ON A SINGULAR PERTURBED PROBLEM IN AN ANNULUS

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1 ON A SINGULAR PERTURBE PROBLEM IN AN ANNULUS SANJIBAN SANTRA AN JUNCHENG WEI Abtact. In thi pape, we extend the eult obtained by Ruf Sikanth 8. We pove the exitence of poitive olution unde iichlet and Neumann bounday condition, which concentate nea the inne bounday and oute bounday epectively of an annulu a 0. In fact, ou eult i independent of the dimenion of R N. 1. Intoduction Thee ha been a conideable inteet in undetanding the behavoi of poitive olution of the elliptic poblem u u + f(u) = 0 in Ω u > 0 in Ω, (1.1) u = 0 o u ν = 0 on Ω whee > 0 i a paamete, f i a upelinea nonlineaity and Ω i a mooth bounded domain in R N. Let F (u) = u f(t)dt. We conide the poblem when 0 f(0) = 0 and f (0) = 0. Thi type of equation aie in vaiou mathematical model deived fom population theoy, chemical eacto theoy ee Gida-Ni-Nienbeg 6. In the iichlet cae, Ni Wei howed in 13 that the leat enegy olution of equation (1.1) concentate, fo 0, to ingle peak olution, whoe maximum point P convege to a point P with maximal ditance fom the bounday Ω. In the Neumann cae, Ni Takagi 11 howed that fo ufficiently mall > 0, the leat enegy olution i a ingle bounday pike and ha only one local maximum P Ω. Moeove, in 1, they pove that H(P ) max P Ω H(P ) a 0 whee H(P ) i the mean cuvatue of Ω at P. A implified poof wa given by del Pino Felme in 3, fo a wide cla of nonlineaitie uing a method of ymmetiation. Highe dimenional concentating olution wa tudied by Amboetti Malchiodi Ni in 1, ; they conide olution which concentate on phee, i.e. on (N 1)- dimenional manifold. They tudied { u V ()u + f(u) = 0 in A (1.) u > 0 in A, u = 0 on A the poblem, in an annulu A = {x R N : 0 < a < x < b}, V () i a mooth adial potential bounded below by a poitive contant. They intoduced a modified potential M() = N 1 V θ (), with θ = p+1 p 1 1, atifying M (b) < 0 (epectively M (a) > 0), then thee exit a family of adial olution which concentate on 1991 Mathematic Subject Claification. Pimay 35J10, 35J35, 35J65. Key wod and phae. iichlet poblem, Neumann poblem, annulu, concentation. The fit autho wa uppoted by the Autalian Reeach Council. 1

2 SANJIBAN SANTRA AN JUNCHENG WEI x = with b (epectively a) a 0. In fact, they conjectued that in N 3 thee could exit alo olution concentating to ome manifold of dimenion k with 1 k N. Moeove, in R, concentation of poitive olution on cuve in the geneal cae wa poved by del Pino Kowalczyk Wei 4. In 9, the aymptotic behavio of adial olution fo a ingulaly petubed elliptic poblem (1.) wa tudied uing the Moe index infomation on uch olution to povide a complete deciption of the blow-up behavio. A a conequence, they exhibit ufficient condition which guaantee that adial gound tate olution blow-up and concentate at the inne o oute bounday of the annulu. In thi pape, we conide the following two ingula petubed poblem, u u + u p = 0 in A (1.3) u > 0 in A u = 0 on A, (1.4) u u + u p = 0 u > 0 in A in A u ν = 0 on A, whee A i an annulu in R N = R M R K with A = {x R N : 0 < a < x < b} and > 0 i a mall numbe and ν denote the unit nomal to A and N. In thi pape, we ae inteeted in finding olution u(x) = u(, ) whee = x 1 + x + x M and = x M+1 + x M+ + x K. Let u conide the conjectue due to Ruf and Sikanth: oe thee exit a olution fo the poblem (1.3) and (1.4), which concentate on R M+K 1 dimenional ubet a 0? Theoem 1.1. Fo > 0 ufficiently mall, thee exit a olution of (1.3) which concentate nea the inne bounday of A. Theoem 1.. Fo > 0 ufficiently mall, thee exit a olution of (1.4) which concentate nea the oute bounday of A.. et up fo the appoximation Note that unde ymmety aumption, A can be educed to a ubet of R whee = {(, ) : > 0, > 0, a < + < b }. Let P = (P 1,, P, ) be a point of maximum of u in A, then u (P ) 1. Fom (1.3) we obtain (.1) u + (M 1) (K 1) u + u + u u + u p = 0 Let 1, ae the inne and oute bounday of epectively and 3, 4 ae the hoizontal and vetical bounday of epectively. If P = (P 1, P ) be a point in uch that dit(p, 1 ) = d, then we can expe, (.) P 1 = (a + d) co θ; P = (a + d) in θ whee θ i the angle between the x axi and the line joining P. Futhemoe, if dit(p, ) = d, then we can expe, (.3) P 1 = (b d) co θ; P = (b d) in θ. See Figue 1 and Figue.

