Direction of bifurcation for some non-autonomous problems

Transcription

1 Direction of bifrction for some non-tonomos problems Philip Kormn Deprtment of Mthemticl Sciences University of Cincinnti Cincinnti Ohio Abstrct We stdy the exct mltiplicity of positive soltions, nd the globl soltion strctre for severl clsses of non-tonomos two-point problems. We present two sittions where the direction of trn cn be compted rther directly. As n ppliction, we consider problem from combstion theory with sign-chnging potentil. We illstrte or reslts by nmericl compttions, sing novel method. Key words: Globl soltion crves, direction of bifrction, contintion in globl prmeter. AMS sbject clssifiction: 34B15, 65L1, 8A25. 1 Introdction In recent yers bifrction theory methods were pplied to stdy the exct mltiplicity of positive soltions, nd the globl soltion strctre of nontonomos two-point problems (1.1) + f(x, ) =, < x < b, () = (b) =, depending on positive prmeter. Let s briefly review the bifrction theory pproch, nd more detils cn be fond in the thor s book [6]. If t some soltion (, ) the corresponding linerized problem (1.2) w + f (x, )w =, < x < b, w() = w(b) = dmits only the trivil soltion, then we cn contine the soltions of (1.1) in, by sing the Implicit Fnction Theorem, see e.g., L. Nirenberg [12]. 1

2 If, on the other hnd, the problem (1.2) hs non-trivil soltions then the Implicit Fnction Theorem cnnot be sed, insted one tries to show tht the Crndll-Rbinowitz [3] bifrction theorem pplies. The crcil condition one needs to verify is (1.3) f(x, )w dx. The Crndll-Rbinowitz theorem grntees existence of soltion crve throgh the criticl point (, ), nd if trn occrs t (, ), its direction is governed by (see e.g., the exposition in P. Kormn [6]) (1.4) I = f (x, )w 3 dx f(x, )w dx. If I > (I < ) the direction of the trn is to the left (right) in the (, ) plne. If one cn show tht trn to the left occrs t ny criticl point, then there is t most one criticl point. Uslly, there is exctly one criticl point, which provides s with the exct shpe of soltion crve, nd the exct mltiplicity cont for soltions. In the present pper we present two sittions in which the sign of I cn be compted in rther direct wy, differently from the previos works. As n ppliction, we obtin three new exct mltiplicity reslts for nontonomos eqtions, inclding one for sign-chnging eqtions of combstion theory. Sign-chnging eqtions present severl new chllenges, which we overcome in cse of symmetric potentils. We lso present some improvements of erlier reslts. In the lst section we develop n lgorithm for the nmericl compttion of globl soltion crves for non-tonomos eqtions. This is ccomplished by contintion in globl prmeter. 2 The direction of bifrction We consider positive soltions of two point non-tonomos bondry vle problem (2.1) + f(x, ) =, < x < b, () = (b) =, depending on positive prmeter. Here f(x, ) C 2 ([, b] R + ), nd we ssme tht f(, ) nd f(b, ) (which implies tht Hopf s bondry lemm holds). The linerized problem corresponding to (2.1) is (2.2) w + f (x, )w =, < x < b, w() = w(b) =. 2

3 When one stdies how soltions of (2.1) chnge in, i.e., the soltion crves, the direction of the trn (or bifrction) depends on the sign of the integrl f (x, )w 3 dx, which is prt of (1.4). We hve the following crcil lemm. Lemm 2.1 Let (x) be positive soltion of (2.1), nd ssme tht the linerized problem (2.2) hs non-trivil soltion w(x), nd moreover w(x) > on (, b). If we hve, for some c >, (2.3) 2 f (x, ) c (f (x, ) f(x, )), for ll >, nd x (, b) then (2.4) If, on the other hnd, for some c >, f (x, )w 3 dx >. (2.5) then (2.6) 2 f (x, ) c (f (x, ) f(x, )), for ll >, nd x (, b) f (x, )w 3 dx <. Proof: We mltiply the eqtion (2.2) by w2, nd sbtrct from tht the eqtion (2.1) mltiplied by w3, then integrte 2 [ f (x, ) ] f(x, ) 2 w 3 dx = ( b w 3 2 w2 w In the lst integrl we integrte by prts. The bondry terms vnish, since () nd (b) re not zero by Hopf s bondry lemm, nd hence (x) is symptoticlly liner ner the end points. We hve ( w 3 w2 2 w ) dx = 2w3 2 4w 2 w 2 +2ww 2 2 dx 4 = If the condition (2.3) holds, then f (x, )w 3 dx c 2w(w w ) 2 3 dx >. [ f (x, ) ) dx. ] f(x, ) 2 w 3 dx >. 3

