Nonlinear problem with subcritical exponent in Sobolev space

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1 Jebrl Journal of Inequaltes and Applcatons 06 06:305 DOI 0.86/s R E S E A R C H Open Access Nonlnear problem wth subcrtcal exponent n Sobolev space Iqbal H Jebrl * * Correspondence: qbal50@hotmal.com Department of Mathematcs, Tabah Unversty, 3 Almadnah Almunawwarah, Saud Araba Abstract Usng Brouwer s fxed pont theorem, we prove the exstence of solutons for some nonlnear problem wth subcrtcal Sobolev exponent n. MSC: Prmary 6E35; 7H0; secondary 35J60 Keywords: Sobolev spaces; subcrtcal exponent; nonlnear problem Introducton and the man result The exponent Lebesgue space L p sdefnedby L p {u L loc : ux } p dx <. Ths space s endowed wth the norm { u L p nf λ >0: ux λ The Sobolev space W,p sdefnedby p } dx. W,p { u W, loc :u Lp and u L p }. The correspondng norm for ths space s u W,p u L p u L p. Defne W0 H 0 astheclosureofc c wth respect to the W,p normwhch s a Hlbert space []. We consder the problem of the scalar curvature on the standard four dmensonal half sphere under mnmal boundary condtons S: S L g u : g u u Ku 3, u >0 n, u ν 0 on S, The Authors 06. Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton.0 Internatonal Lcense whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal authors and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made.

2 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page of where {x R5 / x,x 5 >0}, g s the standard metrc, and K s a C 3 postve Morse functon on. The scalar curvature problem on S n and also on n wasthesubjectofseveralworksn recent years, we can cte for example [ ]. Recall that the embeddng of H ntol s noncompact. For ths reason, we have focused our study on the famly of subcrtcal problems S ε S ε g u u Ku 3 ε, u >0 n, u ν 0 on S, where ε s a small postve parameter. Note that the solutons of problem S can be the lmt as ε 0 of some solutons u ε for S ε. Djadl et al. [3] studed ths problem n the case of the three dmensonal half sphere. Assumng that the crtcal ponts of K verfy K/ νa > 0 they demonstrated that there exst solutons u ε concentratedatthepontsa,...,a p. Moreover, n [], we establshed the exstence of another type of solutons u ε ofs ε suchthatsconcentratedattwo ponts a and a. In ths work, we am to construct some postve solutons of S ε whch are concentrated at two dfferent ponts of the boundary. To state our result, we wll gve the followng notatons. For a and λ >0,let λ δ a,λ xc 0 λ λ cos da, x, where d s the geodesc dstance on, g andc 0 s chosen so that δ a,λ s the famly of solutons of the followng problem: u u u 3, u >0, ns. The space H s equpped wth the norm and ts correspondng nner product, : f f f, and f, g f g fg, f, g H. Theorem Let z and z be a nondegenerate crtcal ponts of K K wth K/ νz > 0,,.Then there exsts ε 0 >0such that, for each ε 0, ε 0, problem S ε has a soluton u ε of the form u ε α δ x,λ α δ x,λ v, where, as ε 0, α Kz / ; v 0; x z ; x ; λ ; λ cλ o. Therestofthsworkssummarzedasfollows.InSecton,wepresentaclasscalprelmnares and we perform a useful estmatons of functonal I ε assocatedtotheproblem S ε forε > 0 and ts gradent. Secton 3 s devoted to the constructon of solutons and the proof of our result.

3 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page 3 of Useful estmatons We ntroduce the structure varatonal assocated to the problem S ε forε >0 I ε u u u K u ε, u H S ε. It s well known that there s an equvalence between the exstence of solutons for S ε and the postve crtcal pont of I ε. Moreover, n order to reduce our problem to we wll perform some stereographc projecton. We denote D, for the completon of Cc R wth respect to the Drchlet norm. Recall that an sometry ı : H D, s nduced by the stereographc projecton π a about a pont a followng the formula ıφy φ π x a y, φ H, y R. 3 For every φ H, one can check that the followng holds true: φ φ ıφ and φ ıφ. Furthermore, usng 3wthπ a,tseasytoseethatıδ a,λ s gven by ıδ a,λ c 0 λ λ x a. δ a,λ wll be wrtten nstead of ıδ a,λ n the sequel. Let { M ε m α, λ, x, v R R S H : v Ex,λ, v < ν 0 ; α Kx αj Kx j < ν 0, λ >, ε log λ < ν 0, ; c 0 < λ < c 0 ν 0 λ ; x x > d 0 ; c K 3 ν x εkx S λ 8 < ε σ where ν 0 s a small postve constant, σ, c 0, d 0 are some sutable postve constants, and }, { E x,λ w H / w, ϕ 0 ϕ Span {δ, δ Here, x j denotes the jth component of x.also λ, δ x j }},,;j. ε : M ε R; m α, λ, x, v I ε α δ x,λ α δ x,λ v. In the sequel, we wll wrte δ nstead of δ x,λ and u α δ α δ v for the sake of smplcty. In the remander of ths secton, we wll gve expansons of the gradent of the functonal I ε assocated to S ε forε > 0. Thus estmatons are needed n Secton 3. We need to recall

