)∫MfuNdvg=1+apN/2,we get(33)1+apinfx∈Mfx2/N≥uN2,and then,(34)∫Mau2dvg≤c0ap1+apinfx∈Mfx2/N.
Therefore,(35)Fu=∫M∇gu2+au2dvg≥∫Mau2dvg≥−c0ap1+apinfx∈Mfx2/N.
Consequently,λM,g is finite.
Now, letumm be a minimizing sequence in A of the functional F; the sequence um is such that(36)λM,g=limm⟶+∞∫M∇gum2+aum2dvg,∫MfumNdvg=1+apN/2.
It is easy to see thatumm is also a minimizing sequence. Hence, we can assume that um≥0; then, for m large enough, we get(37)∫M∇gum2+aum2dvg≤λM,g+1.
IfPg is coercive, by Proposition 1, umm is bounded. If not, we proceed as follows:(38)∫M∇gum2dvg≤λM,g+1−∫Maum2dvg≤λM,g+1+∫Maum2dvg.
By using (34), we still get(39)∫M∇gum2dvg≤λM,g+1+c0ap1+apinfx∈MfxN/2.
On the contrary, by the embeddingLNM⊂L2M and by inequality (33), we get that there exists c>0 such that(40)∫Mum2dvg≤cumN2≤c1+apinfx∈MfxN/2.
This implies in turn thatumm is bounded in H12M, and after restriction to a subsequence still labeled umm, we may assume that there exists u∈H12M, u≥0, such that(i)
um⟶u weakly in H12M(ii)
um⟶u strongly in LqM for all q<N and almost everywhere on M
Furthermore, by Brezis–Lieb lemma applying toumm, we get that(41)∫M∇gum2dvg=∫M∇gu2dvg+∫M∇gum−u2dvg+o1.
Puttingφm=um−u, then φm⟶0 weakly in H12M and strongly in LqM for all q<N; therefore,(42)∫M∇gum2dvg=∫M∇gφm2dvg+∫M∇gu2dvg+2∫M∇gφm.∇gudvg.
We deduce that(43)Fum=Fu+∇gφm22+o1.
By definition ofλM,g, we obtain Fu≥λM,g∫MfuNdvg2/N, and by definition of the sequence umm, we obtain Fum=λM,g+o1 which implies that(44)λM,g∫MfuNdvg2/N+∇gφm22≤λM,g+o1.
Again with Brezis–Lieb lemma applying toumm, we get that(45)1+apN/2=∫MfumNdvg=∫MfuNdvg+∫MfφmNdvg+o1.
Then,(46)1+ap≤∫MfuNdvg2/N+∫MfφmNdvg2/N+o1,and inequality (44) will be written as(47)∇gφm22≤λM,g1−∫MfuNdvg2/N+o1,and multiplying this inequality by 1+ap, then we get(48)1+ap∇gφm22≤λM,g1+ap−1+ap∫MfuNdvg2/N+o1≤λM,g∫MfuNdvg2/N+∫MfφmNdvg2/N−1+ap∫MfuNdvg2/N+o1≤λM,g∫MfφmNdvg2/N−ap∫MfuNdvg2/N+o1≤λM,g∫MfφmNdvg2/N+o1.
Using Sobolev’s inequality, we still get(49)1+ap∇gφm22≤λM,gsupx∈Mfx2/NK02n,1∇gφm22+o1.
Thus,(50)1+ap−λM,gsupx∈Mfx2/NK02n,1∇gφm22≤o1.
Now, if we assume(51)λM,g<1+apK02n,1supx∈Mfx2/N,we find(52)∇gφm22=o1.
Hence,φm converges strongly to 0 in H12M, and then um converges strongly to u in H12M and in LNM. It follows that(53)lim∫Mfum−uNdvg=0,which leads to(54)∫MfuNdvg=1+apN/2,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−2/4n−1Sg. 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). This result is formulated in the following theorem.Theorem 4.