4 Similrly, the condition (2.5) implies f (x, )w 3 dx c [ f (x, ) ] f(x, ) 2 w 3 dx >. Remrks 1. After this pper ws written, we becme wre tht similr reslt ws proved in J. Shi [14]. 2. The condition (2.3), when c = 1, is eqivlent to [ ( ) f() ] ( = f () f() ) >. In T. Oyng nd J. Shi [13] it hs been pointed ot tht the trning direction is sometimes relted to monotonicity of the fnction f()/. This form of (2.3) gin shows connection to the fnction f()/. Exmple f() = + p + q, with constnt. One comptes, with c = 1, 2 f () f () + f() = (p 1) 2 p + (q 1) 2 q + > for ll >. The cse when < p < 1 < q is of prticlr interest. Then f() is concveconvex, i.e., concve on (, ) nd convex on (, ), for some >. Similrly, the condition (2.3) holds for f(x, ) = Σ m i=1 i(x) p i, with i (x) > for ll x, nd ny positive p i. While in the cse of constnt i (x), the direction of bifrction ws known before (see the Theorem 6.1 below), the non-tonomos cse is new. 3 A clss of symmetric nonlinerities We stdy positive soltions of clss of symmetric problems of the type (3.1) + f(x, ) = for 1 < x < 1, ( 1) = (1) =. In severl ppers of P. Kormn nd T. Oyng, clss of symmetric f(x, ) hs been identified, for which the theory of positive soltions is very similr to tht for the tonomos cse, see e.g., [8] nd [9]. Frther reslts in this direction hve been given in P. Kormn, Y. Li nd T. Oyng [7], nd P. 4

5 Kormn nd J. Shi [11]. Nmely, we ssme tht f(x, ) C 1 ( ) [ 1, 1] R + stisfies (3.2) f( x, ) = f(x, ) for ll 1 < x < 1, nd >, (3.3) f x (x, ) for ll < x < 1, nd >. Under the bove conditions the following fcts, similr to those for tonomos problems, hve been estblished. 1. Any positive soltion of (3.1) is n even fnction, with (x) < for ll x (, 1], so tht x = is point of globl mximm. This follows from B. Gids, W.-M. Ni nd L. Nirenberg [4]. 2. Assme, dditionlly, tht f(x, ) >. Then the mximm vle of soltion, α = (), niqely identifies the soltion pir (, (x)), s proved in P. Kormn nd J. Shi [11]. i.e., α = () gives globl prmeter on ny soltion crve. We shll generlize this reslt below, dropping the condition tht f(x, ) >. 3. Any non-trivil soltion of the corresponding linerized problem (3.4) w + f (x, )w = for 1 < x < 1, w( 1) = w(1) = is of one sign on ( 1, 1). We hve the following exct mltiplicity reslt. Theorem 3.1 Consider the problem (3.5) + ( 1 (x) p + 2 (x) q ) = for 1 < x < 1, ( 1) = (1) =. Assme tht < p < 1 < q, while the given fnctions 1 (x) nd 2 (x) stisfy (3.6) i (x) >, i ( x) = i (x) for x ( 1, 1), i = 1, 2, (3.7) i(x) < for x (, 1), i = 1, 2. Then there is criticl >, sch tht for > the problem (3.5) hs no positive soltions, it hs exctly one positive soltion for =, nd exctly two positive soltions for < <. Moreover, ll positive soltions lie on single smooth soltion crve (x, ), which for < < hs two brnches denoted by < (x, ) < + (x, ), with (x, ) =, (x, ) strictly monotone incresing in for ll x ( 1, 1), nd lim + (, ) =. The mximl vle of soltion, (, ), serves s globl prmeter on this soltion crve. 5