3 3 3 3 u ν = 0 u = 0 P d u = 0 u ν = 0 u ν = 0 P d u ν = u ν = 0 4 Figue 1. iichlet cae u ν = 0 4 Figue. Neumann Cae The functional aociated to the poblem i ( (.4) I (u) = M 1 K 1 u + 1 u 1 ) p + 1 up+1 dd. Moeove, (1.3) educe to u + (M 1) (K 1) u + u + u u + u p = 0 in u = 0 on 1 u ν = 0 on 3 4. Re-caling about the point P, we obtain in A (M 1) (.5) u + u + P 1 + u (K 1) + P + u u + u p = 0. The entie olution aociated to (.1) whee U atifie (,) U U + U p = 0 in R (.6) U(, ) > 0 in R U(, ) 0 a (, ). Let z = (, ). Moeove, U(z) = U( z ) and the aymptotic behavio of U at infinity i given by ( ( )) U(z) = A z 1 e z O z (.7) ( ( )) U (z) = A z 1 e z O z fo ome contant A > 0. Let K(z) denote the fundamental olution of (,) + 1 centeed at 0. Then fo z 1, we have ( ( )) 1 U(z) = B + O K(z) z (.8) ( ( )) 1 U (z) = B + O K(z) z

4 4 SANJIBAN SANTRA AN JUNCHENG WEI fo ome poitive contant B. Let U,P (z) = U( z P ). Now we contuct the pojection map fo the iichlet cae a (,) P U,P P U,P + U p,p = 0 in (.9) P U,P (, ) > 0 in P U,P (, ) = 0 on, and the pojection in the Neumann cae a (,) QU,P QU P + U p,p = 0 in (.10) QU,P (, ) > 0 in QU,P (, ) = 0 on. ν If v = U,P P U,P and w = U,P QU,P. Then we have { (,) v v = 0 in (.11) (.1) v = U,P (,) w w = 0 w ν = U,P ν on, in on. Conide the function (θ) = co M 1 θ in K 1 θ in 0, π. Then neithe θ 0 = 0 no θ 0 = π ae point of maxima of. But > 0 and hence θ 0 lie in (0, π ). Fo any θ θ 0 δ, θ 0 + δ we define the configuation pace fo the iichlet and Neumann cae a (.13) Λ, = and (.14) Λ,N = { P : dit(p, 1 ) k ln 1 } { P : dit(p, ) k ln 1 } epectively fo ome k > 0 mall. We develop the following lemma imila to Lin, Ni and Wei 10. Lemma.1. Aume that k ln d(p, 1) δ, then we obtain ( z P ) (.15) v (z) = (B + o(1))k + O( +σ ) whee P = P + d(p, 1 )ν P and P 1 i a unique point, uch that d(p, P ) = d(p, 1 ) and σ i a mall poitive numbe; δ i the ufficiently mall. Moeove, ν P i the oute unit nomal at P. Poof. efine (.16) (,) Ψ Ψ = 0 Ψ > 0 Ψ = 1 in in on, Then fo ufficiently mall, Ψ i unifomly bounded.

