INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BR´EZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL∗

Jing ZHANG（张靖）

Mathematics Science College，Inner Mongolia Normal University，Hohhot 010022，China

E-mail∶jinshizhangjing@eyou.com

Shiwang MA（马世旺）

School of Mathematical Sciences and LPMC，Nankai University，Tianjin 300071，China

E-mail∶shiwangm@163.net



INFINITELY MANY SIGN-CHANGING SOLUTIONS FOR THE BR´EZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL∗

Jing ZHANG（张靖）

Mathematics Science College，Inner Mongolia Normal University，Hohhot 010022，China

E-mail∶jinshizhangjing@eyou.com

Shiwang MA（马世旺）

School of Mathematical Sciences and LPMC，Nankai University，Tianjin 300071，China

E-mail∶shiwangm@163.net

In this article，we give a new proof on the existence of infinitely many signchanging solutions for the following Br´ezis-Nirenberg problem with critical exponent and a Hardy potential

Critical exponent；sign-changing solutions；minimax method；hardy potential

2010 MR Subject Classification35J60；35B33；47J30

1　Introduction and Main Results

In recent years，much attention has been paid to the existence of nontrivial solutions of problem（1.1）.The main reason of interest in Hardy term relies in their criticality.Indeed，Hardy term has the same homogeneity as the Laplacian and does not belong to the Kato class，hence it can not be regarded as a lower order perturbation term.Another reason why we investigate（1.1），in addition to the inverse square potential，is the presence of the critical Sobolev exponent. In last two decades，loss of compactness leads to many interesting existence and nonexistence phenomena for elliptic equations.

Let λi（i=1，2，···）be the eigenvalues of the positive operatorwith zero Dirichlet boundary condition.In view of［11，12］，each eigenvalue λiis positive，isolated，and has finite multiplicity，the smallest eigenvalue λ1is simple，and λi→∞as i→∞.

Cao and Peng［6］also considered problem（1.1）and proved that for N≥7，0≤µ＜µ−4，problem（1.1）possesses at least a pair of sign-changing solutions for any λ∈（0，λ1）.In［5］，Cao and Han showed that ifthen problem（1.1）admits a nontrivial solution for all λ＞0.Jannelli［15］proved that if 0≤µ≤µ−1，problem（1.1）admits a positive solution for all λ∈（0，λ1）；while ifµ−1＜µ＜µand Ω=B（0，1）is a ball，there exits λ∗∈（0，λ1），such that problem（1.1）admits a positive solution if and only if λ∈（λ∗，λ1）.These results in［15］show that any dimension N may be critical for problem（1.1）and is different from the case µ=0，where problem（1.1）turns out to be the well-known Brezis-Nirenberg problem：

Sun and Ma［19］use a combination of invariant sets method and Ljusternik-Schnirelma n theory to prove that the above equation has infinitely many sign-changing solutions.

Recently，using an abstract theorem on the estimate of the Morse index for sign-changing critical points which is introduced by Schechter and Zou［17］，Chen and Zou［9］established the following result.

Theorem 1.1Suppose N≥7，λ＞0，and 0＜µ＜µ−4，problem（1.1）has infinitely many sign-changing solutions.

We will prove Theorem 1.1 by applying the usual Ljusternik-Schnirelman theory in conjunction with invariant set method，essentially from［19］，which is much simpler than the proof of［17］.Note that if λ≥λ1，then any nontrivial solution of（1.1）is sign-changing and by the result in［7］，it has infinitely many sign-changing solutions for this case.Therefore，to prove Theorem 1.1，it suffices to consider the case of 0＜λ＜λ1.

Throughout this article，we denote the norm of ，and we shall work on）equipped with the inner product

and noting that 0≤µ＜µ，it is seen that‖u‖µis equivalent to the usual norm

One of the major difficulty to prove the existence of infinitely many solutions for（1.1）using the variational methods is that I（u）does not satisfy the（PS）condition for large energy level because 2∗is the critical exponent for the Sobolev embedding from H10（Ω）to Lq（Ω）.To overcome the difficulty，we first look at the following perturbed problem：

where ε＞0 is a small constant.

We consider the functional Iε：Hµ→R given by

Because the embedding from Hµto Lq（Ω）is compact for any 1≤q＜2∗，and now Iε（u）is an even functional and satisfies the Palais-Smale condition at all energy level.So，from［1，16］Iε（u）has infinitely many critical points.More precisely，there are positive numbers···，withl→+∞and a critical point uε，lfor Iε（u），satisfying

The next procedure is to show that for each fixed k≥2，‖uε，k‖are uniformly bounded with respect to ε.Then，we can apply the following compactness result（see Theorem 1.1 in［7］），which essentially follows from the uniform bounded theorem due to Devillanova and Solimini［10］，to show that uε，kconverges strongly to ukin Hµas ε→0.Thus，it is easy to check that ukis a solution of（1.1）with

Theorem 1.2Assume that N≥7，λ＞0，µ∈［0，µ−4）.Then，for any sequence un（n= 1，2，...），which is a nontrivial solution of（1.3）with ε=εn→0 and satisfying‖un‖＜C for some positive constant independent of n，unconverges strongly in Hµas n→∞.