LetM,g be a Riemannian compact manifold of dimension n≥3, and let a∈LpM, where p>n/2, and f be a positive C∞M function on M. If u∈H12M is a nontrivial positive weak solution of(55)Δgu+au=fuN−1,then u∈H2pM⊂C1−n/pM and u>0.Proof.
The proof of this theorem is reduced to show thatu∈LN+ϵ for some ϵ>0. Indeed, u verifies the equation(56)Δgu+a−fuN−2u=0,and if u∈LN+ϵ, it follows that a−fuN−2∈LrM, where r=minp,n/2+ϵ>n/2 (see [10] for some details); hence, one has Δgu∈LpM, and by the regularity theorem, we deduce that u∈H2pM. Let l>0 be a real number and H,F be two continuous functions on ℝ+ given by(57)Hx=tγ,if0≤t≤l,lq−1qlq−1t−q−1lq,ift>l,Fx=tq,if0≤t≤l,qlq−1t−q−1lq,ift>l,where γ=2q−1 and 1<q<np−1/np−2. Since u≥0 and u∈H12M, then it follows that H°u and F°u are both in H12M,(58)qHt=FtF′t,F′t2≤qH′t,F2t≥tHt.
Letu be a weak solution of (55); then, for all v∈H12M, one has(59)∫M∇gu∇gvdvg+∫Mauvdvg=∫MfuN−1vdvg.
Now, as in Section2, we define a cutoff function η∈C1M such that(60)ηx=1,onBP,δ,0,onM−BP,2δ.
Chosenv=η2H°u and plugging this function in (59), we get(61)∫Mη2H′∘u∇gu2dvg+2∫MηH∘u∇gu∇gηdvg=∫MfuN−1η2H∘udvg−∫Mauη2H∘udvg.
We puth=F∘u. Now, let us evaluate each of the above integrals by using h and (58). We have ∇gh=F′∘u∇gu; thus, by applying the second relationship of (58), this implies(62)∇gh2=F′∘u2∇gu2≤qH′∘u∇gu2.
We deduce that the first integral of (61) is bounded; then,(63)1qη∇gh22≤∫Mη2H′∘u∇gu2dvg.
The first relationship of (58) and the Cauchy–Schwarz inequality imply that the second integral of (61) is bounded; hence,(64)2∫MηH∘u∇gu∇gηdvg=2q∫Mηh∇gh∇gηdvg≥−2qh∇gη2η∇gh2.
By using the latter relationship of (58), we obtain uH°u≤h2. In the same vain, the two integrals of the right-hand side member in (61) are bounded; thus,(65)∫MfuN−1η2H∘udvg−∫Mauη2H∘udvg≤supx∈MfxuN,2δ4/n−2ηhN2+apηh2p/p−12,where uN,rN=∫BP,ruNdvg. If we group these estimates together, equality (61) becomes(66)η∇gh22−2h∇gη2η∇gf¯2≤qsupx∈MfxuN,2δ4/n−2ηhN2+apηh2p/p−12.
Now, leta1,b1,c1, and d1 be four real numbers; if a12−2a1b≤c12+d12, we easily obtain that a1≤2b1+c1+d1. Then, (66) becomes(67)η∇gh2≤qsupx∈MfxuN,2δ2/n−2ηhN+qapηh2p/p−1+2h∇gη2.
By Sobolev’s embedding, we then get that there exists a constantc>0 depending only on n such that(68)ηhN≤cη∇gh2+h∇gη2+hη2.
Sinceq<N and after using (67), we obtain(69)1−cNsupx∈MfxuN,2δ2/n−2ηhN≤cNapηh2p/p−1+3h∇gη2+hη2.
Forδ sufficiently small, one has(70)uN,2δ2/n−2≤12cNsupx∈Mfx.