5 5 But fo z, we obtain ( ) z P U,P (z) = U Fit, we have = (A + o(1)) 1 z P 1 e z P. ( ) z P U,P (z) = (B + o(1))k. Hence by the compaion pinciple we obtain, fo ome σ > 0, mall v C +σ Ψ wheneve d(p, 1 ) ln. Theefoe, it emain to check whethe (.15) hold in (.17) efine the function k ln d(p, 1) ln. (.18) φ 1 (z) = (B 1 4 )K ( z P Then φ 1 atifie (.19) (,) φ 1 φ 1 = 0. Fo any z in 1 with z P 3 4 (.0) and hence z P Fo any z 1 with z P 3 4 we have ) + +σ Ψ. = (1 + O( 1 ) ln ) z P v φ 1. we have v (z) Ce 1 4 +σ φ 1. Summaizing, we obtain, v φ 1 fo all z 1. Similaly, we obtain the lowe bound fo z 1, (.1) v (z) (B )K ( z P ) +σ Ψ. Coollay.1. Aume that k ln d(p, ) δ whee δ i ufficiently mall. Then ( z P ) (.) w (z) = (B + o(1))k + O( +σ ); whee P = P + d(p, )ν P whee P i a unique point, uch that d(p, P ) = d(p, ) and σ i a mall poitive numbe. Moeove, ν P i the oute unit nomal at P.

6 6 SANJIBAN SANTRA AN JUNCHENG WEI efine H 1 0 () = 3. Refinement of the pojection { u H 1 : u(x) = u(, ), u = 0 in 1 and ; u } ν = 0 in 3 and 4. efine a nom on H0 1 () a (3.1) v = M 1 K 1 v dx + v dd In thi ection, we will efine the pojection, to incopoate the Neumann bounday condition on 3 and 4. We define a new pojection a V,P = ηp U,P whee 0 η 1 i mooth cut off function { 1 in Bd (P ), (3.) η(x) = 0 in \ B d (P ). Hee d = dit(p, ) i dependent on. We will chooe d at the end of the poof. We define (3.3) u = V,P + ϕ,p. Uing thi Anatz, (1.3) educe to (M 1) (K 1) (,) ϕ ϕ + ϕ, + ϕ, +f (V,P )ϕ = h in, ϕ = 0 on 1 ϕ ν = 0 on 3 4 ; whee h = S V,P + N ϕ and (3.4) S V,P = (M 1) (K 1) (,) V,P + V,P, + V,P + f(v,p ) V,P, and Let N ϕ = {f(v,p + ϕ ) f(v,p ) f (V,P )ϕ }. E,P = { ω H0 1 (), ω, V,P = ω, V },P = 0. Lemma 3.1. Then fo any z \ B d (P ) ( ( ) ) z P (3.5) V,P (z) = η U v,p (z). Moeove, we have (3.6) V,P (z) = O( k ).

7 7 Poof. Fo any z \ B d (P ) we have (3.7) V,P (z) ( ) z P U v,p (z) = O(e x P + e x P + 3+σ ) = O(e d(p,p ) + +σ ) = O(e d(p, 1 ) + +σ ) = O( k ). Moeove, V,P i zeo outide B d (P ). Lemma 3.. The enegy expanion i given by ( I (V,P ) = M 1 K 1 V,P + 1 V,P 1 ) p + 1 V p+1,p dd ( P P = γ P1 M 1 P K 1 + γ 1 P1 M 1 P K 1 ) U + o( +k ) whee γ = p 1 (p+1) U p+1 dd and γ R 1 = U p e dd. R Poof. We obtain ( I (V,P ) = M 1 K 1 V,P + 1 V,P 1 ) p + 1 V p+1,p dd ( = η M 1 K 1 P U,P + 1 P U,P 1 ) p + 1 P U p+1,p dd ( ) 1 + M 1 K 1 η η p+1 P U p+1,p p + 1 dd + M 1 K 1 η ηp U P U dd + M 1 K 1 η (P U,P ) dd (3.8)= J 1 + J + J 3 + J 4.

