On uniqueness for nonlinear elliptic equation involving the Pucci s extremal operator

Transcription

1 On uniqueness fo nonlinea elliptic equation involving the Pucci s extemal opeato Paticio L. Felme a,, Alexande Quaas b, Moxun Tang c a Depatamento de Ing. Matemática, F.C.F.M. Univesidad de Chile, Casilla 170, Coeo 3, Santiago, Chile b Depatamento de Matemática, Univesidad Santa Maía Casilla, V-110, Avda. España 1680, Valpaaíso, Chile c Depatment of Mathematics, Michigan State Univesity, East Lansing, MI 48824, USA Abstact In this aticle we study uniqueness of positive solutions fo the nonlinea unifomly elliptic equation M λ,λ + (D2 u) u + u p = 0inR N, lim u() = 0, whee M λ,λ + (D2 u) denotes the Pucci s extemal opeato with paametes 0 <λ Λ and p>1. It is known that all positive solutions of this equation ae adially symmetic with espect to a point in R N, so the poblem educes to the study of a adial vesion of this equation. Howeve, this is still a nontivial question even in the case of the Laplacian (λ = Λ). The Pucci s opeato is a pototype of a nonlinea opeato in no-divegence fom. This featue makes the uniqueness question specially challenging, since two standad tools like Pohozaev identity and global integation by pats ae no longe available. The coesponding equation involving Mλ,Λ is also consideed. Keywods: Uniqueness; Positive adial solutions; Nonlinea elliptic equations; Pucci s extemal opeato 1. Intoduction Let 0 <λ Λ be two given positive eal numbes. Fo a C 2 scala function u defined in R N, the Pucci s extemal opeatos ae given by * Coesponding autho. addesses: pfelme@dim.uchile.cl (P.L. Felme), alexande.quaas@usm.cl (A. Quaas), tang@math.msu.edu (M. Tang).

2 M λ,λ + ( D 2 u ) = λ e i + Λ e i and M ( λ,λ D 2 u ) = Λ e i + λ e i, e i <0 e i >0 e i <0 e i >0 whee e i = e i (D 2 u), i = 1,...,N, ae the eigenvalues of the Hessian matix D 2 u.fomoe details and equivalent definitions see the monogaph of Caffaelli and Cabé [2]. Clealy, in the special case λ = Λ the two opeatos become the same and M + λ,λ( D 2 u ) = M λ,λ( D 2 u ) = λ u, whee u is the usual Laplacian of u. The Pucci s extemal opeatos povide impotant pototypes of fully nonlinea unifomly elliptic opeatos and even though they etain positive homogeneity and some popeties associated to the maximum pinciple, they ae no longe in divegence fom, thus deviating in a fundamental manne away fom the Laplacian. Recently in [8,9], Felme and Quaas studied the nonlinea elliptic equation M ± λ,λ( D 2 u ) + u p = 0, (1.1) fo positive adially symmetic solutions, p>1. Hee, fo convenience, we wite M λ,λ ± in (1.1) to mean the two equations, one with the opeato M λ,λ + and the othe with the opeato M λ,λ.in the special case of the Laplacian, that is λ = Λ, the ange of existence and nonexistence fo the ball o R N is chaacteized by the Sobolev citical numbe p N = (N + 2)/(N 2). Fo a ball of adius R, denoted by B R, Eq. (1.1) has a solution with zeo Diichlet bounday condition in B R if and only if 1 <p<p N. When dealing with adially symmetic positive solutions in all R N the situation is dual, that is, (1.1) has a solution in R N if and only if p p N. These basic facts can be poved, fo example, by doing a phase plane analysis afte the Emden Fowle tansfomation. When 0 <λ<λa simila situation occus as poved in [9]. Fo the opeato M λ,λ + it is shown in [9] that thee exists a numbe p + playing the ole of p N egading existence and nonexistence in the ball and in R N, and fo the opeato Mλ,Λ it is shown that thee exists a coesponding numbe p. Fo these numbes the following inequality holds p <p N <p +. See [9], fo a moe detailed desciption of positive entie solutions in the ange p p + (p p ), whee a new phenomenon appeas. We also notice that no fomula is known fo this numbe p +. We obseve that the nonexistence esults in R N just descibed may allow to find existence esults fo moe geneal nonlineaities via blow-up analysis and degee theoy. This is pecisely the wok done by Felme and Quaas in [10], whee the equation M ± λ,λ( D 2 u ) + f(u)= 0, u>0inb R and u = 0 on B R, (1.2) was studied fo vaious nonlineaities. Fo the canonical model case f(u)= u + u p, p>1, (1.3) they found that wheneve 1 <p<p ± Eq. (1.2) has at least one adially symmetic solution. Moeove they show that the equation possesses a gound state, that is an entie positive solution satisfying lim u() = 0. In view of these esults it would be inteesting to study othe

