Houston Journl of Mthemtics c 999 University of Houston Volume 5, No. 4, 999 ON THE STRUCTURE OF SOLUTIONS OF CLSS OF BOUNDRY VLUE PROBLEMS XIYU LIU, BOQING YN Communicted by Him Brezis bstrct. Behviour of continu of the solution set of both opertor equtions nd clss of boundry vlue problems re obtined, which prtilly nswers n open problem of mbrosetti [].. Introduction In recent pper [],. mbrosetti, H. Brezis nd C. Cermi studied the combined effects of concve nd convex nonlinerities to elliptic boundry vlue problems of the following type u = λu q + u p, x Ω u >, x Ω (.) u =, x Ω with < q < < p. They proved the existence of two positive solutions to (.) for λ smll by upper nd lower solutions nd vritionl techniques when p is subcriticl. In tht pper, they lso indicted severl interesting open problems. See M [] for exmple. One of those is wht the structure of the solutions is in the one-dimensionl cse. The purpose of the present pper is to study this problem. We will give different pproch nd generl setting of the problem. The min feture is the presence of nonlinerity hving subliner nd superliner behvior. By pplying topologicl methods on cones we will show the existence of brnch C of solutions bifurcting from (, ) tht touches bck {} (P \{}). 99 Mthemtics Subject Clssifiction. 35J65, 34B5, 47H5. Key words nd phrses. continu, boundry vlue problems, cones. This work is supported in prt by NSF(Youth) of Shndong Province nd NNSF of Chin. 757
758 XIYU LIU ND BOQING YN s pplictions, we will discuss in detil clss of boundry vlue problems of ordinry differentil equtions. Some further structure theorems re obtined, nd prtil nswer is given to the question rised in [].. Structure of Solutions of Opertor Equtions This section is devoted to the bstrct setting of the problem. We will discuss the behviour of continu of solutions of equtions with prmeter when both superliner nd subliner effects re present. The min results re Theorem..3. Let E be Bnch spce with cone P, nd I : R + P P be completely continuous opertor, where R + = [, ). Let = {(λ, x) R + P : x = I(λ, x)}. Then clerly is closed nd loclly compct. Write B r = {x P : x < r} for r >. First we list the following conditions for this section. (H) (H) (H3) (H4) x I(,x) x <. I(λ,x) x,λ λ x is rbitrry. (H5) x >, for ny λ >. I(λ,x) x >, uniformly for λ R +. λ + I(λ, x) =, uniformly for x P, ε x ε x,λ + I(λ,x) x >. Lemm.. Suppose tht (H) is stisfied. point of I(, x). Moreover, i(i(, ),, P ) =. where ε (, ) Then x = is the isolted fixed Proof. By condition (H) there exists δ > such tht I(, x) λ x for x < δ, where λ <. Hence I(, ) = nd i(i(, ),, P ) = by [3]. Lemm.. Suppose tht (H) is stisfied. Then for ny λ, λ >, there exists τ > such tht ([λ, λ ] B τ ) [λ, λ ] {}. Proof. Suppose tht there exists sequence λ n [λ, λ ], x n, x n such tht (λ n, x n ). ssume without loss of generlity tht λ n λ [λ, λ ]. Then x n = I(λ n, x n ) in contrdiction with condition (H). Now suppose (H) is stisfied. Then (, ). Recll tht continuum is mximl connected set. Let C be the continuum of contining (, ). Clerly C is closed.
