Some indefinite nonlinear eigenvalue problems

May 13, 2004 10:52 WSPC/T im Size: 9in x 6in o P oceedings mawhinsua ez SOME INDEFINITE NONLINEAR EIGENVALUE PROBLEMS A. SU´ AREZ∗ Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico, Fac. Ma em´a icas, C/ Ta ia s/n, C.P. 41012, Uni . Se illa, Spain E-mail: sua [email p o ec ed] Dedica ed o P o . Jean Mawhin o his i s 60 yea s o Nonlinea Analysis In his wo k we s udy he s uc u e o he se o posi i e solu ions o a nonlinea eigen alue p oblem wi h a weigh changing sign. Speci ically, he eac ion e m a ises om a popula ion dynamic model. We use mainly bi u ca ion me hods o ob ain ou esul s. 1. In oduc ion The aim o his wo k is o s udy some nonlinea inde ini e eigen alue p ob- lems o he o m ½−∆u=λm(x) (u) in Ω, u= 0 on ∂Ω, (1) whe e Ω ⊂IRNis a bounded domain wi h a egula bounda y ∂Ω, m∈C(Ω) changes sign, is a egula unc ion and λplays he ole o eal pa ame e . We ocus ou a en ion on he case (0) = 0 and λ > 0; simila esul s can be ob ained o nega i e alues o λ. Depending o he shape o , Eq. (1) models di e en si ua ions: pop- ula ion dynamics, popula ion gene ics, combus ion heo y,... see [10]. In he linea case, i.e., (u) = u, (1) is he eigen alue p oblem ½−∆u=λm(x)uin Ω, u= 0 on ∂Ω. (2) ∗Suppo ed by he Spanish Minis y o Science and Technology unde g an s BFM2000- 0797 and BFM2003-06446. 1 May 13, 2004 10:52 WSPC/T im Size: 9in x 6in o P oceedings mawhinsua ez 2 I is well known (see o ins ance [19] and [23]) ha he e exis wo alues o λ,λ−(m)<0< λ+(m), called p incipal eigen alues because hey ha e associa ed posi i e eigen unc ions. In he p esen wo k, gi en q∈L∞(Ω) we deno e by σΩ 1[−∆ + q] (we dele e he supe sc ip Ω when no con usion a ises) he p incipal eigen alue o he p oblem −∆u+q(x)u=λu in Ω, u= 0 on ∂Ω. When in (1) he weigh does no appea , i.e., m≡1, he nonlinea p oblem ½−∆u=λ (u) in Ω, u= 0 on ∂Ω, (3) has been ex ensi ely s udied. Classical e e ences a e [2] and [21], bu many o he s can be gi en whe e, as well as exis ence esul s, uniqueness ones a e shown: [4], [14], [26], [20], [22] and e e ences he ein. Much less is known o p oblem (1). In [19], assuming o example ha 0(0) >0, he au ho s showed ha he e exis s an unbounded con inuum o posi i e solu ions bi u ca ing om he i ial solu ion a λ=λ+(m)/ 0(0). In [8] he au ho s assumed ha :I7→ IR+,I⊂IR, and 00 <0 and showed ha e e y posi i e solu ion o (1) is s able. I , mo eo e , I= [0,1], (1) = 0 and 0(0) >0 hey p o ed ha he e exis s a posi i e solu ion i , and only i , λ>λ+(m)/ 0(0), and in his case he solu ion is unique. Simila esul was shown in [13], al hough he au ho s’ mo i a ion was o s udy he p oblem in he whole space. Ve y ecen ly, in [9] he au ho s analyze he pa icula cases (u) = gi(u), i= 1,2 wi h g1(u) = u−u2, g2(u) = u+u2.