3 qualitative popeties of the solutions of (1.2) and its associated initial value poblem, fo diffeent values of p. Fo notational convenience, we allow in (1.2) that R = and we intepet the bounday condition as lim u() = 0. In the case of the Laplacian, whee p N = p + = p, it is also known that the numbe p N is optimal fo existence in (1.2), that is, if p p N then (1.2) (1.3) does not have a solution. This nonexistence esult fo solutions fo (1.2), and othe qualitative popeties of the associated initial value poblem, have been histoically poved using the well-known poweful Pohozaev identity. Natually, one may ask if we can establish an analogue of the Pohozaev identity fo the geneal Pucci s opeato, and then use this identity to obtain nonexistence esults. The answe to this question is yes to the fist pat and no to the second pat. We can easily deive the homologue of the Pohozaev identity fo the Pucci s opeato, but this identity is nealy useless. To see this moe clealy, we define the function P()= θñ (u ) 2 () + 2Ñ F ( u() ) + (Ñ 2)θÑ 1 u()u (), (1.4) whee u = u() is a adial solution of (1.2) and θ takes the value λ o Λ accoding to the sign of u and u, as descibed at the beginning of Section 2. We efe to (3.2) fo the pecise definition of the dimension-like paamete Ñ. It is staightfowad to veify that P () = Ñ 1[ 2ÑF(u) (Ñ 2)uf (u) ]. This is the homologue of the classical Pohozaev identity fo the Pucci s opeato. Fo ou function f(u)= u + u p, a futhe calculation gives P () = 2Ñ 1( σu p+1 u 2), σ = 2Ñ (Ñ 2)(p + 1). 2(p + 1) Two fatal factos pevent any effective application of this genealized Pohozaev identity to the Pucci s opeatos. Fist, the function P() has jumps at points whee u and u vanishes, that is at citical o inflection points of u. In paticula, this function could possibly jump fom a positive value to a negative one at those points. Second, fo the ange of p of inteest hee, the coesponding paamete σ is not always positive, and thus it is nealy impossible to undestand how the function P() behaves without any a pioi knowledge of the concavity and citical points changes of u. The discontinuities expeienced by the Pohozaev function P()is indeed an intinsic popety of the Pucci s opeato, posing in this way many inteesting questions about qualitative popeties of positive adial solutions to (1.2). While the qualitative analysis fo equations involving Laplacian may still be nontivial, in the case of Pucci s opeatos the lack of continuity of pope functions o thei deivatives ceates majo technical difficulties fo any agument elying upon a vaiety of applications of the method of integation by pats. In this pape, we shall examine deeply some qualitative popeties of positive adially symmetic solutions to (1.2) and pove uniqueness of the adial solutions found in [10]. This esult, which in paticula includes the uniqueness of the gound state fo (1.2), is impotant by itself and open the fundamental question of the behavio of positive solutions fo (1.2) when p p +. Fo futhe discussion see concluding emaks in Section 4. Now we state ou main theoem.

4 Theoem 1.1. Assume 1 <p<p + in case of the opeato M + λ,λ (D2 u), o1 <p<p in case of M λ,λ (D2 u). Then the poblem (1.2) admits exactly one positive adial solution in each finite ball B R and exactly one gound state (R = ). Let u = u() be this unique solution defined fo (0,R), R. The we have: (i) the maximum value of u, attained at the oigin, is lage than one; (ii) u() and u /u ae stictly deceasing in the adial diection fo (0,R), and (iii) u changes concavity exactly once, that is, thee is a unique c (0,R) with u( c )>1 such that u < 0 fo 0 << c, u > 0 fo c <<R. Remak 1.1. We mention that pat (iii) is new even fo the Laplacian case. It is by no means tivial, as the example of pseudo-slow decaying solutions fo M + λ,λ( D 2 u ) + u p = 0 show in case p + <p<(ñ + 2)/(Ñ 2). These solutions ae deceasing, positive and change concavity infinitely many times. The definition of Ñ is given in (3.2). See details in [9]. Remak 1.2. The Pucci s opeatos etain maximum pinciple and compaison popeties of the Laplacian, so that the moving planes method is applicable to study adial symmety of solutions of (1.2). Da Lio and Siakov in [7] poved, among othe things, that all solutions of (1.2) ae adially symmetic, even in the case R =. The study of uniqueness questions fo Eq. (1.2) in the case of the Laplacian has a long histoy. The main step we can distinguish ae the contibutions of Ni [17] and Ni and Nussbaum [18] who teated the case of a ball. In the case of R N, the study of uniqueness is taced back to Coffman [4], Peletie and Sein [19,20] and McLeod and Sein [15]. Then the fundamental wok by Kwong [11] teating the all ange of exponents. Subsequent contibutions have been given by many authos, see among them the following [3,5,12 14,16]. Moe ecently we mention the wok by Sein and Tang [21] and Tang [22]. The difficult case of the annulus with Diichlet bounday conditions is teated by Tang [23]. In this aticle we use many ideas fom [23]. This aticle is oganized in fou sections. In Section 2 we shall deive the monotonicity of u.in the Laplacian case, this follows fom a vey simple agument of an enegy function. Howeve in ou context this analysis is vey delicate since we do not have an appopiate enegy function fo all. We need to combine diffeent enegies in ode to ovecome the discontinuity each of them may have. In Section 3, we shall pove the monotonicity of u /u and pat (iii) of Theoem 1.1 by showing that the useful function Q() = Ñ [ θu 2 + uf (u) ] + θ(ñ 2)Ñ 1 uu (1.5) is positive on (0,R). Indeed, the poof of the positivity of Q constitutes the majo technical pat of this pape and it is hee whee we intoduce new ideas. In Section 4, we pove the uniqueness of positive adial solutions. Fo this pupose we study the vaiations of the solution with espect to the initial value following an idea of Coffman [4]. Then a majo step is obtained by modifying