Whenl goes to +∞, we then get that there exists a constant C>0 depending only on n,δ,η∞,∇gη∞ap, and f such that(71)uqN,2δ≤Cuq2+uq2p/p−1.
Now, from the boundedness ofu in LNM and as 2p/p−1q<N, we still get(72)uqqN,2δ≤C.
SinceM is compact, it can be covered by a finite number of balls BPi,δi∈I, and let ηii∈I be a partition of unity subordinated to the covering; then,(73)uqNqN=∑i∈IηiuqN,δiqN≤C.
It follows thatu∈LqNM with qN>N.


## 4. Test Functions

The 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 Sr the geodesic sphere centered at P and of radius r with r<rgM, where rgM is the injectivity radius. Let dΩ be the volume element of the n−1-dimensional Euclidean unit sphere Sn−1, and put(74)Gr=1ωn−1∫SrgdΩ,where ωn−1 is the volume of Sn−1 and g is the determinant of the Riemannian metric g. The formula of Taylor’s expansion of Gr in a neighborhood of P is given by(75)Gr=1−SgP6nr2+or2,where SgP is the scalar curvature of M at P. As in Section 2, let η be a smooth function on M such that(76)ηx=1,onBP,δ,0,onM−BP,2δ.Forϵ>0, we define the radial function uϵ as follows:(77)uϵ=ηrr2+ϵ2−n−2/2,where r=dP,x is the distance from P to x and fP=maxx∈Mfx. For further computations, we need the following integrals; then, for any real positive numbers p,q such that p−q>1, we put(78)Ipq=∫0+∞1+t−ptqdt.Furthermore, it can be easily seen that(79)Ip+1q=p−q−1pIpq,Ip+1q+1=q+1p−q−1Ip+1q.Theorem 5.
LetM,g be a Riemannian compact manifold of dimension n≥5. Assume that a∈LpM, where p>n/2, f is a positive C∞M function on M, and P∈M such that fP=supx∈Mfx. If(80)ΔfPfP<SgP3n+2n−4−ap−1ap+1−1,then, (29) is true.Proof.
To proof this theorem, it suffices to show that(81)Euε<1+apK02n,1supx∈Mfx2/N.
The aim of the following is to compute expansions of these integrals:(82)J1=∫MfuεNdvg,J2=∫Mauε2dvg,J3=∫M∇guε2dvg,on the geodesic ball BP,δ. To compute the first term, we need the following limited development of f at P:(83)fx=fP+12∇i,jfPyiyj+or2.
We have(84)J1=∫MfuεNdvg=∫0δuεN∫SrfgdΩrn−1dr,where(85)∫SrfgdΩ=∫SrfP+12∇i,jfPyiyj1−16Ri,jyiyjdΩ+or2=ωn−1fP−ΔfP2n+fPSgP6nr2+or2.
Put(86)L=ΔfP2n+fPSgP6n.
Then,(87)J1=ωn−1∫0δrn−1r2+ϵ2nfP−Lr2dr+or2=ωn−1fP∫0δrn−1r2+ϵ2ndr−L∫0δrn+1r2+ϵ2ndr+orn+1.
Now, we set(88)t=r2ϵ2,dr=ϵdt2dt,r=ϵt.
By changing the variable as above, it follows that(89)J1=ωn−1fP∫0δ/ϵ2tn/22ϵn−21+tndt−L∫0δ/ϵ2tn/22ϵn−21+tndt+oϵn+1=ωn−12ϵnfPInn/2−1−Lϵ2Inn/2+oϵ2.
From(90)Inn/2=nn−2Inn/2−1,ωn=2n−1ωn−1Inn/2−1,we get that(91)J1=ωn−12ϵnInn/2−1fP−ΔfP2n−2+fPSgP6n−2ϵ2+oϵ2=ωn−12ϵnInn/2−1fP1−ΔfP2n−2fP+SgP6n−2ϵ2+oϵ2.
