EXISTENCE OF THREE SOLUTIONS FOR A KIRCHHOFF-TYPE BOUNDARY-VALUE PROBLEM

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1 Electonic Jounl of Diffeentil Eutions, Vol. 20 (20, No. 9, pp.. ISSN: URL: o ftp ejde.mth.txstte.edu EXISTENCE OF THREE SOLUTIONS FOR A KIRCHHOFF-TYPE BOUNDARY-VALUE PROBLEM SHAPOUR HEIDARKHANI, GHASEM ALIZADEH AFROUZI, DONAL O REGAN Abstct. In this note, we estblish the existence of two intevls of positive el pmetes λ fo which the boundy-vlue poblem of Kichhoff-type K` Z b u (x 2 dx u = λf(x, u, u( = u(b = 0 dmits thee wek solutions whose noms e unifomly bounded with espect to λ belonging to one of the two intevls. Ou min tool is thee citicl point theoem by Bonnno.. Intoduction In the litetue mny esults focus on the existence of multiple solutions to boundy-vlue poblems. Fo exmple, cetin chemicl ections in tubul ectos cn be mthemticlly descibed by nonline two-point boundy-vlue poblem nd one is inteested if multiple stedy-sttes exist. Fo ecent tetment of chemicl ecto theoy nd multiple solutions see [, section 7] nd the efeences theein. Bonnno in [3] estblished the existence of two intevls of positive el pmetes λ fo which the functionl Φ λψ hs thee citicl points whose noms e unifomly bounded in espect to λ belonging to one of the two intevls nd he obtined multiplicity esults fo two point boundy-vlue poblem. In the pesent ppe s n ppliction, we shll illustte these esults fo Kichhoff-type poblem. Poblems of Kichhoff-type hve been widely investigted. We efe the ede to the ppes [2, 5, 7, 9, 0,, 5] nd the efeences theein. Riccei [3] estblished the existence of t lest thee wek solutions to clss of Kichhoff-type doubly eigenvlue boundy vlue poblem using [2, Theoem 2]. Conside the Kichhoff-type poblem K ( u (x 2 dx u = λf(x, u, u( = u(b = 0 ( Mthemtics Subject Clssifiction. 35J20, 35J25, 35J60. Key wods nd phses. Kichhoff-type poblem; multiple solutions; citicl point. c 20 Texs Stte Univesity - Sn Mcos. Submitted Apil 26, 20. Published July 6, 20.

2 2 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 whee K : [0, + [ R is continuous function, f : [, b] R R is Cthéodoy function nd λ > 0. In the pesent ppe, ou ppoch is bsed on thee citicl points theoem poved in [3], which is eclled in the next section fo the ede s convenience (Theoem 2.. Ou min esult is Theoem 2.2 which, unde suitble ssumptions, ensues the existence of two intevls Λ nd Λ 2 such tht, fo ech λ Λ Λ 2, the poblem (. dmits t lest thee clssicl solutions whose noms e unifomly bounded in espect to λ Λ 2. Let X the the Sobolev spce H0 ([, b] with the nom ( /2. u = ( u (x dx 2 We sy tht u is wek solution to (. if u X nd K ( u (x 2 dx u (xv (xdx λ f(x, u(xv(xdx = 0 fo evey v X. Fo othe bsic nottions nd definitions, we efe the ede to [4, 6, 8, 4]. 2. Results Fo the ede s convenience, dist we hee ecll [3, Theoem 2.]. Theoem 2.. Let X be sepble nd eflexive el Bnch spce, Φ : X R nonnegtive continuously Gâteux diffeentible nd seuentilly wekly lowe semicontinuous functionl whose Gâteux deivtive dmits continuous invese on X, J : X R continuously Gâteux diffeentible functionl whose Gâteux deivtive is compct. Assume tht thee exists x 0 X such tht Φ(x 0 = J(x 0 = 0 nd tht lim (Φ(x λj(x = + x + fo ll λ [0, + [. Futhe, ssume tht thee e > 0, x X such tht < Φ(x nd x Φ (],[ w J(x < + Φ(x J(x ; hee Φ (], [ w denotes the closue of Φ (], [ in the wek topology (in pticul note J(x 0 since x 0 Φ (], [ w (note J(x 0 = 0 so x Φ w J(x 0. Then, fo ech (],[ the eution Φ(x λ Λ =] J(x x Φ (],[ w J(x, x Φ (],[ w J(x[, Φ (u + λj (u = 0 (2. hs t lest thee solutions in X nd, moeove, fo ech η >, thee exist n open intevl η Λ 2 [0, J(x Φ(x x Φ (,[ w J(x] nd positive el numbe σ such tht, fo ech λ Λ 2, the eution (2.2 hs t lest thee solutions in X whose noms e less thn σ.

