# On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

Authors: Anna Maria Micheletti; Angela Pistoia
Journal: International Journal of Differential Equations
 (2010)

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## Abstract

Given that(M,g)is a smooth compact and symmetric Riemannian n-manifold, n≥2, we prove a multiplicity result for antisymmetric sign changing solutions of the problem −ε2Δgu+u=|u|p−2u in M. Here p>2 if n=2 and 2<p<2∗=2n/(n−2) if n≥3.


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## Body

## 1. Introduction

Let(M,g) be a smooth compact connected Riemannian manifold without boundary of dimension n≥2. Let us consider the problem
(1.1)-ε2Δgu+u=|u|p-2uinM,u∈Hg1(M),
where p>2 if n=2,2<p<2n/(n-2) if n≥3 and ε is a positive parameter. Here Hg1(M) is the completion of C∞(M) with respect to
(1.2)∥u∥g2:=∫M|∇gu|2dμg+∫Mu2dμg.
It is well known that any critical point of the energy functional Jε:Hg1(M)→ℝ constrained to the Nehari manifold 𝒩ε is a solution to (1.1). Here
(1.3)Jε(u)≔1εn∫M(12ε2|∇gu|2+12u2-1p|u|p)dμg,(1.4)𝒩ε:={u∈Hg1(M)∖{0}:Jε′(u)[u]=0}.
In [1] the authors show that the least energy solution of (1.1), that is, the minimum of Jε on 𝒩ε is a positive solution with a spike layer, whose peak converges to the maximum point of the scalar curvature 𝒮g of (M,g) as ε goes to zero. Successively, in [2] (see also [3, 4]) the authors point out that the topology of the manifold M influences the multiplicity of positive solutions of (1.1), that is, (1.1) has at least cat(M) nontrivial solutions provided that ε is small enough. Here cat(M) denotes the Lusternik-Schnirelman category of M. Recently, in [5–7] it has been proved that the existence of positive solutions is strongly related to the geometry of M, that is stable critical points of the scalar curvature 𝒮g generate positive solutions with one or more peaks as ε goes to zero.As far as it concerns the existence of sign changing solutions to (1.1), a few results are known. The first result has been obtained in [7] where it has been constructed solutions with one positive peak and one negative peak, which approach, as ε goes to zero, the minimum point and the maximum point of 𝒮g, provided the scalar curvature is not constant. In [8] the authors assume the following:(S)
the manifoldMis a regular submanifold ofℝNinvariant with respect toτ, whereτ:ℝN→ℝNis an orthogonal linear transformation such thatτ≠Iandτ2=I,Ibeing the identity ofℝN.They prove problem (1.1) has at least Gτ-cat(M-Mτ) pairs of sign changing solutions which change sign exactly once. Here Gτ-cat(M-Mτ) denotes the Gτ-equivariant Lusternik-Schnirelman category for the group Gτ:={I,τ} and Mτ:={x∈M:τx=x}.In this paper we assumeM satisfies (S) in the particular case τ=-I. We look for solutions of the problem
(1.5)-ε2Δgu+u=|u|p-2uinM,u∈Hg1(M),u(-x)=-u(x).
We evaluate the number of solutions of problem (1.5) using Morse theory. Our main result reads as following.Theorem 1.1.
Assume that forε small enough all the solutions to problem (1.5) with energy close to 2m∞ are nondegenerate. Then there are at least P1(M/G) pairs (u,-u) of nontrivial solutions to (1.5) which change sign exactly once, where
(1.6)m∞:=inf∫ℝn|∇u|2+u2=∫ℝn|u|p∫ℝn(12|∇u|2+12u2-1p|u|p)dx.
Here G={I,-I} and P1(M/G) is the Poincaré polynomial Pt(M/g) when t=1.Concerning the assumptions of nondegeneracy of all the critical points with energy close to2m∞, we think that it is true “generically” in some sense with respect to (ε,g) where ε is a positive parameter and g is a Riemannian metric.We point out that problem (1.1) has been widely studied when the manifold M is replaced by an open bounded and smooth domain in ℝN with Dirichlet or Neumann boundary condition. In particular, it has been studied the effect of the domain topology or the domain geometry on the number of solutions. See, for example, [9–19] for the Dirichlet problem and [20–32] for the Neumann problem,The paper is organized as follows. In Section2 we set the problem and we recall some known results; in Section 3 we give the proof of Theorem 1.1; in Section 4 we prove the technical Lemma 4.5, which is crucial for the proof of Theorem 1.1.