4 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page of that [5] proved the followng remark when n 3, but the same argument s avalable for the dmenson. Remark For ε >0andδ a,λ defned n, we have δ ε a,λ x ε log δ a,λ O ε log λ n. Now, explct computatons, usng Remark, yeld the followng propostons. Proposton 3 Let α, λ, x, v M ε. Then, for u α δ x,λ α δ x,λ v, we have the followng expanson: where Iε u, δ α S ε j λ λ j λ j λ λ λ j a a j, dx S 6 x. R α ε Kx O ε log λ λ ε v, Proof We have Iε u, h u h uh Ku 3 ε h. 5 A computaton smlar to the one performed n [6]showsthat,for,, δ δ S R 6 and δ δ j δ δ j δ δ j δ R 3 δ j ε. 7 For the other ntegral, we wrte We also wrte Ku 3 ε δ Kα δ α δ 3 ε δ O ε log ε v. 8 Kα δ α δ 3 ε δ α 3 ε Kδ ε αj 3 ε Kδj 3 ε δ 3 εα ε α j δ j O ε log ε. 9

5 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page 5 of Expansons of K around x and x j gve Kδ ε j δ δ j Kδ ε Kx S ε O log λ λ, 0 j δ ε log λ ε, δ j ε log λ ε. Combnng 5-, we derve our proposton. Proposton Let α, λ, x, v M ε. Then, for u α δ x,λ α δ x,λ v, we have where δ I ε u, λ α j α ε j Kx j α ε Kx ε c λ λ S 6 R α 3 ε c 3 O Proof Observe that see [6] δ δ λ λ δ δ j λ λ λ K ν x O v λ εε log ε dx x, c 6 R α 3 ε λ εs Kx 8 ε log λ ε log λ λ / ε log ε ε log ε /, λ j dx x, c x x R x dx. 5 δ 3 λ δ 0, 3 λ δj 3 λ δ λ c ε λ O ε λ log ε. For the other part, we have the expansons of K around x and usng Remark, Kδ 3 ε δ λ εs Kx c 3 Kx e O ε log λ λ 8 λ λ KPδj 3 ε δ λ Kx j λ c ε λ O λ 3 ε Kδ ε δ δ j λ εε log ε λ j ε λ, 5 O ε log ε, 6 Kx λ c ε λ O εε log ε λ O ε log ε ε log ε. 7 λ j Combnng 5, 3,, 5, 6, and 7, theprooffollows.

6 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page 6 of Proposton 5 Let α, λ, x, v M ε. Then, for u α δ x,λ α δ x,λ v, we have the followng expanson: where I ε u, δ α c α ε Kx α 3 ε Kx εc log λ c 7 λ x α 3 ε c 5 K λ ν x e α j α ε c Kx α 3 ε c 5 T Kx O λ εε log ε x c 3 R x dx, c x R x dx. 5 Proof We have δ δ λ x δ δ j λ x For the other part j δ 3 δ 3 j λ ε x O v λ j x x ε 5 ε log λ T Kx λ ε log ε ε log ε λ j λ ε log λ, λ δ x c e, 8 λ δ δ Kx c e c 5 λ x c x εkx c 7 e O λ δ Kx j λ x c O 3 ε Kδ ε δ R δ j λ λ ε x O ε log ε λ Kx ε log λ Kx c e λ ε a O ε 5 ε log ε Kx x c O 5 ε λ j x x. 9 ε log λ, 0 λ j x x ε log ε, λ j λ ε x O ε 5 ε log ε λ j x x ε log ε. λ Usng 5, 8-, ourpropostonfollows. 3 Constructon of the soluton The methodofthstype oftheoremwasfollowedfrst by Bahr,L andrey [7] when they studed an approxmaton problem of the Yamabe-type problem on domans. Many authors used ths dea to construct some solutons to other problems. The method becomes standard. Here we wll follow the dea of [7] and take account of the new estmates snce