5 ideas fom the ecent wok [23]. Howeve, this is not a simple tivial genealization as a vaiety of technical complexities aise due to the discontinuities of the functions involved. 2. Popeties of the solutions Let u = u(x) be a adial C 2 function in R N. As usual we abuse the notation to wite u(x) = u(), = x, without causing any futhe confusion. As calculated in [8] we have [ D 2 u(x) = u () u ] () Id + 2 u () 3 X, whee Id is the N N identity matix, and X is the matix whose enties ae x i x j. Obseving futhe that D 2 u(x) x = u () x and D 2 u(x)y = u () y, fo evey vecto in the hypeplane x y = 0, we find that the eigenvalues of the Hessian matix D 2 u(x) ae u (), which is simple, and u ()/, which has multiplicity N 1. Theefoe, fo a adial function u() thee holds M ± λ,λ ( D 2 u ) = θu () + N 1 Θu (), whee θ and Θ take the values of eithe λ o Λ, depending on the opeatos M λ,λ ± and the signs of u () and u (). Coesponding to the opeato M λ,λ +,wehave θ = Λ when u > 0 and θ = λ when u < 0; (2.1) Θ = Λ when u > 0 and Θ = λ when u < 0; (2.2) and coesponding to the opeato M λ,λ,wehave θ = λ when u > 0 and θ = Λ when u < 0, Θ = λ when u > 0 and Θ = Λ when u < 0. Consequently, if u = u() is a positive C 2 adial solution of (1.2) and we wite u(0) = α>0, then u is also a solution to the initial value poblem of the odinay diffeential equation θu + N 1 Θu + f(u)= 0, u(0) = α>0, u (0) = 0, (2.3) satisfying additionally the conditions: u() > 0 in the inteval [0,R) and u(r) = 0. Thus, in ode to pove the uniqueness popety fo (1.2) it is sufficient to pove that thee is exactly one α>0 such that the solution of (2.3) satisfies the two additional conditions. The existence and uniqueness of C 2 solution to the initial value poblem (2.3) can be analyzed using the ideas of Ni and Nussbaum [18] and Felme and Quaas [10].

6 2.1. Enegy functions Motivated by the case of the Laplacian, that is, λ = Λ = 1, we may define the enegy function Using (2.3) we find that E θ () = θ 2 (u ) 2 () + F ( u() ). E θ 1)Θ () = (N (u ) 2 (). (2.4) Howeve, this is not sufficient to imply that E θ () is a deceasing function ove the whole ange whee u is defined and positive. In fact E θ () can be discontinuous at points whee u changes concavity. Indeed, let = I be a point of inflection which is not a citical point of u, that is, u changes sign nea = I, u ( I ) = 0 and u ( I ) 0, then E θ () must have a jump at I. Thus E θ () is only a piecewise C 1 function and deceases ove each subinteval whee u has the same concavity. Altenatively, we may use E λ () = λ 2 (u ) 2 () + F ( u() ) o E Λ () = Λ 2 (u ) 2 () + F ( u() ) (2.5) as moe appopiate enegy-type functions, since they ae obviously C 1 functions. In an inteval whee u does not vanish, eithe E λ () o E Λ () agees with E θ () and is theefoe deceasing. Howeve, ove the whole inteval of definition of u, we cannot easily claim the monotonicity of E λ () o E Λ (), since thee ae no simple fomulas like (2.4), available fo the calculation of E λ () o E Λ (). The following lemmas, which suffice fo ou puposes, give some patial esults on the monotonicity of E λ () and E Λ (). Lemma 2.1. Conside the opeato M + λ,λ (D2 u) and let L be an inteval in which u is positive. (i) If u > 0 in L, then both E λ () and E Λ () ae deceasing in L. (ii) If u < 0 in L, then E λ () deceases when f(u)>0, and E Λ () deceases when f(u)<0. Poof. (i) Assume u > 0inL. Conside E λ () fist. At points whee u < 0, θ = λ and then by (2.4) we get that E λ () is deceasing; at points whee u > 0, we have and by (2.4) we obtain E λ () = Λ λ (u ) 2 () + E θ () 2 E λ () (Λ λ)u ()u () < 0. A simila agument yields the same esult fo E Λ (). (ii) Assume u < 0inL.Ifu < 0, then E λ () is deceasing by (2.4) again and, if u > 0 and f(u)>0, then E λ () = λu u + f(u)u < 0. Finally, if f(u)<0, then (2.3) implies u > 0, and so E Λ () < 0 by (2.4).

7 A simila agument establishes the next lemma. Lemma 2.2. Conside the opeato M λ,λ (D2 u) and let L be an inteval in which u is positive. (i) If u < 0 in L, then both E λ () and E Λ () ae deceasing in L. (ii) If u > 0 in L, then E λ () deceases when f(u)<0 and E Λ () deceases when f(u)>0. Remak 2.1. Ou discussion fo the thee enegy-type functions above eveals a athe delicate featue appeaing in ou study on the Pucci s opeatos: functions involving both u and u eithe have discontinuities somewhee, o thei deivatives do not have a univesal fomula ove the whole inteval of definition of u. This fact ceates majo technical difficulties in ou futhe discussion which elies upon the implications of integation by pats Monotonicity The validity of the next esult in the classical Laplacian case can be veified vey easily using the univesal deceasing popety of the enegy function E(). In the cuent case, the poof is nontivial and uses Lemmas 2.1 and 2.2 in a delicate way. Lemma 2.3. Let u = u() be a solution of (2.3). If it attains a positive minimum value at some 0 0, then it is positive and bounded in ( 0, ), and lim inf u>0. Poof. If u is the constant solution, that is, u 1, then the lemma is tivially tue. Suppose u is a nonconstant solution of (2.3). Then u and u do not vanish simultaneously at any >0, as follows fom the uniqueness of solutions to this initial value poblem. Thus if u takes a local minimum value at 0 0, then u ( 0 )>0, and so by (2.3) it holds that f(u( 0 )) < 0 and u( 0 )<1. We shall pove that u is bounded and We have two cases: u() > u( 0 ) fo all > 0. (2.6) Case 1. M + λ,λ (D2 u). Let( 0, 1 ), 0 < 1, be the maximal inteval in which u > 0. Then fo any ( 0, 1 ) it follows fom Lemma 2.1 that fom whee we see that u() is bounded above by F ( u() ) <E λ () E λ ( 0 ) = F ( u( 0 ) ) < 0, (2.7) β = ( (p + 1)/2 ) 1/(p 1, the positive numbe making F(β)= 0. If 1 = we ae clealy done. If 1 <, then u has a stict maximum at 1 and by (2.3) we have f(u( 1 )) > 0 and u( 1 )>1. Let ( 1, 2 ), 1 < 2, be the maximal inteval ove which u < 0. We claim that u() > u( 0 ) fo all ( 1, 2 ). (2.8)

