一类具有临界Sobolev...的椭圆方程的变号解(英文)_王丽霞.pdf
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1、南 开 大 学 学 报(自然科学版)Acta Scientiarum Naturalium Universitatis NankaiensisVol.561Feb.2023第56卷第1期2023年2月Article ID:0465-7942(2023)01-0061-10Sign-changing Solutions for an Elliptic Equation InvolvingCritical Sobolev and Hardy-Sobolev ExponentWang Lixia,Zhao Pingping(School of Sciences,Tianjin Chengjian U
2、niversity,Tianjin 300384,China)Abstract:The existence of sign-changing solution is studied for an elliptic equation involving critical Sobolevexponent and critical Hardy-Sobolev exponent.By using a compactness result,the existence of infinitely manysign-changing solutions are proved by a combination
3、 of invariant sets method and Ljusternik-Schnirelman typeminimax method.Keywords:critical Hardy exponent;sign-changing solutions;critical Sobolev exponentCLC number:O175.25Document code:A0IntroductionLet 0,N 7,0 t 0,0 t 2,2*(t)=2(N-t)N-2,then(1)has infinitely many sign-王丽霞等:一类具有临界Sobolev和Hardy-Sobol
4、ev指数的椭圆方程的变号解Received date:2021-11-21Foundation item:Supported by National Natural Science Foundation of China(11801400);Scientific Research Programof Tianjin Education Commission(2020KJ045)Biography:Wang Lixia(1982-),female,native place:Hebei Shijiazhuang,associate professor,direction of research:d
5、ifferential equations and dynamical system.Corresponding author:Zhao Pingping(1989-),female,native place:Henan Taikang,lecturer.E-mail:ppzhao_.62 南 开 大 学 学 报(自然科学版)第56卷changing solutions.Let1be the first eigenvalue of-u=a(x)u in,u=0 on.(3)Sincea C1(-)and strictly positive,then(3)has infinitely many
6、eigenvalues 1,2,such that0 1()0.DenoteHm span e1,e2,em.ThenHm Hm+1andH10()=-m=1Hm.It is easy to know that if1 1,equation(1)has infinitelymany sign-changing solutions.Indeed,by multiplying the first eigenfunctione1and integrating both sides,then we can check that if1 1,any nontrivial solution of(1)ha
7、s to change sign.Therefore,by theresult of reference 7,to prove Theorem 1 it suffices to consider the case of1 1.In order to prove the result,Bhakta6first used an abstract theorem which is introduced by reference8.Then by combining this with the uniform bounded theorem due to reference 7,the author
8、obtainedinfinitely many sign-changing solutions.The method introduced in references 6-10 sometimes are limited.Because by general minimax procedure to get the Morse indices of sign-changing critical points sometimesare not clear.Another limited condition is that the corresponding functional is also
9、needed to beC2.Before giving our main results,we give some notations first.We will always denote0 t 2.LetE=H10()be endowed with the standard scalar and norm(u,v)=uvdx;|u=(|u2dx)1 2.The norm onLs=Ls()with1 s is given by|us=(|usdx)1s.Lqt()(1 q ,1 t 0is a small constant.The corresponding energy functio
10、nal isJ(u)=12(|u2-a(x)u2)dx-2*-|u2*-dx-12*(t)-|u2*(t)-|xtdx.(6)By the following lemmas,we will knowJ(u)is aC1function onH10()and satisfies the Palais-Smalecondition.It follows from references 12-13,J(u)has infinitely many critical points.More precisely,there are positive numbersc,l,l=2.3,withc,l+asl
11、 +.Moreover,a critical pointu,lforJ(u)satisfyingJ(u,l)=c,l.Next,we will show that for any fixedl 2,u,lare uniformly bounded with respect to,then we can apply the following compactness result Proposition 17which essentially follows from theuniform bounded theorem due to references 11 to show thatu,lc
12、onverges strongly toulinEas 0.Therefore it is easy to prove thatulis a solution of(1)withJ(ul)=cl lim 0c,l.Proposition 17Suppose thata(0)0and0 ,all the principle curvatures ofat 0 arenegative.IfN 7,then for any sequenceun,which is a solution of(1)with=n 0,satisfyingun Cfor some constant independent