STRUCTURE OF SOLUTIONS OF BOUNDRY VLUE PROBLEMS 759 Lemm.3. Suppose tht (H)(H) re stisfied. Let C + = C \ ((, ) {}). Then C + is connected nd closed. Proof. Let (λ n, x n ) C +, (λ n, x n ) (λ, x). Then (λ, x) C. If λ =, then (λ, x) C +. If λ >, then by Lemm. we know x. Hence (λ, x) C + nd C + is closed. Next if there exist closed nonempty sets S, T such tht C + = S T. Let (, ) S. Then C = ([S ([, ) {})] C + ) T. Clerly T ([, ) {}) =, nd [S ([, ) {})] C + is closed, which implies tht C is not connected. Now we re in position to give the structure of. Theorem.. Suppose tht (H)(H) re stisfied. Then the continuum C of contining (, ) hs the following properties. (i) C contins connected closed subset C + [(, ) (P \{})] ({} P ). (ii) λ = is the bifurction point of I if I(λ, ). (iii) There exists λ > such tht [{λ} (P \{})] C + for λ (, λ ). Proof. Let C + be s in Lemm.3. Then C + is closed nd connected by Lemm.3. Thus the projection of C + onto R + is n intervl, nd we need only to show tht there exists λ > such tht [{λ} (P \{})] C. In fct, if [{λ} (P \{})] C = for ny λ >, then C ((, ) {}) ({} P ). Tke λ > nd let Z = [, λ ] P. Then Z is closed nd convex. By Lemm.,. nd condition (H) there exists τ > such tht [{λ } B τ ] (λ, ), [{} B τ ] (, ), nd I(λ, x) > x for x B τ. Write Q = [, λ ] B τ. Then Q = [, λ ] ( B τ P ) in Z. Let X = Q, then X is compct metric spce. Denote S = C Q, S = Q. Thus S, S re compct disjoint subsets of X, nd no subcontinuum of X cn both meet S nd S. By Lemm. of [4] there exist compct disjoint subsets K, K of X such tht X = K K, S K, S K. Thus K Q =, nd we cn choose n open set U of Q with K U, U K =, U K =, hence U =. By the generl homotopy invrince of fixed point index (see mnn [5]) we hve i(i(λ, ), U(λ), P ) = µ = const, λ [, λ ] where U(λ) = {x : (λ, x) U}. By Lemm. µ = when λ =. Since I(λ, x) > x for x B τ, then by Lemm.3.3 of [3] (pge 9) we hve i(i(λ, ), U(λ ), P ) = i(i(λ, ),, P ) =.
76 XIYU LIU ND BOQING YN Theorem.. Suppose tht (H)(H) re stisfied. Let C, C + be s in Theorem.. Then either (i) C + is unbounded, or (ii) C meets {} (P \{}). Proof. Suppose C + is bounded nd C [{} (P \{})] =. Tke R > such tht C + [, R) B R. Write Q R = [, R] B R, Z = [, R] P, X = ( Q R ) (R, ). Then X is compct in Z, nd Q R = [, R] B R in Z. Let S = (C Q R ) (R, ), S = ( [ Q R ({, R} BR )]) \{(R, ), (, )} which re compct disjoint subsets of X by Lemm.,.. By Lemm. of Rbinowitz [4] we get compct disjoint subsets K, K of X such tht X = K K, S K, S K, nd K QR =, K ({R} BR ) = (R, ), K ({} BR ) = (, ). Tke open set U Q R such tht K U, U K =, U Q R =, U K =, U K =. Hence U =, nd U(R) P = {}. Moreover i(i(λ, ), U(λ), P ) = µ = const, λ [, R]. By Lemm. µ = when λ = since U() = {}, while by Lemm.3.3 of [3]. i(i(r, ), U(R), P ) = i(i(r, ),, P ) = Theorem.3. Suppose tht (H) (H5) re stisfied. Then the continuum C of contining (, ) hs the following properties. (i) C contins connected closed subset C + [(, ) (P \{})] ({} P ). (ii) λ = is the bifurction point of I if I(λ, ). (iii) C + meets {} (P \{}). (iv) There exists λ > such tht x = I(λ, x) hs t lest two nontrivil solutions x λ, x λ for λ (, λ ), nd (λ, x λ ), (λ, x λ ) C+. Proof. Let C + be s in Theorem.. First we will prove tht C + is bounded. In fct, by (H3) there exists R > such tht x R for (λ, x). Let (λ n, x n ), λ n. If there exists ε > with x n > ε, then by (H4) we get contrdiction. On the other hnd if x n, then it will contrdicts (H5). Thus C + is bounded nd ssertion (iii) is true. Next we will show tht if there exists λ >, x P such tht C + ({λ} P ) = {x}, then C + ([, λ] P ) is connected. In fct, if there exist nonempty closed disjoint subsets S, S with C + ([, λ] P ) = S S nd (λ, x) S, then C + = S S3, where S 3 = S (C + [λ, )