6 Proof: Conditions (3.6) nd (3.7) imply tht the bove mentioned reslts on symmetric problems pply, nd in prticlr ny non-trivil soltion of the linerized problem, corresponding to (3.5) is positive on ( 1, 1). Then by Lemm 2.1 only trns to the left re possible on the soltion crve. The rest of the proof is similr to tht for similr reslts in P. Kormn nd T. Oyng [8], or P. Kormn nd J. Shi [11], so we jst sketch it. By the Implicit Fnction Theorem, there is crve of positive soltions of (3.5) strting t ( =, = ). By Strm s comprison theorem, this crve cnnot be contined indefinitely in, so tht it will hve to rech criticl point (, ) t which the Crndll- Rbinowitz Theorem [3] pplies. By Lemm 2.1, trn to the left occrs t (, ), nd t ny other criticl point. Hence, there re no other trning points, nd the soltion crve contines for ll decresing >, tending to infinity s +. Remrk This theorem lso holds for more generl f() = Σ m i=1 i(x) p i, with i (x) stisfying (3.6) nd (3.7), nd p i, with t lest one of p i less thn one, nd t lest one of p i greter thn one. 4 Non-symmetric nonlinerities Withot the symmetry ssmptions on f(x, ), the problem is mch hrder. We restrict to sbclss of sch problems, i.e., we now consider positive soltions of the bondry vle problem (4.1) + α(x)f() = for < x < b, () = (b) =, on n rbitrry intervl (, b). We ssme tht f() nd α(x) re positive fnctions of clss C 2, i.e., (4.2) f() > for >, α(x) > for x [, b]. As before, it is crcil for bifrction nlysis to prove positivity for the corresponding linerized problem (4.3) w + α(x)f ()w = for < x < b, w() = w(b) =. The following lemm ws proved in P. Kormn nd T. Oyng [1]. Lemm 4.1 In ddition to the conditions (4.2), ssme tht (4.4) 3 α 2 2 α α for ll x (, b). If the linerized problem (4.3) dmits non-trivil soltion, then we my ssme tht w(x) > on (, b). 6

7 Using Lemm 2.1, we hve the following exct mltiplicity reslt, whose proof is similr to tht of Theorem 3.1. Theorem 4.1 Consider the problem (4.5) + α(x)σ m i=1 i p i = for < x < b, () = (b) =, where α(x) stisfies the conditions (4.2) nd (4.4), i re positive constnts, the constnts p i, with t lest one of p i less thn one, nd t lest one of p i greter thn one. Then there is criticl >, sch tht for > the problem (4.5) hs no positive soltions, it hs exctly one positive soltion for =, nd exctly two positive soltions for < <. Moreover, ll positive soltions lie on single smooth soltion crve (x, ), which for < < hs two brnches denoted by < (x, ) < + (x, ), with (x, ) =, (x, ) strictly monotone incresing in for ll x (, b), nd lim mx x (,b) + (x, ) =. 5 Sign-chnging eqtion of combstion theory We begin with the following generliztion of the corresponding reslt in P. Kormn nd J. Shi [11], which we shll se for n eqtion in combstion theory. Recll tht positive soltions of (3.1) re even fnctions, with (x) < for x >, i.e., () is the globl mximm of soltion (x). Theorem 5.1 Consider the problem (3.1), with f(x, ) stisfying (3.2) nd (3.3), with the ineqlity (3.3) being strict for lmost ll x ( 1, 1) nd >. Then the set of positive soltions of (3.1) cn be globlly prmeterized by the mximm vles (). (I.e., the vle of () niqely determines the soltion pir (, (x)).) Proof: Assme, on the contrry, tht v(x) is nother soltion of (3.1), with v() = (), nd v () = () =. We my ssme tht µ >. Setting x = 1 t, we see tht = (t) stisfies (5.1) + f( 1 t, ) =, () = ( ) =. Similrly, letting x = 1 µ z, nd then renming z by t, we see tht v = v(t) stisfies v + f( 1 µ t, v) =, v () = v( µ) =, 7