8 In fact, if u() 1 in ( 1, 2 ), then (2.8) is obviously valid. Othewise, let ˆ 0 ( 0, 1 ) and ˆ 1 ( 1, 2 ) be the unique numbes such that u = 1. By Lemma 2.1 E λ () is deceasing on (ˆ 0, ˆ 1 ). Hence (u ) 2 (ˆ 0 ) 2 = E λ(ˆ 0 ) F(1) λ > E λ(ˆ 1 ) F(1) λ = (u ) 2 (ˆ 1 ). 2 This, togethe with the deceasing popety of E Λ () ove ( 0, ˆ 0 ) and (ˆ 1,)fo any (ˆ 1, 2 ) we find that F ( u() ) E Λ () < E Λ (ˆ 1 ) = Λ 2 (u ) 2 (ˆ 1 ) + F(1) < Λ 2 (u ) 2 (ˆ 0 ) + F(1) = E Λ (ˆ 0 )<E Λ ( 0 ) = F ( u( 0 ) ), implying (2.8), ou claim. Now if 2 =, then (2.8) implies (2.6) and we ae done. If not, then u assumes a local minimum value at 2 and we can epeat the agument above successively to get u( 2 )<u()<β fo all > 2. This completes the poof fo Case 1. Case 2. M λ,λ (D2 u). We shall use the same notation as in Case 1. Without loss of geneality, we may only discuss the situation when u has citical points at 1 and 2. Since E λ () deceases in ( 0, ˆ 0 ), we have that E λ (ˆ 0 )<E λ ( 0 ) = F(u( 0 )) < 0, and so (u ) 2 (ˆ 0 )< 2F(1)/λ. Hence F ( u( 1 ) ) <E Λ (ˆ 0 ) = Λ ( 2 (u ) 2 (ˆ 0 ) + F(1)< 1 Λ ) F(1), (2.9) λ which clealy povides a uppe bound fo u in (ˆ 0, 1 ). It emains to show that (2.8) holds in this case too. It follows fom Lemma 2.2 that E Λ () is deceasing on (ˆ 0, ˆ 1 ), so we obtain (u ) 2 (ˆ 0 )>(u ) 2 (ˆ 1 ) again. The est of the poof is the same as in Case 1, except one has to eplace Λ with λ thee. Remak 2.2. We obseve that, as the atio Λ/λ, the last tem in (2.9) and hence the uppe bound fo u povided by (2.9) tends to too. This distinguishes fom Case 1 whee we deive a simple estimate u<β.fom λ,λ (D2 u), it emains unclea whethe o not all the positive solutions have a univesal uppe bound independent of λ and Λ. Constucting a shape estimate in this case is nontivial as it is likely that the enegy function E Λ () could become positive in a subinteval of ( 0, 1 ) when Λ is sufficiently lage. Lemma 2.4. If u = u() is a positive adial solution of (1.2), then u(0) >1 and u () < 0 fo (0,R). Poof. If u(0) 1, then u attains a positive minimum value at zeo, and by Lemma 2.3 we have u>u(0) fo all >0, which is impossible fo a positive adial solution of (1.2). Futhemoe, if u(0) >1 then u has a stict maximum at = 0, and by Lemma 2.3 again, u can only be deceasing on (0,R).

9 Remak 2.3. We may obtain a much stonge esult than Lemma 2.3. In fact, it can be poved that if a solution u of (2.3) has a positive minimum then it is oscillatoy, with infinitely many positive minima and maxima in (0, ). The difficult situation to conside is when u is monotonically appoaching 1. In this case the nonlineaity f appoach a linea function, so that one can get a contadiction, by slightly modifying the aguments developed in [1, Lemma 3.1], fo the eigenvalue poblem. Remak 2.4. Summaizing, we have the following classification of the solutions fo the initial value poblem (2.3): evey solutions belongs to one of the following thee classes: (i) u is a cossing solution. Hee by a cossing solution we mean that thee exists a finite numbe R such that u>0fo (0,R), u(r) = 0 and u (R) < 0. (ii) u is a gound state. (iii) u is a positive, oscillatoy solution with infinitely many positive minima and maxima in (0, ). Fom hee we obseve that given 0 <R only cetain values of α>0giveisetoa solution of (1.2). In fact, ou pupose is to show that thee exists a unique α(r) with this popety. In the paticula and impotant case of a gound state and in the ange of p whee uniqueness holds, it can futhe been poved that thee is α such that fo all α (0,α ), solutions of (2.3) ae positive and oscillatoy and if α (α, ) the solutions of (2.3) ae cossing. Remak 2.5. Fo a gound state, we can futhe pove that it decays to zeo exponentially. Fo ou late pupose, we mention given any ε>0 lim u()e(1 ε) = lim u ()e (1 ε) = 0. (2.10) This conclusion can be eached by using compaison techniques associated to the Laplacian, since thee exists 0 such that u () < 0 and u () > 0 fo all A useful function Fo claity of ou discussion, though the est of this pape we will only conside the opeato M λ,λ + (D2 u), as the main idea applies equally well to both M λ,λ + and M λ,λ. Thus the paametes θ and Θ in equation θu + N 1 Θu + f(u)= 0 (3.1) ae detemined by (2.1) and (2.2). As in [6,9], we intoduce the dimension-like paamete Ñ = Θ (N 1) + 1. (3.2) θ Using this notation we can wite (3.1) as θ ( Ñ 1 u ) = Ñ 1 f(u). (3.3)