Remark 1.3According to［7］，to obtain Theorem 1.1，it seems that the condition N≥7 cannot be removed，because when 4≤N≤6，Ω is a ball andµ=0，it is proved in［2］that there is a λ∗such that（1.1）has no radial solution which changes sign if 0＜λ＜λ∗.

In the end，we will distinguish two cases to prove that I（u）has infinitely many sign-changing critical points.

Case IThere are 2≤k1＜···＜ki＜···，satisfying ck1＜···＜cki＜···. Case IIThere is a positive integer l such that ck=c for all k≥l.

The central task in this procedure is to deal with Case II.Indeed，this can be done by showing that the usual Krasnoselskii genus of KcW is at least two，where Kc：=｛u∈Hµ：I（u）= c，I′（u）=0｝.Then，our result is obtained.

2　Preliminaries

Let 0＜λ1＜λ2≤···≤λm≤···be the eigenvalues of∈［0，µ），andnormalized eigenfunction corresponding to λi.Denote Hm：= span｛e1，e2，···，em｝，Br=｛u∈Hµ：‖u‖≤r｝，andFix ξ∈（2，2∗）.In the following，we will always assume that ε∈（0，2∗−ξ）.To construct the minimax values for the perturbed functional Iε，the following two technique lemmas are needed.

Lemma 2.1Assume m≥1.Then，there exists R=R（Hm）＞0 such that for all ε∈（0，2∗−ζ），

ProofNoting that

the following inequality，

holds for any ε∈（0，2∗−ξ），where the auxiliary functional is defined by

Because any norm in finite dimensional space is equivalent，it is easy to check that

Then，the result follows.

Lemma 2.2For any ε∈（0，2∗−ξ），there exists p=p（ε），α=α（ε）＞0 such that

ProofOwing to λ1being the first eigenvalues of the positive operatorwe get

Then，we get the result.

This result also implies that 0 is a strict local minimum critical point.It follows that we can construct invariant sets containing all the positive and negative solutions of（1.3）for the gradient flow of Iε.Then，nodal solutions can then be found outside of these sets.For any ε∈（0，2∗−ξ），let Tε：Hµ→Hµbe given by Tε（u）：=for u∈Hµ.Then，the gradient of Iεhas the form∇Iε（u）=u−Tε（u）.Note that the set of fixed points of Tεis the same as the set of critical points of Iε，which is Kε：=｛u∈E：∇Iε（u）=0｝.It is checked that∇Iεis locally Lipschitz continuous.We consider the negative gradient flow σεof Iεdefined by

For any N⊂Hµand δ＞0，Nδdenotes the open δ-neighborhood of N；that is Nδ：=｛u∈Hµ：dist（u，N）＜δ｝whose closure and boundary are denoted by Nδand∂Nδ.

Here and in the sequel，for u∈Hµ，denote u±（x）：=max｛±u（x），0｝，the convex cones ±P=｛u∈Hµ：±u≥0｝.For ϑ＞0，define（±P）ϑ：=｛u∈Hµ：dist（u，±P）＜ϑ｝.We will show that there exists ϑ0＞0 such that（±P）ϑis an invariant set under the descending flow for all 0＜ϑ≤ϑ0（cf.Lemma 2.4 below）.Note that HµW contains only sign-changing functions， whereThus，it follows from a version of the symmetric Mountain Pass Theorem which provides the minimax critical values on HµW that（1.3）has infinitely many sign-changing solutions.

The following result shows that a neighborhood of±P is an invariant set.This result was proved in Lemma 3.1［3］.

Lemma 2.3There exists ϑ0＞0 such that for any ϑ＜ϑ0，there hold

and

Moreover，every nontrivial solutions u∈（+P）ϑand u∈（−P）ϑof（1.3）are positive and negative，respectively.

To prove our main result to gain nodal solutions，we need to have the order structure and the invariant sets of the gradient flow from the above minimax arguments.SetFrom Lemma 2.3，we may choose an ϑ＞0 sufficiently small such that for∀u∈W，σε（t，u）∈W for all tare invariant set.Note that σε（t，∂W）⊂int（W）and Q：=HµW only contains sign-changing functions.Because Iεsatisfies the Palais-Smale condition，we have the following deformation lemma.