Therefore,(92)J1−2/N=J1−n−2/n=ωn−12ϵnInn/2−1fP−n−2/n1+n−2nΔfP2n−2fP+SgP6n−2ϵ2+oϵ2=2n−2/nϵn−2ωn−1Inn−2−1fPn−2/n1+ΔfP2nfP+SgP6nϵ2+oϵ2.
Let us compute the third integral. First, we have(93)∇guε=∂uε∂r=n−2rr2+ϵ2n/2.
Then, in a similar way, we get(94)J3=ωn−1∫0δn−22r2r2+ϵ2n1−SgP6nr2+or2rn−1dr=n−22ωn−1ϵn−2∫0δ/ϵ2tn/2dt21+tn−∫0δ/ϵ2SgPϵ2tn/2+1dt12n1+tn+oϵ2=n−22ωn−1ϵn−2n2n−2Inn/2−1−SgPϵ2nn+212nn−4n−2Inn/2−1+oϵ2=n−22ϵn−2ωn−1Inn/2−1n2n−2−SgPϵ2nn+212nn−4n−2+oϵ2=n−2ϵn−2ωn−1Inn/2−1n2−SgPϵ2nn+212nn−4+oϵ2.
That is,(95)J3=n−2ϵn−2ωn−1Inn/2−1n21−SgPϵ2n+26nn−4+oϵ2.
Now, compute the second integralJ2. By using Ho¨lder’s inequality, we get(96)J2=∫Mauε2dvg≤∫Mapdvg1/p∫Mu2p/p−1dvgp−1/p≤apuε2p/p−12.
Then, a direct computation shows that(97)uε2p/p−12=∫Muε2p/p−1dvgp−1/p=ωn−1p−1/p∫0δrn−1r2+ϵ2n−2p/p−1−SgP6nrn+1r2+ϵ2n−2p/p−1+orn+1drp−1/p=12p−1/pωn−1p−1/pϵ−n+2+2−n/pIn−2p/p−1n/2−1−SgP3nϵ2In−2p/p−1n/2+oϵ2p−1/p=12p−1/pωn−1p−1/pϵ−n+2+2−n/pIn−2p/p−1n/2−1−SgPn+2p−13npn−8p+4−nϵ2In−2p/p−1n/2−1+oϵ2p−1/p=12p−1/pωn−1p−1/pϵ−n+2+2−n/pIn−2p/p−1n/2−1p−1/p1−SgPn+2p−13npn−8p+4−nϵ2+oϵ2p−1/p=12p−1/pωn−1p−1/pϵ−n+2+2−n/pIn−2p/p−1n/2−1p−1/p1−βϵ2+oϵ2,where(98)β=SgPn+2p−123nppn−8p+4−n.
It follows that(99)uε2p/p−12=12p−1/pωn−1p−1/pϵ−n+2+2−n/pIn−2p/p−1n/2−1p−1/p1−βϵ2+oϵ2,∫Mauε2dvg≤12p−1/pωn−1p−1/pϵ−n+2+2−n/papIn−2p/p−1n/2−1p−1/p1−βϵ2+oϵ2.
Independently, we can easily show that, forφ=∇guε2 or φ=fuεN or φ=auε2, we get(100)∫BP,2δ−BP,δφdvg⟶0.
Let us derive estimates forEuε. Using expansions of J1,J2, and J3, we get that(101)Euε=∫M∇guε2+auε2dvg∫MfuεNdvg−2/N,which yields(102)Euε≤12p−1/pωn−1p−1/pϵ−n+2+2−n/papIn−2p/p−1n/2−1p−1/p1−βϵ2+oϵ2+n−2ϵn−2ωn−1Inn/2−1n21−SgPϵ2n+26nn−4+oϵ2×2n−2/nϵn−2ωn−1Inn/2−1fPn−2/n1+ΔfP2nfP+SgP6nϵ2+oϵ2.