10 Note caefully that Ñ is not a fixed constant, but depends on the concavity and monotonicity of u. In the special case u u > 0, one simply has Ñ = N. Fo the othe cases, Ñ can be eithe lage than o smalle than N. Now we ecall the definition of the functional Q in (1.5), giving pecise meaning to all the constants appeaing in the definition. We shall devote the est of this section to discussing about this Q function. We will see how useful it is in detecting some fundamental qualitative popeties of the solutions of (2.3), and what is the advantage of using this function instead of the homologous fom of the well-known Pohozaev identity Sign-etaining popety As we obseved in Section 2, functions involving with both u and u eithe has discontinuities somewhee, o does not have a univesal fomula fo its deivative with espect to in the whole inteval of definition of u. In paticula, it is easy to see that Q() has jumps at points whee u changes concavity. Howeve, it possesses a cucial popety: it does not change signs at these points. We call such a popety the sign-etaining popety. Lemma 3.1. The function Q() has the sign-etaining popety. Poof. By (3.1) and (3.2) we fist deive Inseting this into (1.5) we obtain f(u)/θ = u + Ñ 1 u. Q() = θñ 1( u 2 uu uu ). (3.4) This fomula gives the sign-etaining popety immediately Concavity Let u = u() be a positive adial solution of (1.2). Recall fom Lemma 2.4 that u(0)>1 and u () < 0fo (0,R). Hence u must be concave down (u < 0) fo close to zeo. On the othe hand, by (3.1) it is clea that u > 0 wheneve f(u)<0, that is, u<1. Consequently, u must change concavity at least once in (0,R). Natually, one may ask how many times u will change its concavity. In fact, to ou knowledge, this question was not addessed befoe, even fo the classical Laplacian equation. Fo the Pucci s opeatos, this question becomes an impotant one since the equation paametes θ and Θ depend on the concavity of u. In the next poposition we show that if Q>0 fo all (0,R), then u changes concavity exactly once. Late we will show that Q is indeed positive in some cases. Poposition 3.1. Let u = u() be a positive adial solution of (1.2). IfQ>0 fo all (0,R), then u changes concavity exactly once.

11 Poof. We fist veify the identity Indeed, by (3.3) we have d d ( u u θñ 2 u ) = Q ( d u ) (θ Ñ 2 u ) ( 2 d θ Ñ 1 u ) (θ = Ñ 2 u ) 2 d u d θñ 1 u 2. (3.5) = θ 2Ñ 3 uf (u) (Ñ 2)θ 2 2Ñ 4 uu θ 2 2Ñ 3 (u ) 2 = θñ 3( Ñ uf (u) + (Ñ 2)θÑ 1 uu + θñ (u ) 2) = θñ 3 Q. Now suppose fo contadiction that u has moe than one points of inflection, and let 0 < c 1 <c 2 be the fist two of them such that u (c 1 ) = u (c 2 ) = 0, u (c 1 ) 0, and u (c 2 ) 0. Since u is always concave up as long as u<1, thee must hold that At = c 1, u = 0 implies Hence we have and then we get u() > 1 fo 0<<c 2. Ñ 1 u = f(u). θ 0 u (c 1 ) = Ñ 1 2 u f (u) u = f(u) f (u) u θ θ θ f(u) uf (u) + u 0. u Similaly, we can show that at = c 2 thee holds f(u) uf (u) + u 0. u On one hand, fom ou hypothesis we have Q>0on(0,R), then by (3.5) we see that u /u is deceasing. On the othe hand, as f(u)= u + u p, with p>1, we have that f(u)/(uf (u)) is an inceasing function of u fo u>1 and thus it is a deceasing function of (c 1,c 2 ).This leads to a contadiction and poves the uniqueness of the points of inflection.

12 3.3. The positivity of Q fo 1 <p<p + By the existence esult of [10] poblem (1.2) admits adial solutions. Given u = u() a positive adial solution of (1.2) we shall pove that indeed Q() > 0fo (0,R). This esult, togethe with Poposition 3.1 and (3.5), eveals some futhe popeties of adial solutions. Moeove, as we see in the next section, the positivity of Q is a cucial condition in ou study of uniqueness of adial solutions. Poposition 3.2. Suppose 1 <p<p +. Let u = u() be a positive adial solution of (1.2). Then fo all (0,R), we have Q>0, and consequently, (i) the negative function u /u is deceasing, and (ii) u changes concavity exactly once: thee is a unique c (0,R)with u( c )>1 such that u < 0 fo 0 << c, u > 0 fo c <<R. The poof of the positivity of Q in the Laplacian case (λ = Λ) follows fom a simple application of the well-known Pohozaev identity, see Tang [23, Lemma 2.2]. Unfotunately, the same appoach does not wok fo the geneal case (λ<λ) as the homologous fom of the Pohozaev identity is no longe useful as obseved in the intoduction. The poof of the positivity of Q constitutes the majo technical pat of the cuent wok. We will begin with showing that Q must be positive when is eithe close to zeo o close to R, so we get the positivity at the two ends of the inteval (0,R). Next, we show that fo some paticula choice of p and Ñ, Q does not admit any local minimum value in the middle of (0,R).This is of couse sufficient to obtain the positivity of Q fo the chosen values of p and Ñ. Finally, we show that wheneve 1 <p<p +, the function Q cannot have a nonnegative minimum value. Using a homotopy type agument we can theefoe establish the positivity of Q as claimed in Poposition 3.2. Lemma 3.2. Let u = u() be a positive adial solution of (1.2). Then Q() > 0 if eithe is close to zeo, o u() 1. Poof. Since u < 0 fo all (0,R)and u < 0fo close to zeo, it eadily follows fom (3.4) that Q>0 as long as >0 is small. Witing we find and so Q() = Ñ uf (u) + θñ 1 u ( u + (Ñ 2)u ), Q () = ÑÑ 1 uf (u) + Ñ uf (u)u + Ñ f(u)u Ñ 1 f(u) ( u + (Ñ 2)u ) + θñ 1 u ( f (u)/θ ) Q () = Ñ u ( uf (u) f(u) ) + 2Ñ 1 uf (u). (3.6)

