An eigenvalue problem for non-bounded quasi-linear operator

P oceedings o he Edinbu gh Ma hema ical Socie y Submi ed Pape Pape 30 Ma ch 2004 AN EIGENVALUE PROBLEM FOR NON-BOUNDED QUASI-LINEAR OPERATOR Jos´e Ca mona1and An onio Su´a ez2 1Dp o. de ´ Algeb a y An´alisis Ma em´a ico, Facul ad de Ciencias, Ca˜nada de San U bano, Alme ´ıa, Spain e-mail: jc[email p o ec ed] 2Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico, Facul ad de Ma em´a icas, Se illa, Spain e-mail: sua [email p o ec ed] (Recei ed ) Abs ac In his pape we s udy he eigen alues associa ed wi h a posi i e eigen unc ion o a quasi- linea ellip ic p oblem wi h a no necessa ily bounded ope a o . Fo ha , we use he bi u ca ion heo y and ob ain he exis ence o posi i e solu ion o a ange o alues o he bi u ca ion pa ame e . AMS 2000 Ma hema ics subjec classi ica ion: P ima y 35J60, 35J25 Seconda y 35D05 1. In oduc ion Le Ω be a bounded open subse o RNwi h su icien ly smoo h bounda y ∂Ω and le A(x, s) be a eal symme ic ma ix which coe icien s, aij : Ω×R+ 0→R, a e Ca a h´eodo y unc ions. We assume ha he e exis s a posi i e cons an αsa is ying o e e y (x, s, ξ)∈Ω× R+×RN, A(x, s)ξ·ξ≥α|ξ|2.(A1) In his pape we analyze he nonlinea eigen alue p oblem (−di (A(x, u)∇u) = λu, x ∈Ω, u= 0, x ∈∂Ω,(Pλ) whe e, we say ha λis an eigen alue o his p oblem i (Pλ) admi s a posi i e and non i ial solu ion, ha is, i he e exis s u∈H1 0(Ω), u≥0, u6≡ 0, such ha A(x, u)∇u∈ (L2(Ω))Nand ZΩ A(x, u)∇u· ∇ =λZΩ u , ∀ ∈H1 0(Ω). 1 2J. Ca mona and A. Su´a ez In addi ion o he in e es i sel in he s udy o (Pλ), his kind o equa ion has been used o model a species inhabi ing in Ω whe e i s di usion depends on he densi y o he species, which a ises in mo e ealis ic models, see [3] and e e ences he ein. P oblem (Pλ) is well known when Adoes no depend on s, i.e., when A(x, s) = B(x) wi h B= (bij) and bij ∈L∞(Ω), bij ≥b0>0 in Ω. In his case, he e exis s he p incipal eigen alue, deno ed by λ1(B), o he p oblem: (−di (B(x)∇u) = λu, x ∈Ω, u= 0, x ∈∂Ω,(1.1) being he unique eigen alue wi h a posi i e eigen unc ion, see o ins ance [5]. In [2], assuming ha Asa is ies (A1) and |A(x, s)| ≤ β, o each (x, s)∈Ω×R, (A2) he au ho p o ed ha o each > 0, he e exis s λ >0 and a posi i e solu ion u ∈H1 0(Ω), o (Pλ ) such ha ku k2= . Mo eo e , deno ing by λ0:= λ1(A(x, 0)), he showed ha i →0, hen λ →λ0and u con e ges o a posi i e eigen unc ion associa ed o λ0in H1 0(Ω). Finally, i Aalso e i ies lim s→∞ A(x, s) = A∞(x),uni o mly in x∈Ω,(A3) hen λ →λ∞and u goes o a posi i e eigen unc ion associa ed o λ∞in H1 0(Ω) as → ∞, whe e λ∞:= λ1(A∞(x)). In [4], a sligh ly modi ica ion o (Pλ) is analyzed. Unde condi ions (A1−3), λu +h(x) o some 0 ≤h∈L2(Ω) is conside ed ins ead o λu. Bu he a gumen s used o p o e he exis ence o solu