(4) Obse e ha he esul o [8] can only be applied o g1. In [9], wi hou he assump ion ha akes only alues in [0,1], he main esul o [8] was imp o ed showing (by a ia ional me hod) ha , assuming some es ic ion in he space dimension, he e exis s posi i e solu ion i λ∈(0, λ+(m)). Fo he case, =g2, hey also p o ed he exis ence o posi i e solu ion o λ∈(0, λ+(m)) and ha he e does no exis posi i e solu ion a λ= λ+(m). In [16] hese esul s ha e been again comple ed. We p o e o =g1 ha he e exis a leas wo posi i e solu ions in λ∈(λ+(m),∞), one o hem linea ly asymp o ically s able and ha o =g2 he e exis s posi i e solu ion i , and only i , λ∈(0, λ+(m)). In his wo k, we a e going o analyze he ollowing nonlinea i ies 1(u) = u−u2−Ku 1 + u, 2(u) = u+u2−Ku 1 + u,(5) May 13, 2004 10:52 WSPC/T im Size: 9in x 6in o P oceedings mawhinsua ez 3 whe e K∈IR. Obse e ha he unc ions in (4) a e included in (5). These las nonlinea i ies a ise in popula ion dynamics. Indeed, when K= 0, 1 is he classical logis ic eac ion e m and o K6= 0 he p eda ion one Ku/(1 + u) is called he Holling-Tanne e m, see o example [7] o an ecological in e p e a ion. In o de o s a e ou main esul s we need some no a ions. Speci ically, assume ha M±:= {x∈Ω : m±>0} a e open and egula se s, whe e m± ep esen he posi i e and nega i e pa o m espec i ely; and suppose ha m±(x)≈[dis (x, ∂M±)]γ± o x close o ∂M±and some γ±≥0. The ollowing condi ion will p o ide us wi h a p io i bounds o he solu ions 2<min ½N+1+γ± N−1,N+ 2 N−2¾.(6) Finally, we de ine o K6= 1 he alues λ+:= λ+(m) 1−Kλ−:= λ−(m) 1−K, and Π : IR ×C(Ω) 7→ IR he p ojec ion map on o IR, i.e. Π(µ, u) = µ. The main esul s a e: Theo em 1.1. Assume ha K6= 1 and (6). (1) The e exis s an unbounded con inuum Co posi i e solu ions o (1) bi u ca ing om he i ial solu ion a λ=λ+i K < 1and λ=λ− i K > 1. (2) The bi u ca ion is supe c i ical o = 1and o = 2and K < −1o K > 1and subc i ical o = 2and K∈[−1,1). (3) I = 1and K < 1( esp. = 2and K > 1), hen Π(C) = (λ+,∞)( esp. (λ−,∞)). Mo eo e , i (λ, uλ)∈ C, hen uλis linea ly asymp o ically and such ha uλ≤√1−K( esp. √K−1). Fu he mo e, he e exis s ano he posi i e solu ion λ o all λ > 0. (4) I = 1and K > 1( esp. = 2and K < −1) hen Π(C) = (0, λ∗] o λ∗> λ−( esp. λ+). Mo eo e , he e exis λ0and λ∗ wi h λ0< λ∗such ha o λ≥λ∗ he p oblem (1) does no admi posi i e solu ions and i possesses a leas wo posi i e solu ions o λ∈(λ−, λ0)( esp. (λ+, λ0)). (5) I = 2and K∈[−1,1) he e exis s posi i e solu ion o λ∈ (0, λ+)and (1) does no admi posi i e solu ions o λ≥λ∗. May 13, 2004 10:52 WSPC/T im Size: 9in x 6in o P oceedings mawhinsua ez 4 (6) In any case, i he e exis s a solu ion λ o λ > 0, hen limλ→0k λk∞= +∞. Theo em 1.2. Assume K= 1 and (6). Then he e exis s a leas a solu- ion uλ o λ > 0and limλ→0kuλk∞= +∞. Rema k 1.1. (1) The exis ence o Cis ue wi hou assuming (6). In he cases (4) and (5) o Theo em 1.1, Ccould “go o in ini y” in a alue λ0. (2) In he pa icula case = 2and K= 0, in [16] i was p o ed using a Picone inequali y ha (1) possesses a posi i e solu ion i , and only i , λ∈(0, λ+). In Figs. 1 and 2 we ha e summa ized hese esul s ( he case = 2and K= 1 is simila o = 1and K= 1). λ || . || || . || || . || a) b) c) +λ−λ λ λ Figu e 1. Bi u ca ion diag ams o = 1: a) K < 1; b) K= 1; c) K > 1. The es o he pape is o ganized as ollows: Secs. 2 and 3 a e de o ed o p o e Theo ems 1.1 and 1.2, espec i ely. 2. P oo o Theo em 1.1 2.1. Local bi