8 nd in view of (3.3) (5.2) v + f( 1 t, v) <, i.e., v(t) is spersoltion of (5.1). We my ssme tht (5.3) v(t) < (t) for t > nd smll. Indeed, the opposite ineqlity is impossible by the strong mximm principle ( spersoltion cnnot toch soltion from bove, nd the possibility of infinitely mny points of intersection of (t) nd v(t) ner t = is rled ot by the Strm comprison theorem, pplied to w = v). Since v(t) is positive on (, ), we cn find point ξ (, ) so tht (ξ) = v(ξ), (ξ) v (ξ), nd (5.3) holding on (, ξ). We now mltiply the eqtion (5.1) by (t), nd integrte over (, ξ). Since the fnction (t) is decresing, its inverse fnction exists. Denoting by t = t 2 (), the inverse fnction of (t) on (, ξ), we hve (5.4) 1 (ξ) 2 2 (ξ) + () f( 1 t 2 (), ) d =. Similrly denoting by t = t 1 () the inverse fnction of v(t) on (, ξ), we hve mltiplying (5.2) by v (t) <, nd integrting (5.5) (ξ) 2 v 2 (ξ) + () Sbtrcting (5.5) from (5.4), we hve ] () [ 2 (ξ) v 2 (ξ) + (ξ) f( 1 t 1 (), ) d >. [ f( 1 t 1 (), ) f( 1 t 2 (), ) ] d <. Notice tht t 2 () > t 1 () for ll ((ξ), ()). Using the condition (3.3), both terms on the left re non-negtive nd the integrl is positive, nd we obtin contrdiction. We now consider bondry vle problem rising in combstion theory, see e.g., J. Bebernes nd D. Eberly [2], S.-H. Wng [15], nd S.-H. Wng nd F.P. Lee [16] (5.6) + α(x)e =, 1 < x < 1, ( 1) = (1) =, depending on positive prmeter. The given fnction α(x) is ssmed to be sign chnging, with α() > (the reslt of this section is known if α(x) is positive on ( 1, 1)). The linerized problem corresponding to (5.6) is (5.7) w + α(x)e w =, 1 < x < 1, w( 1) = w(1) =. 8

9 Lemm 5.1 Assme tht ny non-trivil soltion of (5.7) stisfies w(x) > on ( 1, 1), nd α(x) C[ 1, 1]. Then (5.8) 1 α(x)e w 3 dx >, nd 1 α(x)e w dx >. Proof: Mltiplying the eqtion (5.7) by w 2, nd integrting 1 α(x)e w 3 dx = 2 ww 2 dx >. 1 Integrting (5.7) 1 α(x)e w dx = w ( 1) w (1) >, s climed. We need the following simple lemm. Lemm 5.2 Let (x) be soltion of the problem (5.9) = α(x), < x < 1, () = (1) =. Assme tht α(x) C[, 1] is sign-chnging, nd it stisfies (5.1) A(x) (5.11) x (x ξ)α(ξ) dξ + (1 ξ)α(ξ) dξ > for x [, 1), α(ξ) dξ >. Then (x) > on [, 1), nd (1) <. Proof: The formls (5.1) nd (5.11) give (x) nd (1) respectively. We shll ssme tht α(x) C 1 [ 1, 1] stisfies (5.12) α( x) = α(x) for ll x [, 1], (5.13) α (x) < for lmost ll x [, 1]. As before, we know tht nder these conditions ny non-trivil soltion of (5.7) is positive on ( 1, 1). Observe lso tht nder these conditions signchnging α(x) chnges sign exctly once on (, 1) (nd on ( 1, )). 9