13、ofn,unhas a sequence,which converges strongly inH10()asn +.Now we will distinguish two cases to prove thatJ(u)has infinitely many sign-changing critical points.Case 1:There are2 l1 li,satisfyingcl1 cli.Case 2:There is a positive integerLsuch thatcl=cfor alll L.The central task in this procedure is t
14、o deal with case 2.In fact,we can prove that the usualKrasnoselskii genus ofKcW(Wis denoted in following section)is at least two,whereKc:=u E:J(u)=c,J(u)=0.Then our result is obtained.Throughout this paper,the lettersC,C1,C2,will be used to denote various positive constants whichmay vary from line t
15、o line and are not essential to the problem.The closure and the boundary of setGare denoted by-GandGrespectively.We denoteweak convergence and bystrong convergence.Also if we take a sub-sequence of a sequence un,we shall denote it again un.1PreliminariesNow we give some integrals inequalities,for de
16、tails we refer to reference 14.Lemma 1(Hardy-Sobolev Inequality)LetN 3,0 t 2,then there exists a positive con-stantC=C(N,t)such that()R|u2*(t)|xtdx2 2*(t)CR|u2dx(7)for allu C0(RN).64 南 开 大 学 学 报(自然科学版)第56卷Lemma 210Ifis a bounded subset ofRN,0 t 2,N 3,thenLpt()Lqt()with theinclusion being continuous,
17、whenever1 q p .Remark 1Iff Lpt()for1 p ,thenf Lp()with|fp C|fp,t,.For eachandu E,we define u*=|u2*-+(|u2*(t)-|xtdx)1(2*(t)-).Lemma 310Let1 q 2*(t),0 t 0,such that for all (0,2*(t)-),supBcR HmJ(u)0such that u2*-C u2*-*for allu Hm.ThereforeJ(u)12 u2-2*-|u2*-dx-12*(t)-|u2*(t)-|xtdx12 u2-C u2*-.Since2*-
18、2*(t)-2and1 1,we have thatlim u,u HmJ(u)=-.Lemma 5For any (0,2*(t)-),1 1,there exist=(),=()0such thatinfBJ()u .ProofJ(u)=12(|u2-a(x)u2)dx-2*-|u2*-dx-12*(t)-|u2*(t)-|xtdx12(1-1)a(x)u2dx-C1|u2*-2*-C2|u2*(t)-2*(t)-.Since2*-2*(t)-2and1 1,there exists=(),=()0such thatinfBJ(u).Lemma 5 implies that 0 is a
19、strict local minimum critical point.Then we can construct invariantsets containing all the positive and negative solutions of(1)for the gradient flow ofJ.Therefore,nodal solutions can be found outside of these sets.2AuxiliaryoperatorandinvariantsubsetsofdescendingflowFor any (0,2*(t)-),letA:E Ebe gi
20、ven by第1期王丽霞等:一类具有临界Sobolev和Hardy-Sobolev指数的椭圆方程的变号解 65(u)(-)-1(G(u)+L(u)=(-)-1(|u2*-2-u+|u2*(t)-2-u|xt+a(x)u),whereG(u)=|u2*-2-u+|u2*(t)-2-u|xt,L(u)=a(x)u,foru E.ThenL(u),v=a(x)uvdx,G(u),vH10()=|u2*(t)-2-uvdx+|u2*(t)-2-uv|xtdx,the gradient ofJhas the formJ(u)=u-A(u).Note that the set of fixed point
21、s ofAis thesame as the set of critical points ofJ,which isK u E:J(u)=0.It is easy to check thatJis locally Lipschitz continuous.We consider the negative gradient flowofJdefined byddt(t,u)=-J(t,u)for t 0,(0,u)=0.Here and in the sequel,foru E,denoteu(x)max u(x),0,the convex cones+P=u E:u 0,-P=u E:-u 0
22、.For 0,define(P)=u E:dist(u,P)0suchthat(P)is an invariant set under the descending flow for all0 0,Ndenotes the open-neighborhood ofN,i.e.N u E:dist(u,N)0such that for (0,0,there holdsA(P)(P),and(t,u)(P)for allt 0andu-(P).Moreover,every nontrivial solutionsu (+P)andu (-P)of(5)are positive and negati
23、ve,respectively.To prove our main result,we need to construct nodal solution by using the combination of invariantsets method and minimax method,we need a deformation lemma in the presence of invariant sets.Definition 1A subsetW Eis an invariant set with respect toif,for anyu W,(t,u)Wfor allt 0.From
24、 Lemma 6,we may choose 0sufficiently small such that-(P)are invariant set.SetW-(+P)-(-P).Note that(t,W)int(W)andQ EWonly contains sign-changing functions.SinceJsatisfies the Palais-Smale condition,we have the following deformation lemma whichfollows from references 18-19.DefineK1,c K,c W,K2,c K,c Q,
25、whereK,c u E:J(u)=c,J(u)=0.Let 0besuch that(K1,c)W,where(K1,c)u E:dist(u,K1,c)0such 66 南 开 大 学 学 报(自然科学版)第56卷that for any0 0,there exists C(0,1 E,E)satisfying:(t,u)=ufort=0oru J-1(c-0,c+0)(K2,c).(1,Jc+W(K2,c)3)Jc-Wand(1,Jc+W)Jc-WifK2,c=.HereJd=u E:J 1.For any (0,2*(t)-)small,we define the minimaxval
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