## 2. Setting of the Problem

First of all, we will recall some topological notions which are used in the paper.Definition 2.1 (Poincaré polynomial).
If(X,Y) is a couple of the topological spaces, the Poincaré polynomial Pt(X,Y) is defined as the following power series in t:
(2.1)Pt(X,Y):=∑kdimHk(X,Y)tk,
where Hk(X,Y) is the kth homology group with coefficients in some fields. Moreover, we set
(2.2)Pt(X):=Pt(X,∅)=∑kdimHk(X)tk.
If X is a compact manifold, we have that dimHk(X)<+∞ and in this case Pt(X) is a polynomial and not a formal series.Definition 2.2 (Morse index).
LetJ be a C2-functional on a Banach space X and u∈X an isolated critical point of J with J(u)=c. If Jc:={v∈X:J(v)≤c} then the (polynomial) Morse index it(u) of u is the following series:
(2.3)it(u):=∑kdimHk(Jc,Jc∖{u})tk,
where Hk(Jc,Jc∖{u}) is the kth homology group of the couple (Jc,Jc∖{u}). If u is a nondegenerate critical point of J then it(u)=tμ(u), where μ(u) is the (numerical) Morse index of u and it is given by the dimension of the maximal subspace on which the bilinear form J′′(u)[·,·] is negatively definite.It is useful to recall the following result (see [33]).Remark 2.3.
LetX and Y be topological spaces. If f:X→Y and g:Y→X are continuous maps such that g∘f is homotopic to the identity map on X then Pt(Y)=Pt(X)+Z(t), where Z(t) is a polynomial with non negative coefficients.Now, let us point out that the transformationτ=-I:M→M induces a transformation on Hg1(M). We define the linear operator τ* as follows:
(2.4)τ*:Hg1(M)→Hg1(M),τ*(u(x)):=-u(-x).
The operator τ* is selfadjoint with respect to the following scalar product on Hg1(M), which is equivalent to the usual one:
(2.5)〈u,v〉ε:=1εn∫M(ε2∇gu∇gv+uv)dμg,
which induces the norm
(2.6)∥u∥ε2:=1εn∫M(ε2|∇gu|2+u2)dμg.
In particular, we have
(2.7)∥τ*u∥ε,p=∥u∥ε,p,∥τ*u∥ε=∥u∥ε,Jε(τ*u)=Jε(u).
Here
(2.8)∥u∥ε,pp:=1εn∫M|u|pdμg
denotes the norm in Lp(M), which is equivalent to the usual one. Therefore, in virtue of the Palais Principle, the nontrivial solutions of (1.5) are the critical points of the restriction of Jε to the τ-invariant Nehari manifold
(2.9)𝒩ετ:={u∈𝒩ε:u(-x)=-u(x)}=𝒩ε∩Hgτ,
where Hgτ:={u∈Hg1(M):u(-x)=-u(x)}.In fact, sinceJ-ε(τ*u)=Jε(u) and τ* is a selfadjoint operator, we have
(2.10)〈∇Jε(τ*u),τ*φ〉ε=〈∇Jε(u),φ〉ε∀φ∈Hg1(M)
and so ∇Jε(u)=τ*∇Jε(τ*u)=τ*∇Jε(u) if (τ*u)(x)=u(x)=-u(-x).Let us set(2.11)mε:=inf𝒩εJε,mετ:=inf𝒩ετJε
and let m∞ be as in (1.6).It is easy to verify thatJε satisfies the Palais-Smale condition on 𝒩ετ. Then, there exists vε minimizer of mετ and vε is a critical point of Jε on Hg1(M). Thus vε+ and vε- belong to 𝒩ε, then mετ=Jε(vε)≥2mε. We recall that limε→0mε=m∞ as it has been shown in [2, Remark 5.9].It is well known that there exists a unique positive spherically symmetric (with respect to the origin) functionU∈H1(ℝn) minimizer of m∞. Obviously this fact implies that -ΔU+U=Up-1 in ℝn and for any ε>0 we can define a family of functions Uε(x):=U(x/ε) satisfying the following equation -ε2ΔUε+Uε=Uεp-1 in ℝn.On the tangent bundle of any compact connected Riemannian manifoldM, it is defined the exponential map exp:TM→M which is a C∞-map. Then for ρ sufficiently small (smaller than the injectivity radius of M) the manifold M possesses a special set of charts given by expx:B(0,ρ)→Bg(x,ρ), where TxM is identified with ℝn for x∈M. Here B(0,ρ) denotes the ball in ℝn centered at 0 with radius ρ and Bg(x,ρ) denotes the ball in M centered at x with radius ρ with the distance given by the metric g. The system of coordinates corresponding to those charts are called normal coordinates.