Next,(103)Euε≤12p−1/pωn−1p−1/pϵ2−n/papIn−2p/p−1n/2−1p−1/p1−βϵ2+oϵ2+n−2ωn−1Inn/2−1n21−SgPϵ2n+26nn−4+oϵ2×2n−2/nωn−1Inn/2−1fPn−2/n1+ΔfP2nfP+SgP6nϵ2+oϵ2.
Letϵ be sufficiently small such that(104)12p−1/pωn−1p−1/pϵ2−n/papIn−2p/p−1n/2−1p−1/p2n−2/nωn−1Inn/2−1fPn−2/n≤apA,where A=n−2nωn−1Inn/2−1/22/nfP−2/N, and as 2−n/p>0,(105)ϵ2−n/pβϵ2=oϵ2.
Then, we get that(106)Euε≤apA+oϵ2+n−2ωn−1Inn/2−1n22n−2/nωn−1Inn/2−1fPn−2/n1−SgPϵ2n+26nn−4+oϵ2×1+ΔfP2nfP+SgP6nϵ2+oϵ2.
Therefore,(107)Euε≤apA+oϵ2+n−2nωn−1Inn/2−122/nfP−2/N1−SgPϵ2n+26nn−4+oϵ2×1+ΔfP2nfP+SgP6nϵ2+oϵ2.
Put(108)C=SgPn+26nn−4,D=ΔfP2nfP+SgP6n,i.e.,(109)Euε≤apA+oϵ2+A−ACϵ2+oϵ2×1+Dϵ2+oϵ2.
Direct calculation gives(110)Euε≤Aap+11+apD+D−Cϵ2ap+1+oϵ2.
Knowing that(111)n−2nωn−1Inn/2−122/n=K0−2n,1,we obtain(112)Euε≤K0−2n,1fP−2/Nap+11+apD+D−Cϵ2ap+1+oϵ2.
To ensure assumption (29),(113)λM,g<1+apK02n,1supx∈Mfx2/N,we must take(114)apD+D−C<0.
Then, we get that(115)Euε<1+apK0−2n,1fP−2/N.
It follows that (114) means that(116)ΔfPfP<SgP3n+2n−4−ap−1ap+1−1.


## 5. Application

LetP∈M; we define a function on M by(117)rpq=dp,q,ifdp,q<δM,δM,ifdp,q≥δM,where δM is the injectivity radius of M. For brevity, we denote this function by r. We define the weighted LpM,rγ space as the set of measurable functions u on M such that rγup are integrable, where p≥1. We endow LpM,rγ with the norm(118)up,rγp=∫Mrγupdvg.In this section, we need the Hardy–Sobolev inequality and the Rellich–Kondrakov embedding whose proofs are given in [11].Theorem 6 (Hardy inequality).
For any function inu∈C0∞M, there exists a constant c>0 such that(119)xγup≤cxβ∇uq,where 1≤q≤p≤qn/n−lq, γ=β−l+n1/q−1/p>−n/p and n>lq.This 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 whenβ=0 and l=1. The following theorems were proved by Madani in [11].Theorem 7.
LetM,g be a Riemannian compact manifold of dimension n and p,q and γ be real numbers such that γ+n/p=−1+n/q>0 and 1≤q≤p≤qn/n−q. For any ϵ>0, there exists Aϵ,q,γ such that, for any u∈H1qM,(120)up,rγ≤Kn,q,γ+ϵ∇guq+Aϵ,q,γuq,where the number Kn,q,γ=c is the best constant in Hardy inequality.In particular,Kn,q,0=Kn,q is the best constant in Sobolev’ embedding u∈H1qM⊂LpM.Moreover, in the case ofq=2, we obtain(121)up,rγ2≤K2n,2,γ+ϵ∇gu22+Aϵ,2,γu22,where γ+n/p=−1+n/2>0 and 2≤p≤2n/n−2.Lemma 1.