10 Theorem 5.2 Consider the problem (5.6), nd ssme tht the fnction α(x) stisfies (5.12) nd (5.13). We lso ssme tht α(x) is sign-chnging, nd it stisfies (5.1) nd (5.11). Then there is criticl >, sch tht for > the problem (5.6) hs no positive soltions, it hs exctly one positive soltion for =, nd exctly two positive soltions for < <. Moreover, ll positive soltions lie on single smooth soltion crve (x, ), which for < < hs two brnches denoted by < (x, ) < + (x, ), with (x, ) =, nd lim + (, ) =. The mximl vle of soltion, (, ), serves s globl prmeter on this soltion crve. Proof: We begin with the soltion ( =, = ). This soltion is non-singlr (the corresponding linerized problem (5.7) hs only the trivil soltion), so tht by the Implicit Fnction Theorem we hve crve of soltions (x, ) pssing throgh the point ( =, = ). We clim tht these soltions re positive for smll. Indeed, (x, ) stisfies = α(x), 1 < x < 1, ( 1) = (1) =. Since α(x) is even, so is (x), nd hence (x) stisfies (5.9), nd so the Lemm 5.2 pplies. Hence, for smll, soltion (x) of (5.6) is positive, nd hence it is n even fnction, with (x) < for x (, 1). We show next tht soltions remin positive throghot the soltion crve. Since for positive soltions (x) < for x (, 1), there is only one mechnism by which soltions my stop being positive: (1) = t some = 1 (nd then soltions becoming negtive ner x = 1). We clim tht for ny positive soltion (x) of (5.6) (5.14) (1) <, which will rle ot sch possibility. (Hopf s bondry lemm does not pply here.) Let ξ be the point where α(x) chnges sign, i.e., α(x) > on [, ξ) nd α(x) < on (ξ, 1). Denote (ξ) =. Then from the eqtion (5.6) > e α(x), < x < 1, () = (1) =. Compring this to (5.9), we conclde tht (x) > e A(x) for ll x (, 1), nd then (1) e A (1) <. We clim next tht the crve of positive soltions cnnot be contined in beyond certin point. With ξ denoting the root of α(x), s bove, fix 1

11 ny η (, ξ), nd denote = α(η) >. If we denote ϕ(x) = cos π 2η x nd 1 = π2 4η 2, then We hve η ϕ dx = ϕ + 1 ϕ =, on (, η), ϕ () = ϕ(η) =. η ϕ dx (η)ϕ (η) > η η ϕ dx = 1 ϕ dx. Then mltiplying the eqtion (5.6) by ϕ nd integrting, we hve Also We conclde tht η η η α(x)e ϕ dx < 1 ϕ dx. η α(x)e ϕ dx > ϕ dx. < 1. The next step is to show tht soltions of (5.6) remin bonded, if is bonded wy from zero. Indeed, for lrge, α(x)e > M, with rbitrrily lrge constnt M, when x belongs to ny sb-intervl of ( ξ, ξ). By Strm s comprison theorem, the length of the intervl on which (x) becomes lrge, mst tend to zero. Bt tht is impossible, becse (x) is concve on ( ξ, ξ). We now retrn to the crve of positive soltions, emnting from ( =, = ). Soltions on this crve re bonded, while the crve cnnot be contined indefinitely in. Hence, criticl point (, ) mst be reched on this crve, i.e., t (, ) the corresponding linerized problem (5.7) hs non-trivil soltion. By the reslts on symmetric problems, reviewed bove, ny non-trivil soltion of the corresponding linerized problem (5.7) is of one sign, i.e., we my ssme tht w(x) > on ( 1, 1). By Lemm 5.1, the Crndll-Rbinowitz theorem pplies t ny criticl point, nd trn to the left occrs. Hence, fter the trn t (, ), the crve contines for decresing, withot ny more trns. By the Theorem 5.1, (, ) is globl prmeter on the soltion crve. Along the soltion crve, the globl prmeter (, ) is incresing nd tending to infinity. By bove, this my hppen only s. 11