8 8 SANJIBAN SANTRA AN JUNCHENG WEI Hence we have ( J 1 = M 1 K 1 P U,P + 1 P U,P 1 ) p + 1 P U p+1,p dd ( (1 η ) M 1 K 1 P U,P + 1 P U,P 1 ) p + 1 P U p+1,p dd ( 1 = M 1 K 1 U p,p P U,P 1 ) p + 1 P U p+1,p dd ( P + M 1 K 1 U,P + P U ),P P U,P dd B d (P ) ( P M 1 K 1 U,P + P U ),P P U,P dd B d (P ) ( 1 = 1 ) (P 1 + ) M 1 (P + ) K 1 U p+1 (z)dd p U p,p v M 1 K 1 dd ( 1 = 1 ) P1 M 1 P K 1 U p+1 dd p + 1 R + 1 U p,p v M 1 K 1 dd ( P + M 1 K 1 U,P + P U ),P P U,P dd B d (P ) ( P M 1 K 1 U,P + P U ),P P U,P dd B d (P ) ( + M 1 K 1 \B d P U,P + 1 P U,P 1 ) (3.9) p + 1 P U p+1,p dd + o( ). Now we etimate ( 1 1 ) (P 1 + ) M 1 (P + ) K 1 U p+1 (z)dd p + 1 p 1 (3.10) = (p + 1) P1 M 1 P K 1 U p+1 dd + O( 4 )P1 M 1 P K 1 R Fom Lemma 3.1, we compute the inteaction tem ( U p,p v M 1 K 1 dd = U p U z P P ) (P 1 + ) M 1 (P + ) K 1 dd (3.11) Alo we have J = + O( 4 ) = P M 1 1 P K 1 U = P M 1 1 P K 1 U ( P P ) (γ + o(1)) + O( 4 ) ( ) d(p, 1 ) (γ + o(1)) + O( 4 ). M 1 K 1 ( η η p+1 ) P U p+1,p dd = O( ) (p+1)k,

9 9 Futhemoe, we have ( P M 1 K 1 U,P and B d (P ) J 3 = J 4 = Hence we obtain the eult. + P U,P ) P U,P dd = O( k ); M 1 K 1 η ηp U P U dd = o( k+ ), M 1 K 1 η (P U,P ) dd = o( k+ ). 4. The eduction In thi ection, we will educe the poof of Theoem 1.1 to finding a olution of the fom V,P + ϕ,p fo (1.3) to a finite dimenional poblem. We will pove that fo each P Λ,, thee i a unique ϕ,p E uch that ) I (V,P + ϕ,p, η = 0; η E,P. Let J (ϕ) = I ( V,P + ϕ,p ). Fom now on we conide ϕ,p = ϕ. We expand J (ϕ) nea ϕ,p = 0 a whee (4.1) (4.) and (4.3) l,p (ϕ) = J (ϕ) = J (0) + l,p (ϕ) + 1 Q,P (ϕ, ϕ) + R (ϕ) = Q,P (ϕ, ψ) = R (ϕ) = M 1 K 1 V,P ϕ + V,P ϕ V p,p ϕ dd M 1 K 1 S V,P ϕdd, 1 p + 1 ( (p + 1) M 1 K 1 ϕ ψ + ϕψ pv p 1,P ϕψ dd, p+1 ) p+1 M 1 (V K 1,P + ϕ) (V,P V,P ) p φ ) p 1 p(p + 1) (V,P ϕ dd. We will pove in Lemma 4.1 that l,p (ϕ) i a bounded linea functional in E,P. Hence by the Riez epeentation theoem, thee exit l,p E,P uch that l,p, ϕ = l,p (ϕ) ϕ E,P. In Lemma 4. we will pove that Q,P (ϕ, η) i a bounded linea opeato fom E,P to E,P uch that Q,P ϕ, η = Q,P (ϕ, η) ϕ, η E,P.