u ca ion In his subsec ion we show he di ec ion o bi u ca ion om he i ial solu ion o bo h cases 1and 2. Fo ha , we w i e he nonlinea i y o he ollowing manne (u) = u∓u2−Ku 1 + u=u(1 −K) + u2(K 1 + u∓1). May 13, 2004 10:52 WSPC/T im Size: 9in x 6in o P oceedings mawhinsua ez 5 λ λ λ λλλ a) b) c) + + − || . || || . || || . || Figu e 2. Bi u ca ion diag ams o = 2: a) K < −1; b) K∈[−1,1); c) K > 1. I is clea ha o s udy (1) is equi alen o ind ze os o L(λ)u−N(λ, u) = 0, whe e L(λ)u:= u−λ(−∆)−1m(x)(1 −K)u, N(λ, u) := λ(−∆)−1m(x)u2(K 1 + u∓1). We can p o e ha N(L(λ+)) = Span < ϕ+>and d dλL(λ+)ϕ+/∈R(L(λ+)) (7) whe e, gi en any linea con inuous ope a o L,N[L] and R[L] s and o he null space and he ange o L, espec i ely, and −∆ϕ+=λ+(m)m(x)ϕ+in Ω, ϕ+= 0 on ∂Ω.(8) The i s equali y o (7) is i ial, o he second exp ession we need he ollowing esul . Lemma 2.1. Fo any p≥2we ha e ha ZΩ m(x)(ϕ+)p>0. P oo : Mul iplying (8) by (ϕ+)p−1we ge λ+(m)ZΩ m(x)(ϕ+)p=ZΩ (−∆ϕ+)(ϕ+)p−1= (p−1) ZΩ|∇ϕ+|2(ϕ+)p−2>0. ¦ Now, we show (7). Assume ha he e exis s usuch ha d dλL(λ+)ϕ+=−(−∆)−1m(x)(1 −K)ϕ+=u−(−∆)−1m(x)λ+(1 −K)u, May 13, 2004 10:52 WSPC/T im Size: 9in x 6in o P oceedings mawhinsua ez 6 hen (−∆−λ+(m)m(x))u=−(1 −K)m(x)ϕ+, and so, mul iplying by ϕ+we ge a con adic ion using Lemma 2.1. Now, we can apply he C andall-Rabinowi z Theo em [15] and conclude ha he e exis s δ > 0 such ha in a neighbo hood o (λ+,0) he non i ial solu ions o (1) a e o he o m u(s) = sϕ++s2ϕ2+s3ϕ3+o(s3), λ(s) = λ++sλ1+s2λ2+o(s2). In oducing hese e ms in (1), using (8) and a Taylo exp ession o he unc ion 1/(1 + u(s)), we ge (−∆−λ+(m)m(x))ϕ2=λ+m(x)(ϕ+)2(K∓1) + λ1m(x)(1 −K)ϕ+, and so, λ1=−λ+(K∓1) 1−K ZΩ m(x)(ϕ+)3 ZΩ m(x)(ϕ+)2 .(9) Obse e ha in he pa icula case = 2and K=−1, λ1= 0, and so we ha e o calcula e λ2. I can be p o ed ha λ2=−λ+ 2 ZΩ m(x)(ϕ+)4 ZΩ m(x)(ϕ+)2 .(10) F om (9) and (10), we conclude he pa ag aph (2) o Theo em 1.1. Analo- gously i can be ea ed he case λ−. 2.2. Non-exis ence esul s Lemma 2.2. Assume = 1and K > 1o = 2and K < 1. Then, he e exis s λ∗>0such ha o λ≥λ∗(1) does no ha e posi i e solu ions. P oo : Assume = 1and K > 1. Fi s ly obse e ha h(x) := x(K 1 + x−1) ≤(√K−1)2,∀x≥0.(11) May 13, 2004 10:52 WSPC/T im Size: 9in x 6in o P oceedings mawhinsua ez 7 Le ube a posi i e solu ion o (1). Then, using he mono ony o he p in- cipal eigen alue wi h espec o he domain and (11) we ge 0 = σ1[−∆−λm(x)(1 −K)−λm(x)u(K 1 + u−1)] < < σM− 1[−∆−λm(x)((1 −K)+(√K−1)2)] = =σM− 1[−∆−λm(x)2(1 −√K)], which is an absu dum o λla ge. Now, assume = 2and K < 1. In his case, x(K 1 + x+ 1) ≥0,i K≥ −1, ∀x≥0, x(K 1 + x+ 1) ≥ −(√−K−1)2,i K < −1, ∀x≥0. So, i −1≤K < 1 we ha e 0 = σ1[−∆−λm(x)(1−K)−λm(x)u(K 1 + u+1)] < σM+ 1[−∆−λm(x)(1−K)]; on he o he hand, o K < −1, 0 = σ1[−∆−λm(x)(1−K)−λm(x)u(K 1 + u+1)] < σM+ 1[−∆−λm(x)2√−K], in bo h cases a con adic ion o la ge λ.