12 Exmple The fnction α(x) = 1 cx 2 for 1 < c < 3 is sign-chnging on ( 1, 1), nd it stisfies ll of the bove conditions. Indeed α(t) dt = 1 c/3 >, nd A(x) = cx4 12 x2 2 c >, for < x < 1, 2 becse A(1) =, nd A (x) = cx3 3 x < for < x < 1. 6 Atonomos problems In cse the nonlinerity does not depend explicitly on x, i.e., f = f(), the min reslt on the direction of trn is the following theorem from P. Kormn, Y. Li nd T. Oyng [7] nd T. Oyng nd J. Shi [13]. We present little simpler proof of this key reslt. Atonomos problems cn be posed on ny intervl. We se the intervl ( 1, 1) for convenience, i.e., we consider positive soltions of (6.1) + f() = for 1 < x < 1, ( 1) = (1) =. Corresponding linerized problem is (6.2) w + f ()w = for 1 < x < 1, w( 1) = w(1) =. Both (x) nd w(x) re even fnctions (see e.g., [6]), nd so the direction of bifrction t criticl point (, ) is governed by I = f ( )w 3 dx f( )w dx. Recll tht f() C 2 ( R + ) is clled convex-concve if f () > on (, ), nd f () < on (, ) for some >, nd the definition of concveconvex fnctions is similr. Theorem 6.1 ([7], [13]) (i) Assme f(), nd f() is convex-concve. Then t ny criticl point, with () > nd (1) <, we hve I <, nd hence trn to the right occrs. (ii) Assme f(), nd f() is concve-convex. Then t ny criticl point, with () >, we hve I >, nd hence trn to the left occrs. 12

13 Proof: In cse f(), we hve (1) < by Hopf s bondry lemm. We shll write (, ) insted of (, ). It is known tht ny non-trivil soltion of the linerized problem (6.2) is of one sign, see e.g., [7] or [6], nd so we my ssme tht w(x) > on ( 1, 1), which implies tht (6.3) w (1) <. From the eqtions (6.1) nd (6.2) it is strightforwrd to verify the following identities (6.4) (x)w (x) (x)w(x) = constnt = (1)w (1) ; (6.5) ( w w ) = f () 2 w. Assme tht the first set of conditions hold. Integrting (6.5), (6.6) f () 2 w dx = (1)w (1) = f()w (1). Consider the fnction p(x) w(x). Since p(1) =, nd by (6.4) (x) p (x) = (1)w (1) (x) 2 <, the fnction p(x) is positive nd decresing on (, 1). The sme is tre for p 2 (x) = w2 (x) 2 (x). Let x be the point where f ((x)) chnges sign (i.e., f ((x)) < on (, x ), nd f ((x)) > on (x, 1)). By scling w(x), we my chieve tht w 2 (x ) = 2 (x ). Then w 2 (x) > 2 (x) on (, x ), nd the ineqlity is reversed on (x, 1). Using (6.6), we hve (6.7) Integrting (6.4), we hve (6.8) f ((x))w 3 dx < f ((x)) 2 w dx. f()wdx = 1 2 (1)w (1) >. The formls (6.7) nd (6.8) imply tht I <. The second prt of the theorem is proved similrly. Observe tht, in cse f = f(), this theorem nd the Lemm 2.1 hve intersecting domins of pplicbility, bt neither one is more generl thn the other. 13

14 7 Nmericl compttion of the soltion crves In this section we present compttions of the globl crves of positive soltions for the problem (7.1) + f(x, ) = for 1 < x < 1, ( 1) = (1) =. We ssme tht the fnction f(x, ) stisfies the conditions (3.2) nd (3.3), so tht the Theorem 5.1 pplies, which tells s tht α () is globl prmeter. Since ny positive soltion (x) is n even fnction, we shll compte it on the hlf-intervl (, 1), by solving (7.2) + f(x, ) = for < x < 1, () = (1) =. The stndrd pproch to nmericl compttion involves crve following, i.e., contintion in by sing the predictor-corrector methods, see e.g., E.L. Allgower nd K. Georg [1]. These methods re well developed, bt not esy to implement, s the soltion crve = (x, ) my consist of severl prts, ech hving mltiple trns. Here is locl prmeter, bt not globl one, becse of the trning points. Since α = () is globl prmeter, we shll compte the soltion crve of (7.2) in the form = (α). If we solve the initil vle problem (7.3) + f(x, ) =, () = α, () =, then we need to find, so tht (1) =, in order to obtin the soltion of (7.2). Rewrite the eqtion (7.2) in the integrl form x (x) = α (x t)f(t, (t)) dt, nd then the eqtion for is (7.4) F() (1) = α We solve this eqtion by sing Newton s method n+1 = n F( n) F ( n ). (1 t)f(t, (t)) dt =. We hve F( n ) = α n (1 t)f(t, (t, n )) dt, 14