For all functionu∈H1qM with compact support included in the ball BP,δ, we get(122)up,rγ≤Kn,q,γ+ϵ∇guq,and when δ tends to 0, Kδn,q,γ goes to Kn,q,γ.Theorem 8.
LetM,g be a Riemannian compact manifold of dimension n.(1)
Ifγ+n/p=−1+n/q>0 and 1≤q≤p, then the embedding u∈H1qM⊂LpM,rγ is continuous(2)
Ifγ+n/p=−1+n/q>0, γ≤0 and q≤p, then this embedding is compact
We consider the following second-order equation:(123)Δgu+axrαu=fuN−2u,where a is the smooth function, r denotes the distance function defined as above, and α will be precise later. Let(124)Fαu=∫M∇gu2+axrαu2dvg,be the energy functional, and consider the Sobolev quotient: for any u∈H12M−0,(125)Eαu=Fαu∫MfuNdvg2/N.Let(126)λαM,g=infu∈AFαu,where(127)A=u∈H12Msuch that∫MfuNdvg=1+arαpN/2.Proposition 2.
If0<α<n/p<2 and(128)λαM,g<1+apK0−2n,1fP−2/N,then equation (123) has a positive solution uα∈C1−n/pM such that(129)Fαuα=λαM,g.Proof.
Leta¯=ax/rα; since α<n/p<2, it follows that a¯∈LpM, and then from Theorem 3 and Theorem 4, we get the result.
In the case ofα=2, a¯ does not necessarily belong to LpM, and Theorem 3 is no longer valid, so we look for subcritical cases 0<α<2 and we tend α to 2. This can be done by adding an assumption and using Lebesgue’s theorem. Let Pg be such that(130)Pg,α=Δg+arα.Proposition 3.
IfPg,2 is coercive, the following norm is equivalent to the usual norm on H12M:(131)u=∫M∇gu2+ar2u2dvg1/2=∫MuPg,2udvg1/2.Proof.
IfPg is coercive, then there exits c0>0 such that, for any u∈H12M,(132)∫MuPg,2udvg≥c0uH12M2.
By Hardy–Sobolev inequality (121) with p=2 and γ=−2, we get(133)∫MuPg,2udvg≤∫M∇gu2dvg+a∞∫Mu2r2dvg≤∫M∇gu2dvg+a∞Kn,2,−22+ϵ∇gu22+Aϵ,2,−2u22≤max1,a∞Kn,2,−22,a∞Aϵ,2,−2uH12M2≤cαuH12M2,where(134)cα=max1,a∞Kn,2,−22,a∞Aϵ,2,−2>0.
Therefore,(135)c0uH12M2≤∫MuPg,2udvg≤cαuH12M2.Proposition 4.
LetM,g be a Riemannian compact manifold of dimension n≥5 such that the metric g∈H2pM,T∗M⊗T∗M. Assume that f is a positive C∞M function on M and P∈M such that fP=supx∈Mfx. If μM,g>0 and(136)ΔfPfP<SgP3n+2n−4−ap−1ap+1−1,where a=n−2/4n−1Sg, then there exists a metric g¯=uN−2g conformal to g such that the scalar curvature Sg¯=f.Proof.
As in Section1, if the singular Yamabe invariant μM,g>0, the singular Yamabe operator Pg=Δg+a is weakly conformally invariant, and by Theorem 3, Theorem 4, and Theorem 5, there exists u∈C1−n/pM, u>0, solution of the following equation:(137)Δgu+n−24n−1Sgu=fuN−2u.
On the contrary, by the weak conformal invariance ofPg and if g¯=uN−2g is conformal to g, one has(138)Δgu+n−24n−1Sgu=Sg¯uN−2u.
Then, we deduce that the metricg¯=uN−2g is such that the scalar curvature Sg¯=f.


## 6. The Critical Caseα = 2

Keeping the notations adapted above, in this section, we are going to prove the second main theorem, Theorem2.Proof.Step 1.