## 3. The Main Ingredient of the Proof

Let us sketch the proof of our main result.Sincelimε→0mετ=2m∞ (see Lemma 4.3), given δ∈(0,m∞/4) for ε small enough, we have 0<2(m∞-δ)<mετ<2(m∞+δ). Thus 2(m∞-δ) is not a critical value of Jε for any ε. Fixed ε, if the number of critical points of Jε is finite in Jε2(m∞+δ), we can choose δ such that 2(m∞+δ) is not a critical value of Jε.Let𝒩ετ/ℤ2 be the set obtained by identifying antipodal points of the Nehari manifold 𝒩ετ. It is easy to check that the set 𝒩ετ/ℤ2 is homeomorphic to the projective space P∞:=∂Σ1/ℤ2, which is obtained by identifying antipodal points in un unit sphere ∂Σ1 in the space Hgτ.We are looking for pairs of nontrivial critical points(u,-u) if the functional Jε:Hgτ→ℝ, that is we are searching critical points for the functional J̃ε:Hgτ∖{0}/ℤ2→ℝ defined byJ̃ε([u]):=Jε(u)=Jε(-u). We use the same arguments as in [33]. The following relation can be proved as in [33, 34] (see [33,Lemma 5.2]):(3.1)Pt(J̃ε2(m∞+δ),J̃ε2(m∞-δ))=tPt(J̃ε2(m∞+δ)∩Nετ/ℤ2).
By Lemma 4.5 we deduce that
(3.2)M/G→Φ̃εJ̃ε2(m∞+δ)∩𝒩ετℤ2→β̃MdG,
where β̃∘Φ̃ε is homotopic to the identity map and Md/G is homotopically equivalent to Mg. Therefore by Remark 2.3 we get
(3.3)Pt(J̃ε2(m∞+δ)∩𝒩ετℤ2)=Pt(MG)+Z(t),
where Z(t) is a polynomial with nonnegative integer coefficients.By our assumption we have that forε small enough all the critical points u such that J̃ε(u)<2(m∞+δ) are nondegenerate. Moreover the functional J̃ε satisfies the Palais-Smale condition. Then by Morse theory and relations (3.1) and (3.3) we get at least P1(M/G) pairs (u,-u) of nontrivial solutions for (1.5). By Remark (4.7) these solutions change sign exactly once. That concludes the proof of Theorem 1.1.Remark 3.1.
By [33, Lemma 5.2] we deduce that
(3.4)Pt(Hgτ∖{0}ℤ2,J̃ε2(m∞-δ))=tPt(𝒩ετℤ2).
Since P∞ is homeomorphic to 𝒩ετ/ℤ2 we get Pt(𝒩ετ/ℤ2)=Pt(P∞). Provided the homology is evaluated with 𝒵2-coefficients (see, e.g.,  [35, Theorem 7.4]), we have P1(P∞)=+∞. Then, if all the critical points are nondegenerate, we get infinitely many pairs (u,-u) of nontrivial solutions for (1.5).