7 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page 7 of we have an equaton dfferent from the one studed n [7]. From the dea of [7], usng the coeffcents of Euler-Lagrange, we obtan Proposton 6 Let A pont m α, λ, x, v M ε s a crtcal pont of the functon ε f and only f u α δ α δ v s a crtcal pont of functonal I ε, whch means the exstence of some A, B, C R R R wth the followng: E α 0,,, 3 α E λ ε δ δ B λ λ, v C j x j λ, v,,, E x x B E v v, j δ, v λ x A δ B δ λ δ C j j x j x, v,,, 5 δ C j x j. 6 Now, by a careful study of equaton E v, we get the followng. j Proposton 7 [] For any ε, α, λ, x wth α, λ, x,0 M ε, there exsts a smooth map whch assocates v E x,λ wth v < ν 0 and equaton 6 n the prevous proposton s verfed for some A, B, C R R R. Such a vsunque, mnmzes ε α, λ, x, v wth respect to v n {v E x,λ / v < ν 0 }, and v ε λ λ ε log ε /. 7 Proof of Theorem Once v s defned by Proposton 7, we estmate the correspondng numbers A, B, C by takng the scalar product n H ofe vwthδ, δ / λ, δ / x for,, respectvely. So we get the followng coeffcents of a quas-dagonal system: δ S ; δ δ λ λ λ δ δ λ λ δ δ x λ δ δ, λ λ, δ x δ x n n, ; δ δ 0; λ λ R λ δ δ R λ λ λ ; δ R λ δ R x Ɣ λ ; δ δ ; x δ n δ x δ x wth x x c >0andƔ, Ɣ are postve constants. We have also ε v, δ ; α ε v, δ ; λ α λ λ λ, ε v, δ. x α x Ɣ λ ; δ δ x λ ;

8 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page 8 of Usng Propostons 3, somecomputatonsyeld α S β V α ε, α, λ, x, 8 wth β α /Kz and V α β ε log λ λ x z. 9 In the same way, usng Propostons,weget c3 K λ Kz λ ν x εkx S V λ ε, α, λ, x, 30 8λ where c and c 3 are defned n Proposton and [ V λ λ λ Lastly, usng Propostons 5, we have ε log λ ε log λ β ε x z ε ] λ λ λ. 3 where x c 5 T Kx V x ε, α, λ, x, 3 V x β ε log λ x z x z. 33 λ From these estmates, we deduce α λ β ε log λ λ x z, ε σ / λ ; x x z λ. By solvng the system n A, B,andC,wefnd A β ε log λ λ x z, B ε σ / λ ; C x z. λ λ 3 3 Now, we can evaluate the rght hand sde n E λ ande x, δ B λ, v j δ B, v λ x δ ε σ / C j x j λ, v x z λ λ λ 3 v, 35 δ C j x j x, v ε σ / λ x z λ v, 36 j

9 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page 9 of where Pδ λ λ ; Pδ x λ ; Pδ λ. x Now, we consder a pont z, z S such that z and z are nondegenerate crtcal ponts of K.Weset ε S K Kz λ 6c 3 ν z ζ ; x z ξ, where ζ R and ξ, ξ R 3 R 3 are assumed to be small. Usng 8 and these changes of varables, E α becomes β V α ε, β, ζ, ξo β ε log ε ξ. 37 Also, we use 30, we have c 3 λ K ν z ξ εkz ξ S 8λ K ν z ζ K ν z D Kz e, ξ ε S Kz K 8c 3 ν z ζ O ε ζ ξ ε S Kz K 8c 3 ν z ζ D Kz e, ξ ε S Kz 8c 3 ε S Kz 8c 3 K ν z O ε ζ ξ. Combnng ths wth 3, then E λ becomes K ζ ν z D K z e, ξ V λ ε, β, ζ, ξ Usng 3, 33, and 36, E x sequvalentto ε log ε β ζ ξ. 38 D K z ξ V x ε, β, ζ, ξo ε / β ζ ξ. 39 Observe that the functons V α, V λ,andv x are smooth. We can also wrte the system as β Vε, β, ζ, ξ, Lζ, ξwε, β, ζ, ξ, 0