Lemma 2.4Assume that Iεsatisfies the（PS）-condition，then there exists an δ0＞0 such that for any 0＜δ＜δ0，there exists η∈C（［0，1］×Hµ，Hµ）satisfying：

（iii）η（t，·）is odd and a homeomorphism of Hµfor t∈［0，1］，

（iv）Iε（η（·，u））is non-increasing，

（v）η（t，W）⊂W for any t∈［0，1］.

ProofOur proof is similar to the proof of Lemma 5.1 in［14］（also see Lemma 2.4 in［13］）. Due to the（PS）-condition，we may choose δ0＞0 such that

Consider

Then，σε（t，u）is well-defined and continuous on R×Hµ.We claim that η（t，u）=σε（ρt，u）has all the properties in the lemma.

In fact，（i），（iii），（iv），and（v）are easily checked.To obtain（ii），we suppose by contradiction that η（1，u）

which is a contradiction.This contradiction establishes（ii）.

3　Proof of Theorem 1.1

In the following，λ∈（0，λ1）is fixed.For any ε∈（0，2∗−ξ）small，we define the minimax value cε，kfor the perturbed functional Iε（u）with k=2，3，···.We now define a family of sets for the minimax procedure here.We essentially follow［4］；also see［14］.Define

where R＞0 is given by Lemma 2.1.Note that Gm6=∅，as id∈Gm.Set

for k≥2.From［16］，Γkpossess the following properties：

（1◦）Γk6=∅and Γk+1⊂Γkfor all k≥2.

（2◦）If ϕ∈C（Hµ，Hµ）is odd and ϕ=id on∂BR∩Hm，then ϕ（Y）∈Γkif Y∈Γkfor all k≥2.

（3◦）If Y∈Γk，Z=−Z is open and γ（Z）≤s＜k and k−s≥2，then Y∈Γk−s. Now，for k≥2，we define

Lemma 3.1For any Y∈Γkand k≥2，Y∩Q 6=∅，then cε，kare well defined，and cε，k≥α＞0，where α is given by Lemma 2.2.

ProofConsider the attracting domain of 0 in Hµ：

As 0 is a local minimum of Iεand by the continuous dependence of ODE on initial data，we can note that D is an open set.Moreover，∂D is an invariant set andParticularly，for every，there holds Iε（u）＞0（cf.Lemma 3.4 in［3］）.

Next，we claim that for any Y∈Γkwith k≥2，

If this is true，then we obtain Y∩Q 6=∅and cε，2≥α＞0，because ofby Lemma 2.2.To obtain（3.1），let Y=g（BR∩HmX）with γ（X）≤m−k and k≥2.

Define O：=｛u∈BR∩Hm：g（u）∈D｝.Then，O is a bounded open symmetric set with 0∈O and O⊂BR∩Hm.Thus，it follows from the Borsuk-Ulam theorem that γ（∂O）=m and by the continuity of g，g（∂O）⊂∂D.

In conclusion，g（∂OX）⊂Y∩∂D，and as a result of the“monotone，sub-additive and supervariant”property of the genus（cf.Proposition 5.4 in［18］），we get

Due to（+P）δ∩（−P）δ∩∂D=∅，γ（W∩∂D）≤1.Thus，for k≥2，and according to Y∩∂D⊂（Y∩∂D∩Q）∪（∂D∩W），we deduce that

then the lemma is established.

Lemma 3.2Kε，cε，k∩Q 6=∅.

ProofHere，we deduce by a contradiction.Assume∩Q=∅.Using Lemma 2.4 for the functional Iε，there exist δ＞0，and a map η∈C（［0，1］×Hµ，Hµ）such that η（1，·）is odd，η（1，u）=u for u∈，and

By the definition of cε，k，there exists Y∈Γksuch thatIt follows from（3.2）thatIn contrast，it is easy to check that U∈Γkby Lemma 2.1 and（2◦）above.As a result，cε，k≤cε，k−δ.This is a illogicality to δ＞0.

The above lemma implies that there exists a sign-changing critical point uε，ksuch that Iε（uε，k）=cε，k.As a consequence of Lemma 3.1，we know that cε，kare well defined for all k≥2 and 0＜α≤cε，2≤cε，3≤···≤cε，k≤···.Now，we want to show

Lemma 3.3cε，k→∞as k→∞.

ProofHere，we deduce by a negation.Suppose cε，k→c¯ε＜∞as k→∞.As Iεsatisfies the（PS）condition，it follows that Kε，¯cε6=∅and is compact.In addition，there holdsis a sequence of sign-changing solutions to（1.3）withThen，by Sobolev embedding，where m0is a constant independent of n.This implies that theis still sign-changing.