3 EJDE-20/9 EXISTENCE OF THREE SOLUTIONS 3 Let K : [0, + [ R be continuous function such tht thee exists positive numbe m with K(t m fo ll t 0, nd let f : [, b] R R be Cthéodoy function such tht ξ s f(., ξ L (, b fo ll s > 0. Coesponding to K nd f we intoduce the functions K : [0, + [ R nd F : [, b] R R, espectively s follows nd F (x, t = K(t = t Now, we stte ou min esult. 0 t 0 K(sds fo ll t 0 (2.2 f(x, sds fo ll (x, t [, b] R. (2.3 Theoem 2.2. Assume tht thee exist positive constnts nd θ, nd function w X such tht: (i K( w 2 > 2, (ii (b (b F (x, tdx < + K( w 2 2, (iii (b 2 2m lim t + F (x,t t 2 < θ unifomly with espect to x [, b]. Futhe, ssume tht thee exists continuous function h : [0, + [ R such tht h(tk(t 2 = t fo ll t 0. Then, fo ech λ in the intevl Λ =] (b (b (b F (x, tdx[, (b F (x, tdx, poblem (. dmits t lest thee wek solutions in X nd, moeove, fo ech η >, thee exist n open intevl η Λ 2 [0, R b 2 F (x,w(xdx ] b K( w 2 F (x, tdx (b (b nd positive el numbe σ such tht, fo ech λ Λ 2, poblem (. dmits t lest thee wek solutions in X whose noms e less thn σ. Let us fist give pticul conseuence of Theoem 2.2 fo fixed test function w. Coolly 2.3. Assume tht thee exist positive constnts c, d, α, β nd θ with β α < b such tht Assumption (ii in Theoem 2.2 holds, nd (i K(d 2 ( α+β 4mc2 αβ > b, (ii F (x, t 0 fo ech (x, t ([, + α] [b β, b] [0, d], (iii c,c] F (x, tdx < 2mc2 b R b β +α F (x,ddx b + 2 K(d 2 ( α+β.

4 4 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 Futhe, ssume tht thee exists continuous function h : [0, + [ R such tht h(tk(t 2 = t fo ll t 0. Then, fo ech λ Λ =] K(d 2 2 ( α+β β +α F (x, ddx c,c] F (x, tdx, b c,c] F (x, tdx [, poblem (. dmits t lest thee wek solutions in X nd, moeove, fo ech η >, thee exist n open intevl Λ 2 [0, ( 2ηmc 2 b ( +α 4mc 2 F (x, d α (x dx + β +α F (x, ddx + b β F (x, d β (b xdx b K(d 2 ( α+β c,c] F (x, tdx ] nd positive el numbe σ such tht, fo ech λ Λ 2, poblem (. dmits t lest thee wek solutions in X whose noms e less thn σ. Poof. We clim tht the ll the ssumptions of Theoem 2.2 e fulfilled with d α (x if x < + α, w(x = d if + α x b β, (2.4 d β (b x if b β < x b nd = /(b whee constnts c, d, α nd β e given in the sttement of the theoem. It is cle fom (2.4 tht w X nd, in pticul, one hs w 2 = d 2 ( α + β. (2.5 αβ Moeove with this choice of w nd tking into ccount (2.5, fom (i we get (i of Theoem 2.2. Since 0 w(x d fo ech x [, b], condition (ii ensues tht so fom (iii we hve +α c,c] + b β F (x, tdx < 2mc2 b 2mc2 b = 0, β +α b F (x, ddx + 2 K(d 2 ( α+β b + +, so (ii of Theoem 2.2 holds (note c 2 = (b 2m. Next notice tht (b (b F (x, tdx