ion leads o he i ial one in he case h≡0. In [1], assuming in addi ion he exis ence o an Osgood unc ion ω:R+ 0→Rsuch ha |A(x, s1)−A(x, s2)| ≤ ω(|s1−s2|),(A4) o e e y (x, s1),(x, s2)∈Ω×R, using a bi u ca ion analysis, he au ho s s udy a mo e gene al p oblem (−di (A(x, u)∇u) = (λ, x, s), x ∈Ω, u= 0, x ∈∂Ω, o :R×Ω×R7→ Rand Asa is ying (A1−4). In he pa icula case (λ, x, s) = λs, om hei esul s i can be deduced he exis ence o an unbounded con inuum (closed and connec ed subse ) o posi i e solu ions bi u ca ing om he i ial solu ion a λ=λ0and Non-bounded quasi-linea ope a o 3 mee ing wi h in ini y a he alue λ=λ∞. Thus, as a consequence, he e exis s posi i e solu ion o (Pλ) o λ∈(λ0, λ∞) o (λ∞, λ0). In he ollowing sec ion we comple e his s udy o Asa is ying (A1−4) by gi ing su icien condi ions o he uniqueness o posi i e solu ion. The main goal o his wo k (see Sec ion 3) is o analyze (Pλ) when Ais no neces- sa ily bounded and/o does no sa is y (A3). In his case, we show ha he e exis s an unbounded con inuum o posi i e solu ions bi u ca ing om he i ial one a λ=λ0. I , in addi ion he e exis s a con inuous unc ion g:R+ 0→R, wi h lim s→+∞g(s)=+∞, sa is ying o e e y (x, s, ξ)∈Ω×R+×RN, A(x, s)ξ·ξ≥g(s)|ξ|2≥α|ξ|2.(A∞) hen, he bi u ca ion om in ini y a λ=λ∞(which exis s in he bounded case) “dis- appea s”. Speci ically, he e exis s a leas a posi i e solu ion uλ o λ∈(λ0,∞) and kuλk→∞as λ→ ∞. Howe e , i Ais bounded in a subse o Ω, hen again a bi u ca- ion o in ini y exis s. Along he wo k we will use he ollowing no a ion: •H1 0(Ω) and E=C0(Ω) a e he usual Sobole space and he space o he con inuous unc ions in Ω anishing on ∂Ω endowed wi h he no ms kuk=k∇uk2and kuk0= supΩ|u|, espec i ely. •cl(D) deno es he closu e o he se D. • S deno es he se S= cl{(λ, u)∈R×E:uis solu ion o (Pλ), u ≥0, u 6≡ 0}. Any con inuum subse o Swill be called a con inuum o posi i e solu ions o (Pλ), al hough i may con ain he i ial solu ion (λ, 0) o some alue o λ > 0. •Iwill deno e bo h he iden i y ma ix and he iden i y ope a o . •Gi en squa e ma ices B1, B2we say ha B1>0 ( espec . B1≥0) i he quad a ic o m induced by B1is de ini e posi i e ( espec . semide ini e posi i e). We say ha B1< B2( espec . B1≤B2) i B2−B1>0 ( espec . B2−B1≥0). •The map P ojR:R×E7→ Rs ands o he p ojec ion o he p oduc space R×E on o R. 2. The case o bounded ma ices A In o de o s udy p oblem (Pλ), le us ecall ha , o ma ices Asa is ying (A1,2), i u∈H1 0(Ω) is solu ion o (Pλ) hen using he De Gio gi-S ampacchia Theo em ([8, Th´eo `eme 7.3] and [6, Theo em I] o [7, Theo em 8.29]), u∈C0,γ(Ω) o some 0 < γ < 1. Mo eo e , i he coe icien s o he ma ix Asa is y aij ∈C1,γ0(Ω ×R), o some 0 < γ0<1,(2.1) 4J. Ca mona and A. Su´a ez hen by Theo em 15.17 in [7] we ha e ha u∈C2,γγ0 0(Ω). We also ecall ha o e e y (λ, u)∈ S wi h u∈ C1(Ω) and u6≡ 0, using he Hop maximum p inciple, we ha e ha u > 0 in Ω and he no mal ex e io de i a i e ∂u ∂neis nega i e in ∂Ω. The ollowing lemma p o ides us necessa y condi ions in λ∈R o which (Pλ) admi s solu ion in some special cases. Lemma 2.1. Assume (A1,3)and ha (Pλ)admi s a posi i e solu ion. Then 1. λ0≤λ( espec . <, ≥, >) i o e e y s∈R+,A(x, 0) ≤A(x, s)( espec . <, ≥, >). 