An improved bound for the non-existence of radial solutions of the Brezis-Nirenberg problem in H n

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1 A improved boud for the o-existece of radial solutios of the Brezis-Nireberg problem i H Rafael D. Beguria ad Soledad Beguria To Pavel Exer o the occasio of his 7th birthday Abstract. Usig a Rellich Pohozaev argumet ad Hardy s iequality, we derive a improved boud o the oliear eigevalue for the o existece of radial solutios of a Brezis Nireberg problem, with Dirichlet boudary coditios, o a geodesic ball of H, for < < 4. 1 Mathematics Subject Classificatio. Primary 35XX; Secodary 35B33; 35A4; 35J5; 35J6 Keywords. Brezis Nireberg Problem, Hyperbolic Space, Noexistece of Solutios, Pohozaev Idetity, Hardy Iequality 1. Itroductio For a log time virial theorems have played a key role i the localizatio of liear ad oliear eigevalues. I the spectral theory of Schrödiger Operators, the virial theorem has bee widely used to prove the absece of positive eigevalues for various multiparticle quetum systems (see, e.g., [1, 1, 1]). I 1983, Brézis ad Nireberg [4] cosidered the existece ad oexistece of solutios of the oliear equatio u = λu+ u p 1 u, defied o a bouded, smooth domai of R, >, with Dirichlet boudary coditios, where p = (+)/( ) is the critical Sobolev expoet. I particular, they used a virial theorem, amely the Pohozaev idetity [8], to prove the oexistece of regular solutios whe the domai is star shaped, for ay λ, i ay >. After the classical paper [4] of Brézis ad Nireberg, may people have cosidered extesios of this problem i differet settigs. I particular, the Brézis Nireberg (BN) problem has bee studied o bouded, smooth, domais of the hyperbolic space H (see, e.g., [11,, 6, 3]), where oe replaces the Laplacia by the Laplace Beltrami operator i H. Stapelkamp [11] proved the aalog of the above metioed oexistece result of Brézis Nireberg i H. Namely The work of R.B. has bee supported by Fodecyt (Chile) Projects # ad # ad by the Núcleo Mileio e Física Matemática, RC 1 (ICM, Chile). S.B. would like to thak the E. Schrödiger Istitute i Viea for their hospitality while part of this work was beig doe.

2 R. D. Beguria ad S. Beguria she proved that there are o regular solutios of the BN problem for bouded, smooth, star shaped domais i H ( > ), if λ ( )/4. The purpose of this mauscript is to give a improved boud o λ for the oexistece of radial (ot ecessarily positive) radial, regular solutios of the BN problem o geodesic balls of H for < < 4 (see Theorem.1 below). Notice that for the case of radial solutios of the BN problem o a geodesic ball oe ca cosider oiteger values of, which ca be cosidered just as a parameter. Cosider the Brezis Nireberg problem H u = λu+ u p 1 u, (1) o Ω H, where Ω is smooth ad bouded, with Dirichlet boudary coditios, i.e., u = i Ω. After expressig the Laplace Beltrami operator H i terms of the coformal Laplacia, Stapelkamp [11] proved that (1) does ot admit ay regular solutio for star-shaped domais Ω provided λ ( ). () 4 Here, we cosider the BN problem (1) for radial solutios o geodesic balls of H. We ca prove a differet boud, amely the problem for radial solutios o a geodesic ball Ω does ot admit a solutio if λ ( 1) 4(+) (3) for >. Our boud is better tha () i the radial case, if < < 4. Both bouds coicide whe = 4. I the rest of this mauscript we give the proof of (3).. Noexistece of solutios of the BN problem o geodesic balls i H, for < < 4. I the sequel we cosider (ot ecessarily positive) radial solutios of the BN problem (1) o geodesic balls of H. I radial coordiates, (1) ca be writte as u (x) ( 1)coth(x)u (x) = λu(x)+ u p 1 u(x), (4) with u () = u(r) =, where R is the radius of the geodesic ball. Here, as before, p = (+)/( ). Notice that (4) makes sese also if is ot a iteger. For that reaso heceforth we cosider R, with < < 4. Our mai result is the followig Theorem.1. The Boudary Value problem (1), with u () = u(r) =, has o regular solutios if λ ( 1) 4(+), for < < 4.