10 10 SANJIBAN SANTRA AN JUNCHENG WEI Thu finding a citical point of J (ϕ) i equivalent to olving the poblem in E,P : (4.4) l,p + Q,P ϕ + R (ϕ) = 0. We will pove in Lemma 4.3 that the opeato Q,P i invetible in E,P. In Lemma 4.5, we will pove that if ϕ belong to a uitable et, R (ϕ) i a mall petubation tem in (4.4). Thu we can ue the contaction mapping theoem to pove that (4.4) ha a unique olution fo each fixed P Λ,. Lemma 4.1. The functional l,p : H 1 0 () R defined in (4.1) i a bounded linea functional. Moeove, we have l,p = O( ). Poof. We have l,p l,p (ϕ) = M 1 K 1 S V,P ϕdd = M 1 K 1 (M 1) (,) V,P + V,P, + = M 1 K 1 (M 1) (,) ηp U,P + (ηp U,P ) + ηp U,P + f(ηp U,P ) ϕ = η M 1 K 1 (M 1) (,) P U,P + P U,P, + ϕ + (K 1) V,P, V,P + f(v,p ) ϕ (K 1) (ηp U,P ) (K 1) P U,P, P U,P + f(p U,P ) M 1 K 1 P U,P (,) η + P U,P ηϕ + M 1 K 1 (η η p )P U p,p ϕ (M 1) = M 1 K 1 (K 1) P U,P, + P U,P, ϕ + M 1 K 1 (M 1) (K 1) η P U,P, + η P U,P ϕ + η M 1 f(p K 1 U,P ) f(u,p ) ϕ (M 1) = η M 1 K 1 (K 1) P U,P, + P U,P, ϕdd + η M 1 f(p K 1 U,P ) f(u,p ) ϕ + M 1 K 1 (η η p )P U p,p ϕdd + M 1 K 1 P U,P (,) η + P U,P ηϕdd.

11 11 In ode to etimate all the tem we decompoe the domain into = (\B d (P )) (B d (P ) \ B d (P )) B d (P ). We obtain η M 1 f(p K 1 U,P ) f(u,p ) ϕdx = M 1 f(p K 1 U,P ) f(u,p ) ϕdx + (1 η) M 1 f(p K 1 U,P ) f(u,p ) ϕdx = I 1 + I. Fom I 1, we obtain I 1 + B d (P ) \B d (P ) C ( ( ) p 1 U,P v ϕdx + B d (P ) = O( ) ϕ. Futhemoe, I ( U,P ) p 1 v ϕdx B d (P )\B d (P ) ( U,P ) p 1 v ϕdx ) 1 ϕ M 1 k 1 dd + C +k φ + o(1) +k φ B d (P )\B d (P ) ( P U,P ) p 1 v ϕ = O( ) ϕ. Alo it i eay to check uing the decay etimate in (.15), all the othe tem ae of ode ϕ. Hence we obtain and a a eult l,p (ϕ) = O( ) ϕ. l,p = O( ). Lemma 4.. The bilinea fom Q,P (ϕ, η) defined in (4.) i a bounded linea. Futhemoe, Q,P (ϕ, η) C ϕ η whee C i independent of. Poof. Uing the Hölde inequality, thee exit C > 0, uch that M 1 K 1 V p 1,P ϕη dd C M 1 K 1 ϕ η C ϕ η and M 1 K 1 ϕ η + ϕηdd C ϕ η. Lemma 4.3. Thee exit ρ > 0 independent of, uch that Q,P ϕ ρ ϕ ϕ E,P, P Λ,P.

12 1 SANJIBAN SANTRA AN JUNCHENG WEI Poof. Suppoe thee exit a equence n 0, ϕ n E n,p, P Λ,P uch that ϕ n n = n and Q n ϕ n n = o( n ). Let ϕ i,n = ϕ n ( n z + P ) and n = {y : n z + P } uch that (4.5) M 1 K 1 ϕ i,n + ϕ i,n = n M 1 K 1 ϕ i,n + ϕ i,n = 1. n Hence thee exit ϕ H 1 (R ) uch that ϕ n ϕ H 1 (R ) and hence ϕ n ϕ L loc (R ). We claim that (,) ϕ ϕ + pu p 1 ϕ = 0 in R that i fo all η C0 (R ), (4.6) M 1 K 1 ϕ η + M 1 K 1 ϕη = p M 1 K 1 U p 1 ϕη. R R R Now M 1 K 1 ϕ η + ϕ η pv p 1 ϕ η = Q n,p ϕ n, η which implie whee,p = o( n ) η n M 1 K 1 ϕ η + ϕ η pṽ p 1,P ϕ η = o(1) η, Ṽ n,p n = V n,p n ( n y + P ), η = M 1 η K 1 + η, n { Ẽ n,p = η : M 1 K 1 η W n, + M 1 K 1 η W n, n = 0 = M 1 K 1 η W n, + M 1 K 1 η } W n,, n and W Ṽn (ny+pn) n, = n chooe a 1, a R uch that, W n, = n Ṽn (ny+p ). Let η C0 (R ). Then we can η n = η a 1 Wn, + a Wn,. Note that W n, atifie the poblem (,) Wn, + W n, =pηu p 1 (y) U + Φ n(y) in n (4.7) W n, = 0 on 1,n,n W n, = 0 on 3,n 4,n ν η whee Φ n (y) = n U p + η P U,P + η P U,P.