5 EJDE-20/9 EXISTENCE OF THREE SOLUTIONS 5 nd In ddition note tht Finlly note tht K(d 2 2 ( α+β αβ b β +α F (x, ddx c,c] F (x, tdx (b = (b F (x, tdx K(d 2 2 ( α+β αβ b β +α F (x, ddx c,c] F (x, tdx < = b c,c] F (x, tdx. K(d 2 2 ( α+β ( b + K(d 2 2 ( α+β c,c] F (x, tdx b b c,c] F (x, tdx. h R b 2 F (x,w(xdx K( w 2 (b (b F (x, tdx = ( 2hmc 2 b ( +α 4mc 2 F (x, d α (x dx + β +α F (x, ddx + b β F (x, d β (b xdx b K(d 2 ( α+β c,c] F (x, tdx, nd tking into ccount tht Λ Λ we hve the desied conclusion diectly fom Theoem 2.2. It is of inteest to list some specil cses of Coolly 2.3. Coolly 2.4. Assume tht thee exist positive constnts c, d, p, p 2, α, β nd θ with β α < b such tht Assumption (ii of Coolly 2.3 holds, nd (i p d 2 ( α+β αβ + p2 2 d4 ( α+β αβ 2 > 4pc2 b, (ii c,c] F (x, tdx < 2p c 2 b (iii (b 2 F (x,t 2p lim t + t 2 Then, fo ech λ Λ =] < θ 2p c 2 b p 2 d 2 ( α+β αβ + p2 4 d4 ( α+β αβ 2 β +α F (x, ddx c,c] F (x, tdx, β F (x, ddx +α + p 2 d2 ( α+β αβ + p2 4 d4 (, α+β αβ 2 unifomly with espect to x [, b]. 2p c 2 b c,c] F (x, tdx [,

6 6 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 the poblem (p + p 2 u (x 2 dxu = λf(x, u, u( = u(b = 0 (2.6 dmits t lest thee wek solutions in X nd, moeove, fo ech η >, thee exist n open intevl Λ 2 [0, ( 2ηp c 2 b ( 4p c 2 b c,c] +α F (x, d α (x dx + β +α F (x, ddx + b β F (x, d β (b xdx F (x, tdx ] p d 2 ( α+β αβ + p2 2 d4 ( α+β αβ 2 nd positive el numbe σ such tht, fo ech λ Λ 2, poblem (2.6 dmits t lest thee wek solutions in X whose noms e less thn σ. Poof. Fo fixed p, p 2 > 0, set K(t = p + p 2 t fo ll t 0. Being in mind tht m = p, fom (i (iii, we see tht (i (iii of Coolly 2.4 hold espectively. Also we note tht thee exists continuous function h : [0, + [ R such tht h(tk(t 2 = t fo ll t 0 becuse the function K is nondecesing in [0, + [ with K(0 > 0 nd t tk(t 2 (t 0 is incesing nd onto [0, + [. Hence, Coolly 2.3 yields the conclusion. Coolly 2.5. Assume tht thee exist positive constnts c, d, α, β nd θ with β α < b such tht Assumption (ii in Coolly 2.3 holds, nd (i d 2 ( α+β αβ > 4c2 b, (ii c,c] F (x, tdx < (iii (b 2 F (x,t 2 lim t + t < 2 θ Then, fo ech λ Λ =] the poblem d 2 2 ( α+β αβ β 2c2 F (x, ddx +α b 2c 2 b + d2 2 ( α+β αβ, β +α F (x, ddx c,c] F (x, tdx, unifomly with espect to x [, b]. u = λf(x, u, u( = u(b = 0 2c 2 b c,c] F (x, tdx [, (2.7 dmits t lest thee wek solutions in X nd, moeove, fo ech η >, thee exist n open intevl Λ 2 [0, ( 2ηc 2 b ( +α 4c 2 F (x, d α (x dx + β +α F (x, ddx + b β F (x, d β (b xdx b d 2 ( α+β αβ