¦ 2.3. Mul iplici y esul s To ob ain mul iplici y esul s, we include (1) in he mo e gene al equa ion ½−∆u=µm(x)(1 −K)u+λm(x)g(u) in Ω, u= 0 on ∂Ω, (12) whe e gsa is ies (Hg)g(0) = g0(0) = 0, g00(u)<0,lim s→+∞ g(s) s2=β < 0. P oblem (12) has a ac ed a g ea deal o a en ion du ing las yea s (see o example [1], [3], [5], [6], [18] and [24]) when m≡1 in he i s e m on he igh -hand side o (12) and in [11], [12] and [13] wi h he igh -hand side o he o m µh(x)u+g(x)upand es ic i e condi ions on hand gwhich a e no sa is ied in ou case. In [16] was p o ed (see Fig. 3): P oposi ion 2.1. Assume ha gsa is ies (Hg),(6),K6= 1 and ix λ > 0. Deno e by Λ+:= λ+(m(x)(1 −K)),Λ−:= λ−(m(x)(1 −K)). May 13, 2004 10:52 WSPC/T im Size: 9in x 6in o P oceedings mawhinsua ez 8 Then, (12) possesses a posi i e solu ion i µ > Λ−. Mo eo e , om he i - ial solu ion u= 0 emana e wo unbounded in IR×C(Ω) con inua o posi i e solu ions C+:= {(µ, uµ)}and C−:= {(µ, wµ)}a µ= Λ+and µ= Λ−, espec i ely. Bo h con inua bi u ca e o he igh and Π(C−)⊃(Λ−,+∞), Π(C+) = (Λ+,+∞). Finally, o µ > Λ+,uµis linea ly asymp o ically s able and uµ6=wµ. Rema k 2.1. Obse e ha o K < 1, Λ+=λ+and Λ−=λ−, and o K > 1, Λ+=λ−and Λ−=λ+. Indeed, o example o K > 1, i ollows ha Λ+=λ+(m(x)(1 −K)) = λ+(−m(x)) K−1=−λ−(m(x)) K−1=λ−(m(x)) 1−K=λ−. µλ λ C C || . || + + − − Figu e 3. Bi u ca ion diag am o (12) and K < 1. 2.4. P oo o Theo em 1.1: Be o e p o ing he esul , we gene alize a well-known esul o m≡1. The p oo is coming om [8]. Lemma 2.3. Assume ha is a egula unc ion and (0) = 0. Le u0be a posi i e solu ion o (1) such ha (u0)>0, i holds: May 13, 2004 10:52 WSPC/T im Size: 9in x 6in o P oceedings mawhinsua ez 9 (1) I 00(u0)<0, hen u0is linea ly asymp o ically s able. (2) I 00(u0)>0, hen u0is uns able. P oo : We ha e o calcula e he sign o he eigen alue σ1[−∆− λm(x) 0(u0)]. Take ψ:= (u0)>0, hen (−∆−λm(x) 0(u0))ψ=− 00(u0)|∇u0|2. So, i is conca e ( esp. con ex) he unc ion ψis a supe solu ion ( esp. subsolu ion) o −∆−λm(x) 0(u0), and hen (see [23]) σ1[−∆− λm(x) 0(u0)] >0 ( esp. <0). ¦ The ollowing esul is p o ed in Theo em 3.4 o [3] and p o ides us wi h a p io i bounds o he posi i e solu ions o (1). Lemma 2.4. Assume (6). I (λ, u)is a posi i e solu ion o (1) and λ∈J, whe e Jis a compac subse such ha J⊂(0,∞), hen he e exis s a posi i e cons an C(independen om λ) such ha kuk∞≤C. Finally, he ollowing esul is p o ed in [17]. Lemma 2.5. Assume ha Σ⊂I×C2 0(Ω),I⊂IR an in e al, is a con- nec ed se o posi i e solu ions o (1). Conside u:I7→ C2 0(Ω) a con inuous map o supe solu ion o each λ∈I, bu no a solu ion. I u0< u(λ0) o some (λ0, u0)∈Σ, hen u < u(λ) o all (λ, u)∈Σ. We a e eady o p o e he esul . By subsec. 2.1 we know ha he e exis s bi u ca ion om he i ial solu ion a λ=λ+o λ=λ−when K < 1 o K > 1, espec i ely. Mo eo e , we can apply Theo em 6.4.3 o [25], and conclude ha om λ=λ+o λ=λ−bi u ca es an unbounded con inuum Co posi i e solu ions o (1). We would like o ema k ha he a de ailed p oo ha Cis unbounded and i does no sa is y he o he al e na i es o he abo e men ioned esul will be p esen ed elsewhe e. Now assume = 1and K < 1. I is clea ha u:= √1−K is a supe solu ion o (1). So, we can apply Lemma 2.5 ( aking λ0=λ+) and conclude ha o all (λ, uλ)∈ C, we ha e ha uλ<√1−K. (13) Mo eo e , 1(uλ)>0 and 00 1(uλ)<0, and so by Lemma 2.3 we ge ha uλis linea ly asymp o ically s able.