Now，using Lemma 2.4 for the functional Iε，there exist δ＞0 and a map η∈C（［0，1］×Hµ，Hµ）such that η（1，·）is odd，，and

Because cε，k→¯cεas k→∞，we can choose k large enough such thatholds. Indeed，By the definition ofwe can find a setthat is Y=，where g∈Gm，m≥k+l，γ（X）≤m−（k+l），such that

Let X1=X∪g−1（M）.Then，X1is symmetric and open，and γ（X1）≤γ（X）+γ（g−1（M））≤m−（k+l）+l=m−k.Then，it is easy to check that~Y：=η（1，g（BR∩HmX1））∈Γkby（2◦）and（3◦）above.In the end，by（3.4），we obtain

Lemma 3.4For any fixed k≥2，‖uε，k‖is uniformly bounded with respect to ε，and then uε，kconverges strongly to ukin Hµas ε→0.

ProofAs a matter of fact，using the same Γkabove，we can also define the minimax value for the auxiliary functional I∗（see（2.1）），

Now，choosing R＞0 sufficiently large if necessary，Lemma 2.2 also applies to I∗.Then，from a Z2version of the Mountain Pass Theorem（see Theorem 9.12 in［16］），for each k≥2，βk＞0 is well defined and βk→∞as k→∞.Because

holds for any ε∈（0，2∗−ξ），by the definition of cε，kand βk，we have

Consequently，for any fixed k≥2，cε，kis uniformly bounded for ε∈（0，2∗−ξ），that is，there exists C=C（βk，Ω）＞0 independent on ε，such that cε，k≤C uniformly for ε.Because uε，kis a nodal solution of（1.3）and Iε（uε，k）=cε，k，and the fact that λ1is the first eigenvalue of the operator，one concludes that

Proof of Theorem 1.1Now，we prove Theorem 1.1.Noting that ckis non-decreasing with respect to k，we have the following two cases：

Case I.There are 2≤k1＜···＜ki＜···，satisfying ck1＜···＜cki＜···.Obviously，in this case，equation（1.1）has infinitely many sign-changing solutions such that I（ui）=cki.

Case II.There is a positive integer τ such that ck=c for all k≥τ.

So，our main task is to deal with Case II.We also begin the proof with a absurdity.Suppose that there exists a δ＞0，such that I（u）has no sign-changing critical point u with I（u）∈［c−δ，c）∪（c，c+δ］.Otherwise，we are done.

In this case，first of all，we claim that γ（K2c）≥2，where Kc：=｛u∈E：I（u）=c，I′（u）=0｝and=Kc∩Q.Then，by the property of the genus，we obtain I（u）that has infinitely many sign-changing critical points.

In the following，we utilize a technique in the proof of Theorem 1.1 in［8］.Assume，to the opposite，that=1（because of K2c6=∅）.Moreover，we assume thatcontains only finitely many critical points，otherwise we are done.Then，it follows thatis compact. Obviously，0/∈Then，there exists a open neighborhood N in Hµwith⊂N such that γ（N）=γ）.Define

Secondly，we now claim that if ε＞0 small，Iε（u）has no sign-changing critical point u∈Vε. Factually，arguing indirectly，suppose that there exist εn→0 and un∈Vεnsatisfying0，with6=0，and un/∈N.Then，by Theorem 1.2，up to a subsequence，unconverges strongly to u in Hµ.Therefore，I′（u）=0，I（u）∈［c−δ，c+δ］，and u/∈This is a contradiction to our assumption and the fact that u is still sign-changing.

The following proof is similar to that of Lemma 3.2.Using Lemma 2.1 for the functional Iε，there exist δ＞0 and a map η∈C（［0，1］×E，E）such that η（1，·）is odd，η（1，u）=u for u∈Ic−2δε，and

Let~X=X∪g−1（N）.Then，~X is symmetric and open，and γ（~X）≤γ（X）+γ（g−1（N））≤m−（k+1）+1=m−k.Then，it is easy to check thatbY：=η（1，g（BR∩Hm~X））∈Γkby（2◦）and（3◦）above.As a result，by（3.6），

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October 30，2014；revised October 22，2015.Research supported by the Specialized Fund for the Doctoral Program of Higher Education and the National Natural Science Foundation of China.

where Ω is a smooth open bounded domain of RNwhich contains the origin，is the critical Sobolev exponent.More precisely，under the assumptions that N≥7，µ∈［0，µ¯−4），and，we show that the problem admits infinitely many sign-changing solutions for each fixed λ＞0.Our proof is based on a combination of invariant sets method and Ljusternik-Schnirelman theory.