15 Alph Lmbd Figre 1: The soltion crve for problem from combstion theory F ( n ) = (1 t)f(t, (t, n )) dt n (1 t)f (t, (t, n )) dt, where (x, n ) nd re respectively the soltions of (7.5) + n f(x, ) =, () = α, () = ; (7.6) + nf (x, (x, n )) + f(x, (x, n )) =, () =, () =. (As we vry, we keep () = α fixed, tht is why () =.) This method is very esy to implement. It reqires repeted soltions of the initil vle problems (7.5) nd (7.6) (sing the NDSolve commnd in Mthemtic). Exmple Consider problem from combstion theory with sign-chnging potentil + (1 2x 2 )e = for < x < 1, () = (1) =. The globl soltion crve is presented in Figre 1. For ny point (, α) on this crve, the ctl soltion (x) is esily compted by shooting (or NDSolve commnd), see (7.3). In Figre 2 we present the soltion (x) for , when () = (This soltion lies on the pper brnch, shortly fter the trn.) References [1] E.L. Allgower nd K. Georg, Nmericl Contintion Methods. An Introdction. Springer Series in Compttionl Mthemtics, 13. Springer- Verlg, Berlin, (199). [2] J. Bebernes nd D. Eberly, Mthemticl Problems from Combstion Theory, Springer-Verlg, New York (1989). 15

16 x Figre 2: The soltion (x), corresponding to α = () = 1.25 [3] M.G. Crndll nd P.H. Rbinowitz, Bifrction, pertrbtion of simple eigenvles nd linerized stbility, Arch. Rtionl Mech. Anl., 52, (1973). [4] B. Gids, W.-M. Ni nd L. Nirenberg, Symmetry nd relted properties vi the mximm principle, Commn. Mth. Phys. 68, (1979). [5] S.P. Hstings nd J.B. McLeod, Clssicl Methods in Ordinry Differentil Eqtions. With pplictions to bondry vle problems. Grdte Stdies in Mthemtics, 129. Americn Mthemticl Society, Providence, RI (212). [6] P. Kormn, Globl Soltion Crves for Semiliner Elliptic Eqtions, World Scientific, Hckensck, NJ (212). [7] P. Kormn, Y. Li nd T. Oyng, Exct mltiplicity reslts for bondry-vle problems with nonlinerities generlising cbic, Proc. Royl Soc. Edinbrgh, Ser. A 126A, (1996). [8] P. Kormn nd T. Oyng, Exct mltiplicity reslts for two clsses of bondry vle problems, Differentil Integrl Eqtions 6, no. 6, (1993). [9] P. Kormn nd T. Oyng, Mltiplicity reslts for two clsses of bondry-vle problems, SIAM J. Mth. Anl. 26, no. 1, (1995). [1] P. Kormn nd T. Oyng, Soltion crves for two clsses of bondryvle problems, Nonliner Anl. 27, no. 9, (1996). [11] P. Kormn nd J. Shi, Instbility nd exct mltiplicity of soltions of semiliner eqtions. Proceedings of the Conference on Nonliner Differentil Eqtions (Corl Gbles, FL, 1999), (electronic), Elec- 16

17 tron. J. Differ. Eq. Conf., 5, Sothwest Texs Stte Univ., Sn Mrcos, TX (2). [12] L. Nirenberg, Topics in Nonliner Fnctionl Anlysis, Cornt Institte Lectre Notes, Amer. Mth. Soc. (1974). [13] T. Oyng nd J. Shi, Exct mltiplicity of positive soltions for clss of semiliner problems, II, J. Differentil Eqtions 158, no. 1, (1999). [14] J. Shi, Topics in Nonliner Elliptic Eqtions, Ph.D. Disserttion, Brighm Yong University (1998). [15] S.-H. Wng, Rigoros nlysis nd estimtes of S-shped bifrction crves in combstion problem with generl Arrhenis rection-rte lws, R. Soc. Lond. Proc. Ser. A Mth. Phys. Eng. Sci. 454, no. 1972, (1998). [16] S.-H. Wng nd F.P. Lee, Bifrction of n eqtion from ctlysis theory, Nonliner Anl. 23, no. 9, (1994). 17