13 Let 1 (0,R) be the unique numbe whee u = 1, then fo 1 <<Rwe have 0 <u()<1, and Q () < 0 by (3.6). Since Q(R) = 0, it follows eadily that Q() > 0 as long as 1 < <R. Lemma 3.3. Suppose p Ñ/(Ñ 2). Let u = u() be a positive adial solution of (1.2), then Q() > 0 fo 0 <<R. Poof. In view of Lemma 3.2, it suffices to show that Q does not admit any nonpositive minimum value in (0, 1 ), whee 1 was defined in the poof above. We do this by showing that wheneve Q = 0 it holds that Q < 0. Stating with (3.6) we have Q () = d d ( (p 1)Ñ u p u + 2Ñ 1 uf (u) ) = (p 1)Ñ u p u + (p 1)ÑÑ 1 u p u + p(p 1)Ñ u p 1 (u ) 2 + 2Ñ 1 f(u)u + 2Ñ 1 uf (u)u + 2(Ñ 1)Ñ 2 uf (u). At a citical point of Q,weuseQ = 0 to educe Q to Q () = (p 1)Ñ u p u + Ñ 1 u [ (p 1)Ñu p 2pf (u) + 2f(u) + 2uf (u) (Ñ 1)(p 1)u p] = (p 1)Ñ u p u + Ñ 1 u [ (p + 1)u p + 2(p 2)u ]. Now if p Ñ/(Ñ 2), then p + 1 (p 1)(Ñ 1) and (p + 1)u p + 2(p 2)u > (p 1)(Ñ 1)u p, yielding [ Q <(p 1)Ñ u p u + Ñ 1 ] u = (p 1)Ñ u p f(u)/θ <0 fo (0, 1 ). The poof is completed. Lemma 3.4. Let u = u() be a positive adial solution of (1.2). Then Q() does not have any nonnegative minimum value in (0,R). Poof. Defining Q 1 () = u () u() + 2f(u) uf (u) f(u) (3.7) we can ewite Q () as Q () = Ñ 1( uf (u) f(u) ) uq 1 ().

14 Diffeentiating Q 1 () we obtain Q 1 () = Q θñ 1 u + 2 2u p u. (3.8) Suppose, fo contadiction, that Q has a nonnegative minimum value at 0 (0,R). Then thee exists a set I R having 0 as accumulation point and Q 1 () 0fo I and < 0, and Q 1 () 0fo I and > 0. It follows then that Q 1 ( 0) 0. On the othe hand, fom (3.8) and (3.5) we have Q 1 ( 0)<0asQ( 0 ) 0 and u ( 0 )<0 yielding a contadiction. Poof of Poposition 3.2. Fo a given p (1,p + ),ifp Ñ/(Ñ 2), then Lemma 3.3 implies Q() > 0fo0<<Ras needed. Assume Ñ/(Ñ 2) p<p +.IfQ() is zeo o negative somewhee in the inteval (0,R), then Lemmas 3.2 and 3.4 imply that Q must assume a negative minimum value in the inteval (0,R). Recall fom the existence esult of [10] that fo each p (1,p + ) thee exists a positive adial solution of (1.2). Moeove, in the poof of existence in [10], we may think of p as a paamete and use homotopy popeties of the degee to pove that given 1 <p 1 <p 2 <p +,thesetof solutions of (1.2) contains a connected subset having inside a solution fo p 1 and a solution fo p 2. Thus we can use a continuity agument to find a numbe p [Ñ/(Ñ 2), p) such that the coesponding Q function possesses a nonnegative minimum value in (0,R). This gives a contadiction to Lemma 3.4. Hence Q>0 fo all (0,R). We just ecall that Q>0if u() 1, so that the case R = is also well coveed. 4. Uniqueness The uniqueness in Theoem 1.1 follows if we can show that thee is at most one α>0 such that the solution of (2.3) satisfies, fo R<, and fo R =, u() > 0 fo (0,R) and u(r) = 0, (4.1) u() > 0 fo >0 and lim u() = 0. (4.2) In eithe case we have α>1 and u () < 0fo (0,R)as a consequence of Lemma 2.3, and in the fist case u (R) < 0 by the uniqueness theoem to the initial value poblem (2.3). Theefoe, wheneve the case (4.1) occus, the cossing numbe R is a C 1 function of α. Denote the solution u by u(, α) to emphasize its dependence on α. Diffeentiating the identity u(r(α), α) = 0 with espect to α, and denoting by the vaiation of u, we obtain v(,α) = u(,α), α R (α) = v(r,α)/u (R, α).