STRUCTURE OF SOLUTIONS OF BOUNDRY VLUE PROBLEMS 76 P ). Evidently S 3 nd S re disjoint. This contrdicts with the fct tht C + is connected. Now suppose tht there exist λ n >, λ n such tht the set C + ({λ n } P ) is single-pointed for n >. Let Cn = C + ([, λ n ] P ). Then Cn is connected nd closed. By (iii) there exists x >, (, x ) C +. Let C = n Cn = {z : there exist subsequence n k with z nk Cn k, z nk z}. Hence (, x ) C. By Liu [6] we know tht C is connected nd closed. Moreover C, nd by definition C {} P. Hence x = could not be n isolted fixed point of I(λ, ). 3. utonomous nd Non-utonomous Boundry Vlue Problems In this section, we will use the results obtined in section to study clss of utonomous nd non-utonomous boundry vlue problems of ordinry differentil equtions. First we consider the following non-utonomous problem { (Lx)(t) = f(λ, t, x(t)), t (, ) αx() β p(t)x (t) = γx() + δ p(t)x (3.) (t) = t t where (Lx)(t) = p(t) (p(t)x (t)), p C[, ] C (, ), p(t) > for t (, ), α, β, γ, δ, βγ +αδ +αγ >, nd f C[R + (, ) R +, R + ]. We will ssume p(t) dt < throughout this section. Denote τ (t) = t ρ = βγ + αδ + αγ p(t) dt, nd ρ >. Define p(t) dt, τ (t) = t p(t) dt, u(t) = ρ [δ + γτ (t)], v(t) = ρ [β + ατ (t)], (3.) Then γv + αu ρ. Let E = C[, ] nd { u(t)v(s)p(s), s t k(t, s) = v(t)u(s)p(s), t s (3.3) Then problem (3.) is equivlent to the opertor eqution x = I(λ, x), x P [7], where I(λ, x) = k(t, s)f(λ, s, x(s))ds (3.4) nd P = P (, b) = {x E : min x(t) m(, b) x }, where m(, b) is determined t [,b] by the next lemm, nd, b (, ) be fixed ( = 4, b = 3 4 for exmple). Lemm 3.. The following estimtes hold. min k(t, s) m(, b) mx k(t, s) t [,b] t [,]
76 XIYU LIU ND BOQING YN mx t [,] k(t, s)ds mx{v() up, u(b) where m(, b) = min{ u(b) u(), v() v() }, nd the opertor I mps R+ P (, b) into P (, b) nd is completely continuous. Proof. It is stright forwrd. Now we will list the conditions used in this section. f(,t,x) (F): x x λ, uniformly for t (, ) nd λ u()v() mx p(t) <. t [,] f(λ,t,x) F(): x,λ λ x λ (λ ), uniformly for t (, ), where λ (λ )C(, b) >, C(, b) = m(, b) mx{v() up, u(b) vp}, nd λ > is rbitrry. f(λ,t,x) (F3): x x λ 3, uniformly for λ R +, t (, ) where λ 3 C(, b) >. (F4): f(λ, t, x) = +, uniformly for x [x, x ], t (, ), nd λ + x, x >. f(λ,t,x) (F5): x λ 5, uniformly for t (, ), where λ 5 C(, b) >. x,λ + Lemm 3.. Let (F)(F) be stisfied. Then conditions (H)(H) re vlid. Proof. Choose r > such tht f(, t, x) (λ +ε)x for x < r. Then for x < r we hve I(, x) uvpf(, s, x)ds (λ + ε) x Thus condition (H) is true. Similrly choose r > such tht f(λ, t, x) (λ (λ ) ε)x for λ λ < r, x < r. Then for λ λ < r, x < r, x P (, b) we hve (λ (λ ) ε) I(λ, x)(t) = k(t, s)f(λ, s, x)ds k(t, s)x(s)ds (λ (λ ) ε)m(, b) x vp} uvp k(t, s)ds Lemm 3.3. Let (F) (F5) be stisfied. Then conditions (H) (H5) re vlid. Proof. () Let R > such tht f(λ, t, x) (λ 3 ε)x for x R, λ. Then for x P (, b), x > we hve R m(.b) I(λ, x)(t) k(t, s)f(λ, s, x)ds