## 4. Technical  Results

Letχr be a smooth cut-off function such that
(4.1)χr(z)=1ifz∈B(0,r2),χr(z)=0ifz∈ℝN∖B(0,r),|∇χr(z)|≤2∀z∈ℝN.
Fixing a point q∈M and ε>0, let us define the function wε,q on M as(4.2)wε,q(x):=Uε(expq-1(x))χr(expq-1(x))ifx∈Bg(q,r)wε,q(x):=0otherwise.
We choose r smaller than the injectivity radius of M and such that Bg(q,r)∩Bg(-q,r)=∅ for any q∈M. For any ε>0 we can define a positive number t(wε,q) such that
(4.3)Φε(q):=t(wε,q)wε,q∈Hg1(M)∩𝒩εforanyq∈M.
Namely, t(wε,q) verifies
(4.4)t(wε,q)=[∫M(ε2|∇gwε,q|2+wε,q2)dμg∫Mwε,q2dμg]1/p-2.
In [2, Proposition 4.2] the following lemma has been proved.Lemma 4.1.
Givenε>0 the map Φε:M→Hg1(M)∩𝒩ε is continuous. Moreover, given δ>0 there exists ε0(δ) such that if ε∈(0,ε0(δ)) then Φε(q)∈𝒩ε∩Jεm∞+δ.Now, fixing a pointq∈M let us define the function
(4.5)Φετ(q):=t(wε,q)wε,q-t(wε,τq)wε,τq.
It holds that
(4.6)∫M|wε,q|2p=∫M|wε,τq|2p,∫M|∇gwε,q|2dμg=∫M|∇gwε,τq|2dμg.
By (4.4) and (4.6), we deduce
(4.7)t(wε,q)=t(wε,τq).
The proof of the next results follows the same arguments as in [8].Lemma 4.2.
Givenε>0 the map Φετ:M→Hg1(M)∩𝒩ετ is continuous. Moreover, given δ>0 there exists ε0(δ) such that if ε∈(0,ε0(δ)) then Φετ(q)∈𝒩ετ∩Jε2(m∞+δ).Proof.
SinceUεχr is a radially symmetric function, we set Ũε(|z|):=Uε(z)χr(z). Moreover, since we have
(4.8)|expτq-1(τx)|=dg(-x,-q)=dg(x,q)=|expq-1(x)|,|expq-1(τx)|=dg(-x,q)=dg(x,-q)=|expτq-1(x)|,
we get
(4.9)τ*Φετ(q)(x)(4.10)=-t(wε,q)wε,q(-x)+t(wε,τq)wε,τq(-x)(4.11)=-t(wε,q)Ũε(|expq-1(-x)|)+t(wε,τq)Ũε(|expq-1(-x)|)=t(wε,τq)Ũε(|expq-1(x)|)-t(wε,q)Ũε(|expq-1(τx)|)=t(wε,q)Ũε(|expq-1(x)|)-t(wε,q)Ũε(|expq-1(x)|)(4.12)=Φετ(q)(x),
because by (4.7) we have t(wε,q)=t(wε,τq). Hence Φετ(q)∈𝒩ετ.
To get thatΦετ(q)∈Jε2(m∞+δ), it is enough to prove that Jε(Φετ(q))=2Jε(Φε(q)), because by Lemma 4.1 the statement will follow. Since the support of the function Φετ(q) is Bg(q,r)∪Bg(-q,r) and Bg(q,r)∩Bg(-q,r)=∅, by (4.6) and the definition of the function Φετ, we get
(4.13)Jε(Φετ(q))=(12-1p)1εn∫M|Φετ(q)|pdμg=(12-1p)1εn(∫Bg(q,r)|Φε(q)|pdμg+∫Bg(-q,r)|Φε(τq)|pdμg)=2Jε(Φε(q)).
That concludes the proof.Lemma 4.3.
One has thatlimε→0mετ=2m∞.Proof.