13 13 Then we claim that W n, i bounded in H0 1 ( n ). Uing the Hölde inequality, we have M 1 N 1 W n, + W n, = p M 1 N 1 p 1 U ηu n n W n, + M 1 N 1 Φ n W n, n ( ) 1 C M 1 k 1 W n, n ( ) 1 C M 1 N 1 W n, + W n, (4.8). n Hence n M 1 N 1 W n, + W n, i unifomly bounded and a a eult thee exit W uch that up to a ubequence. Hence W n, W in H 1 (R ) W n, W in L loc. Note that W atifie the poblem, p 1 U (,) W + W = pu in R (4.9) M 1 K 1 W + W = p M 1 K 1 p 1 U U R R W. We claim that W n, W in H 1 (R ). Fit note that M 1 K 1 W n, + W n, = p M 1 K 1 p 1 U U n n W n, + M 1 K 1 Φ n Wn, n p M 1 K 1 p 1 U U R W (4.10) = M 1 K 1 W + W dd. R Hee we have ued that W n, convege weakly in L. Hence W n, W = U in H 1 tongly. Similaly, we can how that W n, W = U in H1 tongly. Now if we plug the value η n in (4.7) we obtain and letting n, we have M 1 K 1 ϕ η pu p 1 ϕη + ϕη R ( = a 1 M 1 K 1 ϕ U R + ϕ U pu p 1 ϕ U ) ( + a M 1 K 1 R ϕ U + ϕ U pu p 1 ϕ U ).

14 14 SANJIBAN SANTRA AN JUNCHENG WEI Uing the non-degeneacy condition we obtain M 1 K 1 ϕ η + ϕη pu p 1 ϕη = 0. R N Hence we have (4.6). Since ϕ H 1 (R ), it follow by non-degeneacy U ϕ = b 1 + b U. Since ϕ n Ẽ n,p, letting n in (4.7), we have M 1 K 1 ϕ U R = 0 M 1 K 1 ϕ U R = 0, which implie b 1 = b = 0. Hence ϕ = 0 and fo any R > 0 we have M 1 K 1 ϕ ndd = o( n). Hence B nr (P ) o( n) Q n,p (ϕ n ), ϕ n n ϕ n n p (V n,p) p 1 ϕ n n o(1) n which implie a contadiction. Lemma 4.4. Let R (ϕ) be the functional defined by (4.3). Let ϕ H 1 0 (), then (4.11) and (4.1) fo ome τ > 0 mall. R (ϕ) o(1) ϕ + o(1) (p 1)k ϕ = τ ϕ R (ϕ) o(1) ϕ + o(1) (p 1)k ϕ = τ ϕ. Poof. We have ( ) R (ϕ) o M 1 K 1 V p 1,P ϕ ( ) o(1) M 1 K 1 V p 1,P ϕ + o V p 1,P ϕ B d (P ) \B d (P ) Moeove, by the exponential decay of V,P we obtain, ( ) o M 1 K 1 V p 1,P ϕ Co(1) p 1 k M 1 K 1 ϕ o(1) p 1 k ϕ. \B d (P ) The econd etimate follow in a imila way. Lemma 4.5. Thee exit 0 > 0 uch that fo (0, 0, thee exit a C 1 map ϕ,p : E,P H, uch that ϕ,p Λ, atifying ) I (V,P + ϕ,p, η = 0, η Λ,.