15 (Hee the pime in u is indeed the patial deivative u/.) Hence the sign of v(r,α) detemines the monotonicity of R. As in the Laplacian case, to complete the poof of ou uniqueness esult it is sufficient to pove the following lemma. Lemma 4.1. Thee is a τ (0,R)such that v>0 in (0,τ), v(τ)= 0 and v<0 in (τ, R). (4.3) Moeove, v(r) < 0 if R< and lim v() = if R =. In fact, if this lemma is established, then R (α) < 0 wheneve R(α) <. This implies that fo all α >α, R( α) is defined and R( α) < R(α), yielding the uniqueness in the finite ball immediately. That this lemma implies the uniqueness of gound states is not so obvious. We notice that fo all lagewehaveu < 0 and u > 0, so that thee is no moe change in θ and Θ. Thus we may apply the classical aguments used fo the Laplacian to conclude. See, fo example, the pape by Peletie and Sein [19] o the pape by Kwong [11]. Remak 4.1. By using the aguments in [11] one can pove that the unique solution u is nondegeneate in the sense that the lineaized equation θh + N 1 Θh + f (u)h = 0, h(r)= 0, h (0) = 0. (4.4) has only the tivial solution. The est of this section is theefoe devoted to the poof of Lemma 4.1. The basic stategy hee is to modify the appoaches ecently developed by Tang in [23] fo the Laplacian case, which lagely simplified the technicalities of pevious woks in the study of the uniqueness poblem fo the semi-linea elliptic equations involving the Laplace opeatos. We notice that the appoach hee is not simply a tivial genealization of the wok of [23] as new technical complexities aise due to the discontinuities of vaious functions. In the fist pat that follows, we shall genealize some key functions and identities fom [23], which will be used in the second pat to complete the poof of Lemma Seveal functionals We stat with a diffeentiation of (2.3) with espect to α to get θv + N 1 Θv + f (u)v = 0, v(0) = 1, v (0) = 0. (4.5) Simila to (3.3) we can wite (4.5) as Let θ ( Ñ 1 v ) = Ñ 1 f (u)v. (4.6) ξ() = θñ 1 (u v uv )

16 denote the Wonskian of u and v. Using (3.3) and (4.6) we obtain Then we intoduce the following function To find the deivative of δ(), we fist compute Using this and (3.3) we obtain and simplifying ξ () = Ñ 1[ uf (u) f(u) ] v. (4.7) δ() = Ñ [ θu v + f(u)v ] + (Ñ 2)θÑ 1 u v. (4.8) θ [ v + (Ñ 2)v ] = f (u)v. δ () = [ θñ 1 u ( v + (Ñ 2)v ) + Ñ f(u)v ] Finally, we intoduce the functions T and g = Ñ 1 f(u) ( v + (Ñ 2)v ) Ñ f (u)u v + ÑÑ 1 f(u)v+ Ñ f (u)u v + Ñ f(u)v, δ () = 2Ñ 1 f(u)v. (4.9) T()= g(u)ξ() δ(), g(u) = As in [23], (4.7) and (4.9) we obtain the useful identity 2f(u) uf (u) f(u). (4.10) T () = g (u)u ()ξ(), (4.11) which can be futhe simplified, using that f(u)= u + u p, to obtain 4.2. Poof of Lemma 4.1 T () = 2u p u ()ξ(). (4.12) We fist pove that v must vanish somewhee in (0,R). Suppose this is not tue, then v emains positive in (0,R). Thus ξ inceases ove all subintevals whee u 0 (notice that as u < 0in (0,R),wehaveΘ = λ). Denote by 1 the numbe such that u( 1 ) = 1. Then u > 0in( 1,R), in which ξ is C 1 and inceasing. It follows by the sign-etaining popety of ξ that ξ(r) > 0, o lim ξ() > 0 when R =. But the evaluation of ξ(r) by the definition of ξ gives ξ(r) 0. In case R = we have lim ξ() = 0 when R =, whee we use the exponential decay of u. In both cases we each a contadiction. Denote the fist zeo of v by τ. We next pove that v must stay negative in the emaining inteval (τ, R). We pove this again by contadiction. Suppose this is not tue then thee is a

17 numbe τ (τ, R) such that v( τ)= 0 and v<0in(τ, τ). Then, using the definition of ξ (4.7) we find that ξ(τ) < 0 and ξ( τ)<0. Then, thee must be a numbe t (τ, τ) such that ξ(t) = 0 and ξ() > 0 fo (0,t). (4.13) By (4.12) we see that T()is deceasing on the subintevals of (0,t) in which u 0. To continue, using (4.7), (4.8) and (4.10), we ewite T()as T()= θg(u)ñ 1 (u v uv ) Ñ [ θu v + f(u)v ] (Ñ 2)θÑ 1 u v whee we used the substitution = θg(u)ñ 1 (u v uv ) θñ u v + θñ u v + θñ 1 u v = θñ 1[ g(u)(u v uv ) u v + u v ] + θñ u v, f (u) = θ ( u + (Ñ 1)u ), which follows fom (2.3) and (3.2). It is theefoe clea that T also has the sign-etaining popety, since θu is continuous whee u vanishes. This, togethe with (4.10) and (4.13), allows us to conclude δ(t) = T(t)>0. A futhe calculation using (4.8) and (4.13) yields δ(t) = tñ [ θu v + f(u)v ] + (Ñ 2)θtÑ 1 u v = [ tñ ( θu v u/v + uf (u) ) + (Ñ 2)θtÑ 1 uu ] v/u = [ tñ ( θu 2 + uf (u) ) + (Ñ 2)θtÑ 1 uu ] v/u = Q(t)v(t)/u(t). Since v(t)/u(t) < 0, we obtain Q(t) < 0, which contadicts Poposition 3.2 poving (4.3). To each the emaining conclusion of Lemma 4.1, we fist establish lim δ() > 0. (4.14) R If τ> 1, then v<0 and f(u)<0in(τ, R), ecalling that u( 1 ) = 1. Hence u > 0 and δ() is C 1 and inceasing in (τ, R), yielding lim R δ() > δ(τ) = θτñu (τ)v (τ) > 0 as desied. If τ 1, then the same agument shows that δ() is C 1 and inceasing ove ( 1,R). Moeove, obseving that the agument in last paagaph shows that ξ() > 0 fo all (0,R), by (4.12) and the sign-etaining popety of T we conclude that T<0in(0,R), giving in paticula This veifies (4.14) in the second case. δ( 1 ) = T( 1 )>0.