STRUCTURE OF SOLUTIONS OF BOUNDRY VLUE PROBLEMS 763 (λ 3 ε) k(t, s)x(s)ds (λ 3 ε)m(, b) x k(t, s)ds () Let x P (, b), ε x ε. Then for t (, b) we hve εm(, b) x(t) ε. Let λ > such tht f(λ, t, x) > T for λ > λ, εm(, b) x ε. Then I(λ, x)(t) k(t, s)f(λ, s, x)ds T k(t, s)ds (3) Let f(λ, t, x) (λ 5 ε)x for x < r, λ > λ. Then for λ > λ, x < r we hve (λ 5 ε) I(λ, x)(t) k(t, s)f(λ, s, x)ds k(t, s)x(s)ds (λ 5 ε)m(, b) x k(t, s)ds Theorem 3.. Suppose tht (F)(F) re stisfied. Then the continuum C contining (, ) of the solution set of problem (3.) hs the following properties. (i) C contins connected closed subset C + [(, ) (P \{})] ({} P ). (ii) λ = is the bifurction point of I if f(λ, t, ). (iii) There exists λ > such tht [{λ} (P \{})] C + for λ (, λ ). Theorem 3.. Suppose tht (F) (F5) re stisfied. Then the continuum C of contining (, ) hs the following properties. (i) C contins connected closed subset C + [(, ) (P \{})] ({} P ). (ii) λ = is the bifurction point of I if f(λ, t, ). (iii) C + meets {} (P \{}). (iv) There exists λ > such tht problem (3.) hs t lest two nontrivil solutions for λ (, λ ). Corollry 3.. Let f(λ, t, x) = λg(t, x) + h(t, x) where g, h : [, ] R + R + re continuous nd g(t, x) > for t [, ], x >. If h(t, x) g(t, x) h(t, x) =, = +, = + x x x x x + x Uniformly for t [, ], then the conclusions of Theorem 3. 3. hold. Now we consider more specil type of utonomous problems, nmely { x (t) = λg (x(t)) + h (x(s)), t (, ) x() = x() =, x C[, ] (3.5)
764 XIYU LIU ND BOQING YN where g, h C [, ), g() = h() =, g (x), h (x) > for x >. Let λ nd x be nontrivil solution to (3.5); i.e.; x(t) > for t (, ), nd x = x(t) =, x(ω) =. Then x (t) for t (, ω) nd x (t) for mx t [,] t (ω, ). By integrtion we hve Hence x (t) = λg(x) + h(x) λg() h() x (t) = ö λg(x) h(x) + λg() + h() where ö= for t (, ω) nd ö= for t (ω, ). Write F,λ (x) = E(λ, ) = x du λ(g() g(u)) + (h() h(u)), x (, ] (3.6) x λ (t) = { F,λ (t), t (, ω) F,λ ( t), t (ω, ) (3.7) du λ(g() g(u)) + (h() h(u)), > (3.8) If x is nontrivil solution of (3.5), then by (3.7) we know ω =. Thus we hve the following: Lemm 3.4. Let λ nd x be nontrivil solution of (3.5), then E(λ, x ) =. Conversely, if there exists λ, > such tht E(λ, ) =, then x λ is solution of (3.5), where x λ is determined by (3.7). Lemm 3.5. E : R + (, ) (, ) is continuous function. Moreover, E is strictly decresing with respect to λ. Proof. Let u = t, then E(λ, ) = λ(g() g(t)) + (h() h(t)) dt, > (3.9) Thus for t (, ) by the men vlue theorem we hve λ(g() g(t)) + (h() h(t)) = λg (θ + ( θ )t) + h (θ + ( θ )t) t C() t where θ, θ [, ] nd C() is constnt. Hence E(λ, ) is continuous.
STRUCTURE OF SOLUTIONS OF BOUNDRY VLUE PROBLEMS 765 g Lemm 3.6. Suppose (x) x x = +. Then dt = + g() g(t) Proof. Note tht g increses, hence we hve θ such tht dt g() g(t) θ g (θ ) g (θ ) g() g( ) Similrly we hve t θ such tht dt g() g(t) θ dt t g (θ ) dt g (θ ) t Let M >, > be such tht g () > M for < <. Consequently dt g() g(t) M dt t Lemm 3.7. Suppose h (x) x + x = +. Then dt = + h() h(t)
766 XIYU LIU ND BOQING YN Proof. Similr to the proof of Lemm 3.6, we hve h() h(t) dt dt = h() h( ) h (θ ) h() h(t) dt h (θ ) t Theorem 3.3. Suppose tht the following conditions re stisfied h (x) g (x) = +, = + (3.) x + x x x Then there exists λ (, ) such tht problem (3.5) hs t lest two nontrivil solutions for < λ < λ, nd no nontrivil solutions for λ > λ. Proof. By Lemm 3.5 3.7 we know tht E(λ, ) is continuous with respect to, E(λ, ) > for >, nd for fixed λ >, E(λ, ) = E(λ, ) =. Let > be such tht for > + h() h(t) dt < ε Then E(λ, ) < ε for >. Let C > be such tht h() h(t) dt C, < + Then E(λ, ) C λ for <. Hence E(λ, ) = uniformly for λ >. s result, eqution E(λ, ) = hs no solutions for λ lrge enough. In order to consider continu of the solution set, we need the following lemm. Let, C, C + be s before, nd Ω = {(λ, ) R : E(λ, ) =, λ, > }. Lemm 3.8. Let S E be closed nd connected subset of. Denote S R = {(λ, x ) : (λ, x) S E }. Then S E is closed nd connected in R. Conversely, if S R Ω is closed nd connected. Let S E = {(λ, x λ ) : x λ is determined by (3.7)}. Then S R is closed nd connected in R + E.