10 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page 0 of where L s a fxed lnear operator on R 8 defned by K K Lζ, ξ ζ ν z D K z e, ξ ; ζ ν z D K z e, ξ ; D K z ξ ; D K z ξ, and V, W are smooth functons satsfyng Vε, β, ζ, ξoε / β ξ, Wε, β, ζ, ξoε β ζ ξ. Now, by an easy computaton, we see that the determnant of the lnear operator L s not 0. Hence L s nvertble, and accordng to Brouwer s fxed pont theorem, there exsts a soluton β ε, ζ ε, ξ ε of0forε small enough, such that β ε ε / ; ζ ε ε / ; ξ ε ε /. Hence, we have constructed m ε α ε, αε, λε, λε, xε, xε suchthatu ε : α εδ x ε,λε v ε,verfes 3-7. From Proposton 6, u ε s a crtcal pont of I ε, whch mples that u ε verfy u ε u ε K u ε ε u ε n, u ε/ ν 0 on. We multply equaton byu ε max0, u εandwentegrateon,weget u ε u ε We know also from the Sobolev embeddng theorem that u ε ε : K u ε ε. K u ε ε ε C u ε. 3 Equatons and3 mply that ether u ε 0, or u ε ε s far away from zero. Snce m ε M ε,wehave v ε < ν 0,whereν 0 s a small postve constant see the defnton of M ε. Ths mples that u ε ε s very small. Thus, u ε 0forε small enough. Then u ε s a nonnegatve functon whch satsfes. Fnally, the maxmumprncple completesthe proof of our theorem. Concluson Thus t has been concluded that under some assumptons on the functon K, there exst solutons of the nonlnear problem S ε whch are concentrated at two dfferent ponts of the boundary. Competng nterests Theauthordeclarestohavenocompetngnterests.

11 JebrlJournal of Inequaltes and Applcatons 06 06:305 Page of Acknowledgements I would lke to thank Deanshp of Scentfc Research at Tabah Unversty for the fnancal support of ths research project. Receved: 8 July 06 Accepted: November 06 References. Denng, L, Harjulehto, P, Hasto, P, Ruzcka, M: Lebesgue and Sobolev Spaces wth Varable Exponents. Lecture Notes n Mathematcs, vol. 0. Sprnger, Hedelberg 0 MR7905. Ambrosett, A, Garca Azorero, J, Peral, A: Perturbaton of u u n n 0, the scalar curvature problem n R n and related topcs. J. Funct. Anal. 65, Bahr, A, Coron, JM: The scalar curvature problem on the standard three dmensonal spheres. J. Funct. Anal. 95, Banch, G, Pan, XB: Yamabe equatons on half spheres. Nonlnear Anal. 37, Chang, SA, Yang, P: A perturbaton result n prescrbng scalar curvature on S n.dukemath.j.6, Cherrer, P: Problèmes de Neumann non lnéares sur les varétés remanennes. J. Funct. Anal. 57, Escobar, J: Conformal deformaton of Remannan metrc to scalar flat metrc wth constant mean curvature on the boundary. Ann. Math. 36, Escobar, J, Schoen, R: Conformal metrcs wth prescrbed scalar curvature. Invent. Math. 86, Han, ZC, L, YY: The exstence of conformal metrcs wth constant scalar curvature and constant boundary mean curvature. Commun. Anal. Geom. 8, Hebey, E: The sometry concentraton method n the case of a nonlnear problem wth Sobolev crtcal exponent on compact manfolds wth boundary. Bull. Sc. Math. 6, L, YY: Prescrbng scalar curvature on S n andrelatedtopcs,parti.j.dffer.equ.0, ; Part II. Exstence and compactness. Comm. Pure Appl. Math Ould Bouh, K: Blowng up of sgn-changng solutons to a subcrtcal problem. Manuscr. Math. 6, Djadl, Z, Malchod, A, Ould Ahmedou, M: Prescrbng the scalar and the boundary mean curvature on the three dmensonal half sphere. J. Geom. Anal. 3, Ben Ayed, M, Ghoud, R, Ould Bouh, K: Exstence of conformal metrcs wth prescrbed scalar curvature on the four dmensonal half sphere. Nonlnear Dffer. Equ. Appl. 9, Rey, O: The topologcal mpact of crtcal ponts at nfnty n a varatonal problem wth lack of compactness: the dmenson 3. Adv. Dffer. Equ., Bahr, A: An nvarant for Yamabe-type flows wth applcatons to scalar curvature problems n hgh dmenson. A celebraton of J. F. Nash jr. Duke Math. J. 8, Bahr, A, L, YY, Rey, O: On a varatonal problem wth lack of compactness: The topologcal effect of the crtcal ponts at nfnty. Calc. Var. Partal Dffer. Equ. 3,