We haveF2u>−∞ and(139)limα⟶2λαM,g=λ2M,g.
LetδM be the injectivity radius of M. For any ϵ>0, there exists δ such that 0<δ<min1,δM, and if Q∈BP,δ, we get aQ−aP<ϵ, and also, if u∈H12M with ∫MfuNdvg=1+a/r2pN/2, then we have(140)∫Mar2u2dvg≥aP−ϵ∫BP,δr−2u2dvg−a∞δ2u22.
By applying Theorem7, the following inequality holds:(141)∫BP,δu2r2dvg≤K2n,2,−2+ϵ∇gu22+Aϵ,2,−2u22.
Combining (140) and (141), we get(142)∫Mau2r2dvg≥minaP,0−ϵK2n,2,−2+ϵ∇gu22+minaP,0−ϵAϵ,2,−2−a∞δ2u22.
Independently, by Ho¨lder’s inequality, one has(143)u22≤1+ar2pinfx∈Mfx−2/NVM1−2/N.
Then, we get that(144)∫Mau2r2dvg≥minaP,0−ϵK2n,2,−2+ϵ∇gu22+minaP,0−ϵAϵ,2,−2−a∞δ21+ar2pinfx∈Mfx−2/NVM1−2/N.
Hence,(145)F2u≥∇gu22+minaP,0−ϵK2n,2,−2+ϵ∇gu22+minaP,0−ϵAϵ,2,−2−a∞δ21+ar2pinfx∈Mfx−2/NVM1−2/N,which yields(146)F2u≥1+minaP,0−ϵK2n,2,−2+ϵ∇gu22+minaP,0−ϵAϵ,2,−2−a∞δ21+ar2pinfx∈Mfx−2/NVM1−2/N.
Now, if1+aPK2n,2,−2>0, then we can chose ϵ sufficiently small such that the first term of the right-hand side of the latter equality will be strictly positive; then, it follows that(147)F2u>minaP,0−ϵAϵ,2,−2−a∞δ21+ar2pinfx∈Mfx−2/NVM1−2/N.
Consequently,(148)F2u>−∞.
On the contrary, by Lebesgue’s dominated convergence theorem, we get that(149)∫Marαu2dvg⟶∫Mar2u2dvg,whenα⟶2.
Hence,(150)limα⟶2Eαu=E2u,and by passing to the infimum over u such that ∫MfuNdvg=1+a/rαpN/2, we obtain(151)limα⟶2λαM,g=λ2M,g,which implies that there exists α0 such that, for all α∈α0,2, we also have(152)λαM,g≤1+ar2pK0−2n,1fP−2/N.Step 2.
We claim that the sequenceuαα is bounded in H12M and converges to a weak solution.
In a similar way, by Proposition4 and from Ho¨lder’s inequality, we get then for all α∈α0,2, the solution uα of equation (123) satisfies(153)uα22≤1+a/rαpinfx∈Mfx−2/NVM1−2/N.
Knowing that(154)Fαuα=∫M∇guα2+arαu2dvg≥∫M∇guα2+∫BP,δarαu2−uα∞δ2uα22dvg,we get by the assumption made in (128) that(155)1+ar2pK0−2n,1fP−2/N≥Fαuα≥∇guα22+∫BP,δarαu2−uα∞δ2∫Muα2dvg,which leads to(156)1+ar2pK0−2n,1fP−2/N+uα∞δ2∫Muα2dvg≥∇guα22+∫BP,δarαu2dvg,and by letting(157)A=1+ar2pK0−2n,1fP−2/N,the last inequality will be written as(158)A+uα∞δ2∫Muα2dvg≥∇guα22+∫BP,δarαuα2dvg.