7 EJDE-20/9 EXISTENCE OF THREE SOLUTIONS 7 c,c] F (x, tdx ] nd positive el numbe σ such tht, fo ech λ Λ 2, the poblem (2.7 dmits t lest thee wek solutions in X whose noms e less thn σ. We conclude this section by giving n exmple to illustte ou esults pplying by Coolly 2.4. Exmple 2.6. Conside the poblem ( u (x 2 dxu = λ(e u u (2 u, u(0 = u( = 0 (2.8 whee λ > 0. Set p = 28, p 2 = 64 nd f(x, t = e t t (2 t fo ll (x, t [0, ] R. A diect clcultion yields F (x, t = e t t 2 fo ll (x, t [0, ] R. Assumptions (i nd (ii of Coolly 2.4 e stisfied by choosing, fo exmple F (x,t d = 2, c =, [, b] = [0, ] nd α = β = /4. Also, since lim t + t = 0, 2 Assumption (iii of Coolly 2.4 is fulfilled. Now we cn pply Coolly 2.4. Then, fo ech λ Λ 33 =] 2 4 e 2 8e, 64e [ poblem (2.8 dmits t lest thee wek solutions in H0 ([0, ] nd, moeove, fo ech η >, thee exist n open intevl Λ 2 [0, ( e 8t t 2 dt + 2 e η 4 ] e 8( t ( t 2 dt 64e nd positive el numbe σ such tht, fo ech λ Λ 2, poblem (2.8 dmits t lest thee wek solutions in H 0 ([0, ] whose noms e less thn σ. We begin by setting 3. Poof of Theoem 2.2 Φ(u = 2 K( u 2, (3. J(u = F (x, u(xdx (3.2 fo ech u X, whee K nd F e given in (2.2 nd (2.3, espectively. It is well known tht J is Gâteux diffeentible functionl whose Gâteux deivtive t the point u X is the functionl J (u X, given by J (uv = f(x, u(xv(xdx fo evey v X, nd tht J : X X is continuous nd compct opeto. Moeove, Φ is continuously Gâteux diffeentible nd seuentilly wekly lowe semi continuous functionl whose Gâteux deivtive t the point u X is the functionl Φ (u X, given by Φ (uv = K( u (x 2 dx u (xv (xdx

8 8 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 fo evey v X. We clim tht Φ dmits continuous invese on X (we identity X with X. To pove this fct, guing s in [3] we need to find continuous opeto T : X X such tht T (Φ (u = u fo ll u X. Let T : X X be the opeto defined by { h( v v T (v = v if v 0 0 if v = 0, whee h is defined in the sttement of Theoem 2.2. Since, h is continuous nd h(0 = 0, we hve tht the opeto T is continuous in X. Fo evey u X, tking into ccount tht inf t 0 K(t m > 0, we hve since h(t K(t 2 = t fo ll t 0 tht T (Φ (u = T (K( u 2 u = h(k( u 2 u K( u 2 u K( u 2 u = u K( u 2 u K( u 2 u = u, so ou clim is tue. Moeove, since m K(s fo ll s [0, + [, fom (3. we hve Φ(u m 2 u 2 fo ll u X. (3.3 Futhemoe fom (iii, thee exist two constnts γ, τ R with 0 < γ < /θ such tht (b 2 2m F (x, t γt2 + τ fo ll x (, b nd ll t R. Fix u X. Then F (x, u(x 2m (b 2 (γ u(x 2 + τ fo ll x (, b. (3.4 Fix λ ]0, + [. Then thee exists θ > 0 with λ ]0, θ]. Now since mx x [,b] fom (3.3, (3.4 nd (3.5, we hve Φ(u λj(u = 2 K( u 2 λ m 2 u 2 (b /2 u(x u, ( θm ( (b 2 γ F (x, u(xdx u(x 2 + τ(b m 2 u 2 2θm ( (b 2 (b 2 γ u 2 + τ(b 4 = m 2 ( γθ u 2 2θτm b, nd so lim (Φ(u λj(u = +. u + Also fom (3. nd (i we hve Φ(w >. Using (3.3 nd (3.5, we obtin Φ (], [ = { u X; Φ(u < } { u X; u < 2/m }

9 EJDE-20/9 EXISTENCE OF THREE SOLUTIONS 9 so, we hve { u X; u(x (b /(2m, fo ll x [, b] }, u Φ (],[ w J(u Theefoe, fom (ii, we hve u Φ (],[ w J(u (b (b F (x, tdx. (b (b F (x, tdx < + K( w 2 2 = + Φ(w J(w. Now, we cn pply Theoem 2.. Note fo ech x [, b], u Φ (],[ w J(u nd Φ(w J(w u Φ (],[ w J(u Note lso tht (ii immeditely implies Also < = J(w Note fom (ii tht ( + η (b (b (b (b (b (b F (x, tdx. (b F (x, tdx (b F (x, tdx. (b F (x, tdx (b F (x, tdx Φ(w u Φ (,[ w J(u η R b 2 F (x,w(xdx = ρ. b K( w 2 F (x, tdx (b (b 2 K( w 2 (b F (x, tdx (b