15 15 Moeove, we have ϕ,p = O( ). Poof. We have l,p + Q,P ϕ + R (ϕ) = 0. A Q 1,P exit, the above equation i equivalent to olving efine Q 1,P l,p + ϕ + Q 1,P R (ϕ) = 0. G(ϕ) = Q 1,P l,p Q 1,P R (ϕ) ϕ Λ,. Hence the poblem i educed to finding a fixed point of the map G. Fo any ϕ 1 Λ and ϕ E with ϕ 1 τ, ϕ τ Fom Lemma 4.4, we have Hence we have Hence G i a contaction a G(ϕ 1 ) G(ϕ ) C R (ϕ 1 ) R (ϕ ). R (ϕ 1 ) R (ϕ ), η o(1) ϕ 1 ϕ η. R (ϕ 1 ) R (ϕ ) o(1) ϕ 1 ϕ. G(ϕ 1 ) G(ϕ ) Co(1) ϕ 1 ϕ. Alo fo ϕ E with ϕ τ, and τ > 0 ufficiently mall (4.13) G(ϕ) C l,p + C R (ϕ) C + C τ+τ C. Hence G : Λ, B τ (0) Λ, B τ (0) i a contaction map. Hence by the contaction mapping pinciple, thee exit a unique ϕ Λ, B k(0) uch that ϕ,p = G(ϕ,P ) and ϕ,p = G(ϕ,P ) C. We wite u = V,P + ϕ,p. Then we have I (u ) = I (V,P ) + M 1 K 1 ( V,P ϕ V,P ϕ + f(v,p )ϕ )dd + 1 ( ) M 1 K 1 ϕ ϕ + f (V,P )ϕ,p dd M 1 F K 1 (V,P + ϕ ) F (V,P ) f(v,p )ϕ,p 1 f (V,P )ϕ,p dd

16 16 SANJIBAN SANTRA AN JUNCHENG WEI which can be expeed a I (u ) = I (V,P ) + E (V,P )ϕ,p M 1 K 1 dd + 1 ( ) ϕ dx f (V,P )ϕ M 1 K 1 dd M 1 F K 1 (V,P + ϕ ) F (V,P ) f(v,p )ϕ 1 f (V,P )ϕ dd ( ) = I V,P + O( l,p ϕ,p + ϕ + R (ϕ,p )) ( ) (4.14) = I V,P + O( 4 ). 5. The educed poblem: min-max pocedue Poof of Theoem 1.1. Let G (P ) = G (d, θ) = I (u ). Conide the poblem min d Λ,P max G (d, θ). θ 0 δ θ θ 0+δ To pove that G (P ) = I ( V,P + ϕ,p ) i a olution of (1.1), we need to pove that P i a citical point of G, in othe wod we ae equied to how that P i a inteio point of Λ,. Fo any P Λ,P, fom Lemma 4.3 we obtain ( ) G (P ) = I V,P + O( l,p ϕ,p + ϕ + R (ϕ,p )) ( ) = I V,P + o(1) k+ (5.1) = γp M 1 1 P K 1 + γ 1 P M 1 1 P K 1 U ( d(p, 1 ) ) + o( k+ ). We have the expanion ( ) d(p, G (d, θ) = γ a M+K + a M+K 1 d + γ 1 γ 1 a M+K 1 ) U + O(d ) co M 1 θ in K 1 θ + o( +k ). It i clea that the maximum i attained at ome inteio point of θ (θ 0 δ, θ 0 +δ). Now we pove that fo that θ the minimum i attained at a citical point of Λ,P. Let P Λ,P, be a point of minimum of G (d, θ ), then we obtain G (d, θ ) = γ a M+K + a M+K 1 d + O(d ) co M 1 θ in K 1 θ + O( +k ). Chooe P uch that the d = d( P, 1 ) k ln. Then P Λ,P. But by definition, we have (5.) G (d, θ ) G (d, θ ).