By applying (153), we still get(159)A+a∞δ21+arαpinfx∈Mfx−2/NVM1−2/N≥∇guα22+∫BP,δarαuα2dvg,and as(160)1+ar2p≤1+arαp,by letting(161)B=1+arαpK0−2n,1fP−2/N+a∞δ2infx∈Mfx−2/NVM1−2/N,we also get(162)B≥∇guα22+∫BP,δarαuα2dvg.
Now, by Lemma1 and for δ sufficiently small, we obtain(163)∫BP,δarαuα2dvg≥minaP,0−ϵK2n,2,−2∇guα22.
Then, inequality (162) gives(164)1+minaP,0−ϵK2n,2,−2∇guα22≤B.
Then, the sequenceuαα is bounded in H12M, and then there exists a sequence αm∈α0,2 which converges to 2 such that the sequence uαmm=umm converges weakly in H12M, and after restriction to a subsequence still labeled umm, we may assume that there exists u∈H12M such that(i)
um⟶u weakly in H12M.(ii)
um⟶u strongly in LpM for all p<N and almost everywhere on M. Then, from Hardy–Sobolev’s embedding Theorem 8, we have um⟶u weakly in L2M,r−2, and we deduce that, for any φ∈L2M,(165)∫Mar2umφdvg=∫Mar2uφdvg+o1.
In particular, for anyφ∈H12M, we have(166)∫MΔgum+arαmumφdvg=λmM,g∫MfumN−2umφdvg.
By the weak convergence inH12M, we get(167)∫MφΔgumdvg=∫MφΔgudvg+o1,∫Mφarαmum−ar2udvg=∫Mφarαmum−ar2um+ar2um−ar2udvg=∫Maφum1rαm−1r2dvg+∫Mar2φum−udvg.
Again with the weak convergence inL2M,r−2 and Lebesgue’s dominated convergence theorem, we get that(168)∫Maφum1rαm−1r2dvg+∫Mar2φum−udvg⟶0.
On the contrary, sinceum is bounded in LNM, the sequence umN−2umm is bounded in LN/N−1M; hence,(169)λmM,g∫MfumN−2umφdvg⟶λM,g∫MfuN−2uφdvg.Step 3.
u is nonidentically null.
By Sobolev’s inequality applying toumm, we can easily have(170)1+arαmpfP−2/N≤umN2≤K02n,1∇gum22+Bum22,and since um satisfies (123), we also have(171)∇gum22=λmM,g−∫Mum2arαmdvg.
Writing(172)∫Maum2rαmdvg≥minaP,0−ϵK2n,2,−α+ϵ∇gum22+minaP,0−ϵAϵ,2,−α−a∞δ2um22,then(173)∇gum22≤λmM,g−minaP,0−ϵK2n,2,−α+ϵ∇gum22−minaP,0−ϵAϵ,2,−α−a∞δ2um22.
It follows that(174)1+minaP,0−ϵK2n,2,−α+ϵ∇gum22≤λmM,g−minaP,0−ϵAϵ,2,−α−a∞δ2um22,which implies(175)∇gum22≤λmM,g−minaP,0−ϵAϵ,2,−α−a∞/δ2um221+minaP,0−ϵK2n,2,−α+ϵ.
Plugging this in (170), we get(176)1+arαmpfP−2/N≤λmM,g−minaP,0−ϵAϵ,2,−α−a∞/δ2um221+minaP,0−ϵK2n,2,−α+ϵK02n,1+Bum22.
Hence,(177)1+arαmpfP−2/N−λmM,gK02n,11+minaP,0−ϵK2n,2,−α+ϵ≤B−K02n,1minaP,0−ϵAϵ,2,−α−a∞/δ21+minaP,0−ϵK2n,2,−α+ϵum22.
In the end, if (22) is satisfied, as above for m large enough, we can chose ϵ sufficiently small such that(178)0<1+arαmpfP−2/N−λmM,gK02n,11+minaP,0−ϵK2n,2,−α+ϵ.
Then, it follows that the solutionu is not trivial.



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*Source: 4589864-2020-05-20.xml*