18 Now, if R< then δ(r) > 0, which is incompatible with v(r) = 0, implying that v(r) < 0. Fo the case R =, we fist notice that as, eithe v() o v() 0; see McLeod [16, Lemma 2(b)], ecalling that the asymptotic behavio hee is the same as in the Laplacian case, since eventually u > 0. If v() occus, then lim δ() = 0, as follows fom (4.8) and (2.10), poviding an obvious contadiction to (4.14). Thus lim v() =, and the poof is completed. Remak 4.2. In the case of gounds states, ou theoem assues that fo evey 1 <p<p +, thee is a numbe α sepaating the ange of α in cossing and positive oscillating solutions fo (2.3). The main open question left in this pape is the analysis of the solutions beyond p +. We believe that the function Q defined hee contains cucial infomation about this question. On the othe hand, we ecall that thee is not fomula known fo p +. In the seach fo one we may also obtain valuable infomation about the question open hee. Acknowledgments The fist autho was patially suppoted by Fondecyt Gant # and FONDAP de Matemáticas Aplicadas. The second autho was patially suppoted by Fondecyt Gant # The thid autho was patially suppoted by Fondecyt Gant # fo Intenational Coopeation. Refeences [1] J. Busca, M.J. Esteban, A. Quaas, Nonlinea eigenvalues and bifucation poblems fo Pucci s opeato, Ann. Inst. H. Poincaé Anal. Non Linéaie 22 (2) (2005) [2] L.A. Caffaelli, X. Cabé, Fully Nonlinea Elliptic Equations, Ame. Math. Soc. Colloq. Publ., vol. 43, Ame. Math. Soc., Povidence, RI, [3] C.C. Chen, C.S. Lin, Uniqueness of the gound state solutions of u + f(u)= 0inR n, n 3, Comm. Patial Diffeential Equations 16 (1991) [4] C. Coffman, On the positive solutions of bounday poblems fo a class of nonlinea diffeential equations, J. Diffeential Equations 3 (1967) [5] C. Cotáza, M. Elgueta, P. Felme, Uniqueness of positive solutions of u+f(u)= 0inR N, N 3, Ach. Ration. Mech. Anal. 142 (1998) [6] A. Cuti, F. Leoni, On the Liouville popety fo fully nonlinea equations, Ann. Inst. H. Poincaé Anal. Non Linéaie 17, 2 (2000) [7] F. Da Lio, B. Siakov, Symmety esults fo viscosity solution of fully nonlinea elliptic equation, pepint. [8] P.L. Felme, A. Quaas, Citical exponents fo the Pucci s extemal opeatos, C. R. Math. Acad. Sci. Pais 335 (2002) [9] P.L. Felme, A. Quaas, On citical exponents fo the Pucci s extemal opeatos, Ann. Inst. H. Poincaé Anal. Non Linéaie 20 (2003) [10] P.L. Felme, A. Quaas, Positive adial solutions to a semilinea equation involving the Pucci s opeato, J. Diffeential Equations 199 (2004) [11] M.K. Kwong, Uniqueness of positive solutions of u u + u p = 0inR n, Ach. Ration. Mech. Anal. 105 (1989) [12] M.K. Kwong, Y. Li, Uniqueness of adial solutions of semilinea elliptic equations, Tans. Ame. Math. Soc. 333 (1992) [13] M.K. Kwong, K. McLeod, L.A. Peletie, W. Toy, On gound state solutions of = u p u q, J. Diffeential Equations 95 (1992) [14] M.K. Kwong, L. Zhang, Uniqueness of the positive solution of u + f(u)= 0 in an annulus, Diffeential Integal Equations 4 (1991) [15] K. McLeod, J. Sein, Uniqueness of positive adial solutions of u+f(u)= 0inR n, Ach. Ration. Mech. Anal. 99 (1987)

19 [16] K. McLeod, Uniqueness of positive adial solutions of u + f(u)= 0inR n II, Tans. Ame. Math. Soc. 339 (1993) [17] W.M. Ni, Uniqueness of solutions of nonlinea Diichlet poblems, J. Diffeential Equations 50 (1983) [18] W.M. Ni, R. Nussbaum, Uniqueness and nonuniqueness fo positive adial solutions of u + f(u,)= 0, Comm. Pue Appl. Math. 38 (1985) [19] L.A. Peletie, J. Sein, Uniqueness of positive solutions of semilinea equations in R N, Ach. Ration. Mech. Anal. 81 (1983) [20] L.A. Peletie, J. Sein, Uniqueness of nonnegative of semilinea equations in R N, solutions of semilinea equations in R N, J. Diffeential Equations 61 (1986) [21] J. Sein, M. Tang, Uniqueness of gound states fo quasilinea elliptic equations, Indiana Univ. Math. J. 49 (2000) [22] M. Tang, Uniqueness and global stuctue of positive adial solutions fo quasilinea elliptic equations, Comm. Patial Diffeential Equations 26 (2001) [23] M. Tang, Uniqueness of positive adial solutions fo u u + u p = 0 on an annulus, J. Diffeential Equations 189 (2003)