17 17 Fom thi we obtain γa M+K + a M+K 1 d + O(d ) co M 1 θ in K 1 θ + O( k ) γ a M+K + a M+K 1 d + γ 1 γ 1 e d + O(d ) co M 1 θ in K 1 θ + o( k ) Hence thi implie that d ln. Hence d 0. Thi finihe the poof. 6. The educed poblem: max-max pocedue Poof of Theoem 1.. Hee we obtain the citical point uing a max-max pocedue. The pojection in the Neumann cae i jut Q,P. Hence the educed poblem ( ) d(p, (6.1) R (P ) = γp1 M 1 P K 1 γ 1 P1 M 1 P K 1 ) U + o( k+ ). Conide (6.) max max R (d, θ). d Λ,N θ 0 δ θ θ 0+δ We have the expanion ( ) d(p, R (d, θ) = γ a M+K + a M+K 1 d γ 1 γ 1 a M+K ) U + O(d ) co M 1 θ in K 1 θ + o( +k ). It i clea that the maximum in θ i attained at ome inteio point of θ (θ 0 δ, θ 0 + δ). Now we pove that fo that θ the minimum i attained at a citical point of Λ,N. Let P Λ,N, be a point of maximum of R (d, θ ), then we obtain R (d, θ ) = γ a M+K + a M+K 1 d + O(d ) co M 1 θ in K 1 θ + o( +k ). Chooe P uch that the d = d( P, 1 ) k ln. Then P Λ,P. But by definition, we have (6.3) R (d, θ ) R (d, θ ). Thi implie γa M+K + a M+K 1 d + O(d ) co M 1 θ in K 1 θ + o( k ) γ a M+K + a M+K 1 d γ 1 γ 1 e d + O(d ) co M 1 θ in K 1 θ + o( k ) Hence d ln. Hence d 0. Theoem 1. i poved. Refeence 1 A. Amboetti, A. Malchiodi, W. Ni; Singulaly petubed elliptic equation with ymmety: exitence of olution concentating on phee. I. Comm. Math. Phy. 35 (003), no. 3, A. Amboetti, A. Malchiodi, W. Ni; Singulaly petubed elliptic equation with ymmety: exitence of olution concentating on phee. II. Indiana Univ. Math. J. 53 (004), no., M. el Pino, P. Felme; Spike-layeed olution of ingulaly petubed elliptic poblem in a degeneate etting. Indiana Univ. Math. J. 48 (1999), no. 3,

18 18 SANJIBAN SANTRA AN JUNCHENG WEI 4 M. el Pino, M. Kowalczyk, J. Wei; Concentaton on cuve fo nonlinea Schödinge equation. Comm. Pue Appl. Math. 60 (007), no. 1, M. Eteban, P. Lion; Exitence and nonexitence eult fo emilinea elliptic poblem in unbounded domain. Poc. Roy. Soc. Edinbugh Sect. A 93 (198/83), B. Gida, W. Ni, L. Nienbeg; Symmety and elated popetie via the maximum pinciple. Comm. Math. Phy. 68 (1979), no. 3, F. Pacella, P.N.Sikanth; A eduction method fo emilinea elliptic equation and olution concentating on phee. (Pepint 01). 8 B. Ruf, P. Sikanth; Singulaly petubed elliptic equation with olution concentating on a 1-dimenional obit. J. Eu. Math. Soc. 1 (010), no., P. Epoito, G.Mancini, S.Santa, P. Sikanth; Aymptotic behavio of adial olution fo a emilinea elliptic poblem on an annulu though Moe index. J. iff. Equation 39 (007), no. 1, F. Lin, W. M. Ni, J. Wei; On the numbe of inteio peak olution fo a ingulaly petubed Neumann poblem. Comm. Pue Appl. Math. 60 (007), no., W. M. Ni, I. Takagi; On the hape of leat-enegy olution to a emilinea Neumann poblem. Comm. Pue Appl. Math. 4 (1991), no. 7, W. M. Ni, I. Takagi; Locating the peak of leat-enegy olution to a emilinea Neumann poblem. uke Math. J. 70 (1993), no., W. M. Ni, J. Wei; On the location and pofile of pike-laye olution to ingulaly petubed emilinea iichlet poblem. Comm. Pue Appl. Math. 48 (1995), no. 7, Sanjiban Santa, School of Mathematic and Statitic, The Univeity of Sydney, NSW 006, Autalia. adde: anjiban.anta@ydney.edu.au Juncheng Wei, epatment of Mathematic, The Univeity of Bitih Columbia, Vancouve. adde: jcwei@math.ubc.ca

On the quadratic support of strongly convex functions

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