By Lemma4.2 and (4.12) we have that for any δ>0 there exists ε0(δ) such that for any ε∈(0,ε0(δ)) it holds that
(4.14)2mε≤mετ≤Jε(Φετ(q))=2Jε(Φε(q))≤2(m∞+δ).
Since limε→0mε=2m∞ (see [2,Remark 5.9]) we get the claim.For any functionu∈𝒩ετ we can define a point β(u)∈ℝN by
(4.15)β(u)≔∫Mx|u+(x)|pdμg∫M|u+(x)|pdμg.Lemma 4.4.
There existsδ0>0 such that for any δ∈(0,δ0), for any ε∈(0,ε0(δ)) (as in Lemma 4.2), and for any function u∈𝒩ετ∩Jε2(m∞+δ), it holds that β(u)∈Md, where Md:={x∈ℝN:d(x,M)<d}.Proof.
Letu∈𝒩ετ∩Jε2(m∞+δ). Since u(x)=-u(-x) we set M+:={x∈M:u(x)>0} and M-:={x∈M:u(x)<0}. It is easy to see that M+={-x:x∈M-}. Then we have
(4.16)Jε(u)=(12-1p)1εn∫M|u|pdμg=(12-1p)1εn(∫M+|u+|pdμg+∫M-|u-|pdμg)=2Jε(u+).
Since Jε(u)≤2(m∞+δ), we have Jε(u+)≤m∞+δ and by [2, Proposition 5.10] we get the claim.It is easy to check thatΦετ(-q)=-ϕετ(q) and β(-u)=-β(u). Moreover, by Lemmas 4.1 and 4.2, we can define a map Φ̃ε:M/G→J̃ε2(m∞+δ)∩𝒩ετ/ℤ2 by
(4.17)Φ̃ε([q]):=[Φετ(q)]={Φετ(q),Φετ(-q)}.
By Lemma 4.4 we can define a map β̃:J̃ε2(m∞+δ)∩𝒩ετ/ℤ2→Md/G by
(4.18)β̃([u]):=[β(u)]={β(u),β(-u)}.Lemma 4.5.
There existsε0>0 such that for any ε∈(0,ε0) the map
(4.19)Iε:=β̃∘Φ̃ετ:MG→MdG
is well defined, continuous, and homotopic to the identity map.Proof.
By Lemmas4.2 and 4.4, Iε is well defined. In order to show that Iε is homotopic to the identity, we estimate the following difference:
(4.20)|βΦετ(q)-q|=∫M(x-q)|(Φετ(q))+|pdμg∫M|(Φετ(q))+|pdμg=∫B(0,r)y|U(y/ε)χr(|y|)|p|gq(y)|1/2dy∫B(0,r)|U(y/ε)χr(|y|)|p|gq(y)|1/2dy=ε∫B(0,r/ε)z|U(z)χr(|εz|)|p|gq(εz)|1/2dμg∫B(0,r/ε)|U(z)χr(|εz|)|p|gq(εz)|1/2dμg.
Hence |βΦετ(q)-q|,|βΦετ(-q)+q|≤cε, because βΦετ(-q)=-βΦετ(q), for a constant c which does not depend on the point q. Therefore |Iε(q)-q|<cε; that concludes the proof.Remark 4.6.
We have only to prove that any solutionu of (1.5) such that Jϵ(u)<2(m∞+δ) changes sign exactly once. In fact, assume that the set {u∈M:u(x)>0} has h connected components M1,…,Mh. Set ui(x):=u(x) if x∈Mi∪(-Mi) and ui(x):=0 otherwise. We have ui∈𝒩ετ and
(4.21)32hm∞≤mετ≤Jε(u)=∑i=1hJε(ui)≤2(m∞+δ)<3m∞.
Then h=1. This concludes the proof.



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