10 0 S. HEIDARKHANI, G. A. AFROUZI, D. O REGAN EJDE-20/9 ( 2 > K( w 2 + K( w 2 2 ( 2 K( w 2 2 = 0 K( w 2 since 0 (note F (x, 0 = 0 so (b (b F (x, tdx 0 nd now pply (ii. Now with x 0 = 0, x = w fom Theoem 2. (note J(0 = 0 fom (2.3 it follows tht, fo ech λ Λ, the poblem (. dmits t lest thee wek solutions nd thee exist n open intevl Λ 2 [0, ρ] nd el positive numbe σ such tht, fo ech λ Λ 2, the poblem (. dmits t lest thee wek solutions tht whose noms in X e less thn σ. Refeences [] R. P. Agwl, H. B. Thompson nd C.C. Tisdell; On the existence of multiple solutions to boundy vlue poblems fo second ode, odiny diffeentil eutions. Dynm. Systems Appl. 6 ( [2] C. O. Alves, F. S. J. A. Coê nd T. F. M; Positive solutions fo usiline elliptic eutions of Kichhoff type, Comput. Mth. Appl., 49 (2005, [3] G. Bonnno; A citicl points theoem nd nonline diffeentil poblems, J. Globl Optimiztion, 28 ( [4] G. Bonnno, G. Molic Bisci nd V. Rădulescu; Infinitely mny solutions fo clss of nonline eigenvlue poblem in Olicz-Sobolev spces, C. R. Acd. Sci. Pis, Se. I 349 ( [5] M. Chipot nd B. Lovt; Some emks on non locl elliptic nd pbolic poblems, Nonline Anl., 30 (997, [6] M. Ghegu nd V. Rădulescu; Singul Elliptic Poblems. Bifuction nd Asymptotic Anlysis, Oxfod Lectue Seies in Mthemtics nd Its Applictions, vol. 37, Oxfod Univesity Pess, [7] X. He nd W. Zou; Infinitely mny positive solutions fo Kichhoff-type poblems, Nonline Anl. 70 ( [8] A. Kistály, V. Rădulescu nd C. Vg; Vitionl Pinciples in Mthemticl Physics, Geomety, nd Economics: Qulittive Anlysis of Nonline Eutions nd Uniltel Poblems, Encyclopedi of Mthemtics nd its Applictions, No. 36, Cmbidge Univesity Pess, Cmbidge, 200. [9] T. F. M; Remks on n elliptic eution of Kichhoff type, Nonline Anl., 63 (2005, e957-e977. [0] A. Mo nd Z. Zhng; Sign-chnging nd multiple solutions of Kichhoff type poblems without the P. S. condition, Nonline Anl. 70 ( [] K. Pee nd Z. T. Zhng; Nontivil solutions of Kichhoff-type poblems vi the Yng index, J. Diffeentil Eutions, 22 (2006, [2] B. Riccei; A futhe thee citicl points theoem, Nonline Anl. 7 ( [3] B. Riccei; On n elliptic Kichhoff-type poblem depending on two pmetes, J. Globl Optimiztion 46 ( [4] E. Zeidle; Nonline functionl nlysis nd its pplictions, Vol. II, III. Belin-Heidelbeg- New Yok 985. [5] Z. T. Zhng nd K. Pee; Sign chnging solutions of Kichhoff type poblems vi invint sets of descent flow, J. Mth. Anl. Appl., 37 ( Shpou Heidkhni Deptment of Mthemtics, Fculty of Sciences, Rzi Univesity, 6749 Kemnshh, In E-mil ddess: s.heidkhni@zi.c.i

11 EJDE-20/9 EXISTENCE OF THREE SOLUTIONS Ghsem Alizdeh Afouzi Deptment of Mthemtics, Fculty of Bsic Sciences, Univesity of Mzndn, Bbols, In E-mil ddess: Donl O Regn Deptment of Mthemtics, Ntionl Univesity of Ielnd, Glwy, Ielnd E-mil ddess: donl.oegn@nuiglwy.ie