3 No existece of radial solutios of the Brezis-Nireberg problem i H 3 Remark.. Notice that our boud ( 1)/(4(+)) is strictly bigger tha ( )/4 for < 4. Notice, o the other had, that Stapelkamp s boud holds for all regular solutios, while our improved boud oly holds for regular radial solutios. We do ot kow whether our boud is optimal, i.e., we do ot kow if there are solutios for λ > ( 1)/(4( + ). I view of [3], there ca be o positive solutios if λ < µ() (see [3] for the defiitio of µ()). It is importat to otice that for = 4, we have that, ( ) 4 so at least our result is optimal as 4. = ( 1) 4(+) = µ(4), Proof. We use a Rellich Pohozaev argumet [9, 8]. Multiplyig equatio (4) by u(x)sih 1 (x) ad itegratig, we obtai = λ u (x)(u(x)sih 1 (x)) ( 1) u sih 1 (x)+ u(x) p+1 sih 1 (x). u(x)u (x)cosh(x)sih (x) Itegratig the first term by parts, we ca write this equatio as u sih 1 (x) = λ Now let G(x) = sih 1 (s)ds. Multiplyig equatio (4) by u G ad itegratig, we obtai x u sih 1 (x)+ (5) u p+1 sih 1 (x). (6) = λ u R G ( 1) coth(x)u G u R u G+ p+1 G. p+1 After itegratig by parts, ad sice G() =, we obtai u (R)G(R) + = λ u G + 1 p+1 u ( 1)Gcoth(x) G u p+1 G. (7)

4 4 R. D. Beguria ad S. Beguria Substitutig equatio (6) ito equatio (7), ad sice 1/ 1/(p + 1) = 1/, it follows that ( ) u ( 1)Gcoth(x) G G + u (R)G(R) p+1 (8) = λ u sih 1 (x). Notice that i equatio (9) we have writte sih 1 (x) as G (x). Thus, sice the boudary term is positive, ad sice 1/+1/(p+1) = ( 1)/, we have ( 1) u Gcoth(x) G λ. (9) u G Now let L(x) = Gcoth(x) G. The L. I fact, we ca write L(x) = m(x)/ sih(x), where m(x) = Gcosh(x) sih (x)/. The, sice G(), we have m() =. Also, m (x) = Gsih(x)+G cosh(x) sih 1 (x)cosh(x) = Gsih(x). It follows that m, ad therefore that L. We ow use a Hardy type iequality to write the deomiator itegral i terms of u. For a review o Hardy s iequa;ities see, e.g., [7, 5]. Itegratig by parts, we ca write u G = (usih 1 (x) ) ( Gu sih 1 ) (x). The, usiog Cauchy-Schwarz, it follows that ( R u G ) 4 u G G u G. That is, u G u G 4 G. (1) Usig iequality (1) i the quotiet (9), we coclude that ( 1) u Gcoth(x) G λ u G 4 G. (11)