STRUCTURE OF SOLUTIONS OF BOUNDRY VLUE PROBLEMS 767 Proof. It suffices to note tht the following mps re continuous, where x λ is determined by (3.7). R + E R : (λ, x) (λ, x ) Ω R + E : (λ, ) (λ, x λ ) Theorem 3.4. Suppose (3.) is stisfied, then there exist λ, > such tht \((, ) {}) [, λ ] B R, nd ny continuum of will either meet {} P twice, or lie in {} P. Proof. By the proof of Theorem 3.3 we know there exists λ > such tht E(λ, ) 4 for λ > λ, >. By Lemm 3.7 there exists > such tht E(λ, ) 4 for λ, >. Therefore Ω [, λ ] [, ], \((, ) {}) [, λ ] B R. Let Ω = { > : E(, ) > }. Then Ω is n open set composed of open intervls. Let J Ω be one of its mximl open intervls, then the implicit function theorem implies tht there exists continuous curve λ = λ() : J [, λ ] such tht E(λ(), ) =. Hence {(λ(), ) : J} Ω is connected. Let (λ, x), λ >, then (, x ) Ω since E(λ, ) is strictly decresing with respect to λ. Theorem 3.5. Suppose the following conditions re stisfied h (x) h (x) g (x) =, = +, = + (3.) x x x + x x x xh (x) h(x)is strictly incresing for x > (3.) Then = C nd C + meets {} P exctly twice. Proof. By Theorem 3. nd Corollry 3. we know tht C + meets {} (P \{}). Thus by Lemm 3.4, (iii) of Theorem 3. nd Corollry 3. there exists λ > with E(λ, x λ ) =, where (λ, x λ) C +, < λ < λ. By Lemm 3.5 we know E(, x λ ) > for < λ < λ. Hence (, λ ) Ω. Thus by Lemm 3.4 we need only to prove tht E(, ) is strictly decresing. In fct, let t (, ), φ() = h() h(t), then by (3.) 3 φ () = h () h() + h(t) th (t) >, t (, ) Hence φ() is strictly incresing, nd E(, ) = [ h() h(t) ] dt is strictly decresing. Therefore Ω is n open intervl.
768 XIYU LIU ND BOQING YN Corollry 3.. Consider problem (.) in the sclr cse, i.e., x = λx q + x p, t (, ) x(t) >, t (, ) x() = x() =, (3.3) with < q < < p. Then ll the conclusions of Theorem 3. 3.5 hold for (3.3). cknowledgements.the finl version of this work is ccomplished while the first uthor is visiting Institute of Mthemtics, cdemi Sinic. The uthor is grteful to Professor Shujie Li for his hospitlity. References [] mbrosetti., Brezis H., nd Cermi C., Combined effects of concve nd convex nonlinerities in some problems, J. Func. nlysis,, No.4,(994), 59 543. [] Ruyun M, On conjecture concerning the multiplicity of positive solutions of elliptic problems, Nonliner nl., 7, No.7,(996), 775 78. [3] Guo Djun nd Lkshmiknthm, Nonliner Problems in bstrct Cones, cdemic Press, Orlndo, FL(988). [4] Rbinowitz, Some globl results for nonliner eigenvlue problems, J. Func. nlysis, 7, (97), 487 53. [5] mnn H., Fixed point equtions nd nonliner eigenvlue problems in ordered Bnch spces, SIM Review, 8, No.4,(976), 6 79. [6] Liu Xiyu, Structure of solutions of some singulr opertors with pplictions to impulsive integrodifferentil boundry vlue problems, SUT J. Mth., 3, No.,(996), 9. [7] Liu Xiyu, Some existence nd nonexistence principles for clss of singulr boundry vlue problems, Nonl. nl., 7, No.,(996), 47 64. Received June 3, 997 Revised version received November 6, 997 Finl version for publiction received ugust 3, 999 (Xiyu Liu) Deprtment of Mthemtics, Shndong Norml University, Ji-Nn, Shndong 54, People s Republic of Chin E-mil ddress: Yliu@jn-public.sd.cninfo.net (Current ddress) Deprtment of Computer Sciences, School of Informtion nd Mngement, Shndong Norml University, Ji-Nn, Shndong 54, People s Republic of Chin (Boqing Yn) Deprtment of Mthemtics, Shndong Norml University Ji-Nn, Shndong 54, People s Republic of Chin E-mil ddress: ynbq@sdnu.edu.cn