5 No existece of radial solutios of the Brezis-Nireberg problem i H 5 I the Lemma.3 below, we show that L(x) c G G, where c =. With this, + we coclude that λ ( 1) 4(+). (1) Lemma.3. Let x ad let L(x) = Gcoth(x) G. The Here, as above, G(x) = x L(x) sih 1 (s)ds. G (+) G. Proof. Let f(x) = L(x)G (x) cg (x), where c =. It suffices to show that + f. As before, we write L(x) = m(x) sih(x), where m(x) = Gcosh(x) sih (x) m (x) = Gsih(x). The, ad f(x) = sih (x)m(x) cg (x). Notice that sice sih() = G() =, oe has that f() =, so it suffices to show that f. We have that f (x) = sih 3 (x) ( ( )cosh(x)m(x)+gsih (x)(1 c) ). Let σ = c 1 = 1/p, where p = (+)/( ) is the critical Sobolev expoet; ad let g(x) = ( )cosh(x)m(x) σgsih (x). It suffices to show that g. Sice m() =, the g() =. Also, ( ) g (x) = sih(x)cosh(x)g(x)( σ) sih +1 (x) +σ ad i particular g () =. Sice σ = ( )(+1) (+) ad ( ) +σ = (+1)( ) (+)

6 6 R. D. Beguria ad S. Beguria we ca write g (x) = (+1)( ) (+) sih(x)[gcosh(x) sih (x)]. Fially, let h(x) = Gcosh(x) sih (x). If we show h(x), the we will have g, which will imply g, ad thus, that f, as desired. Notice that h() =. Also, sice G (x) = sih 1 (x), we have h (x) = Gsih(x). That is, h, which cocludes the proof of Lemma.3. Remark.4. I the proof of Lemma.3, the costat σ = 1/p plays a crucial role, where p is the critical Sobolev expoet. It is worth otig that for small x ad g as i the proof above, ( 1 g(x) = x + p σ ) +O(x +4 ). It follows that if σ 1/p, the g is positive i a eighborhood of the origi. It was this observatio that led us to realize that σ = 1/p would yield the optimal estimate. 3. Refereces [1] E. Balslev, Absece of positive eigevalues of Schrödiger Operators, Archive Ratioal Mechaics ad Applicatios, 59 (1975), [] C. Badle ad Y. Kabeya, O the positive, radial solutios of a semiliear elliptic equatio i H N. Adv. Noliear Aal., 1 (1), 1 5. [3] S. Beguria, The solutio gap of the Brezis-Nireberg problem o the hyperbolic space. Moatsh. Math., DOI 1.17/s (16). [4] H. Brézis ad L. Nireberg, Positive solutios of oliear elliptic equatios ivolvig critical Sobolev expoets. Comm. Pure Appl. Math., 36 (1983), [5] E. B. Davies, A review o Hardy iequalities. Operator Theory Advaces ad Applicatios, 11, pp , Birkhäuser Verlag, Basel, [6] D. Gaguly ad K. Sadeep, Sig chagig solutios of the Brezis-Nireberg problem i the hyperbolic space. Calc. Var. Partial Differetial Equatios, 5 (1-) (14), [7] B. Opic ad A. Kufer, Hardy type iequalities. Pitma Research Notes i Math., 19, Logma, 199. [8] S. I. Pohozaev, O the eigefuctios of the equatio u +λf(u) =. Dokl. Akad. Nauk., 165 (1965), [9] F. Rellich, Darstellug der Eigewerte vo u + λu = durch ei Raditegral, Math. Z., 46 (194),

7 No existece of radial solutios of the Brezis-Nireberg problem i H 7 [1] B. Simo, Absece of positive eigevalues i a class of multiparticle quatum systems, Math. A., 7 (1974), [11] S. Stapelkamp, The Brézis-Nireberg problem o H. Existece ad uiqueess of solutios. Elliptic ad parabolic problems (Rolduc/Gaeta, 1), World Sci. Publ., River Edge, NJ, [1] J. Weidma, The virial theorem ad its applicatio to the spectral theory of Schrödiger operators, Bull. Amer. Math. Soc. 77 (1967), Rafael Beguria, Istituto de Física, Potificia Uiversidad Católica de Chile, Avda. Vicuña Mackea 486, Satiago, Chile rbeguri@fis.puc.cl Soledad Beguria, Mathematics Departmet, Uiversity of Wiscosi - Madiso, 48 Licol Dr, Madiso, WI, USA beguria@math.wisc.edu