2. λ∞≥λ( espec . >, ≤, <) i o e e y s∈R+,A∞(x)≥A(x, s)( espec . >, ≤, <). P oo . The esul ollows om he ac ha o gi en symme ic ma ices B1(x), B2(x) o which he e exis λ1(B1) and λ1(B2), wi h 0 < B1≤B2 hen λ1(B1) = in ½ZΩ B1(x)∇u· ∇u, u ∈H1 0(Ω),kuk2= 1¾≤λ1(B2). Thus, i u∈H1 0(Ω) is a solu ion o (Pλ), we conclude by aking in o accoun ha λ=λ1(A(x, u)). u The main esul o his sec ion is he ollowing: Theo em 2.2. Assume (A1−4). We ha e ha λ0and λ∞a e he only bi u ca ion poin s om he i ial solu ion and om in ini y, espec i ely, and he e exis s a con- inuum Σ⊂ S o posi i e solu ions mee ing (λ0,0) and (λ∞,∞), in pa icula , (Pλ) possesses a posi i e solu ion o e e y λ∈(λ0, λ∞)o λ∈(λ∞, λ0). Mo eo e , • he bi u ca ion om λ0is subc i ical ( esp. supe c i ical) i he e exis s s0>0such ha A(x, s)< A(x, 0),( espec . A(x, s)> A(x, 0)),∀s∈(0, s0), • he bi u ca ion om λ∞is subc i ical ( esp. supe c i ical) i A(x, s)< A∞(x),( esp. A(x, s)> A∞(x)),∀s∈R+. Fu he mo e, •i A(x, 0) < A(x, s)< A∞(x) o e e y s∈R+, hen he e exis s non i ial solu ion o (Pλ)i , and only i , λ∈(λ0, λ∞), in pa icula P ojRΣ = [λ0, λ∞). I , in addi ion, A(x, s)is inc easing in sand i e i ies (2.1), he solu ion is unique. •I A(x, 0) > A(x, s)> A∞(x) o e e y s∈R+, hen he e exis s non i ial solu ion o (Pλ)i , and only i , λ∈(λ∞, λ0), in pa icula P ojRΣ = (λ∞, λ0]. Non-bounded quasi-linea ope a o 5 P oo . The exis ence o he con inuum Σ o posi i e solu ions ollows by Theo em 5.1 in [1], and so he exis ence o posi i e solu ions o e e y λin (λ0, λ∞) o in (λ∞, λ0). The desc ip ion P ojRΣ, in he cases A(x, 0) < A(x, s)< A∞(x) o A(x, 0) < A(x, s)< A∞(x) o e e y s∈R+, ollows di ec ly om Lemma 2.1. Mo eo e , a guing as in ha lemma we ge he la e ali y o he bi u ca ions. Now, assume ha A(x, s) is inc easing in sand (2.1) is sa is ied. In o de o p o e he uniqueness o solu ion o (Pλ), le us suppose ha he e exis λ∈(λ0, λ∞) and u1, u2∈E, solu ions o (Pλ) wi h u16≡ u2. We claim ha u1, u2can be chosen such ha u1≤u2. Indeed, his is a consequence o he exis ence o a sequence (λn, un) wi h λn→λ0and un→0 in E. In ac , by egula i y esul s, un→0 in C1(Ω). Thus, o λn< λ,unis a subsolu ion o (Pλ) and o la ge n,un≤min{u1, u2}. Then, by he sub and supe solu ion me hod, he e exi s w∈Esolu ion o (Pλ) wi h un≤w≤u1, un≤w≤u2. This implies ha w6≡ u1o w6≡ u2, and he claim is p o ed by aking u1=wand u2=ui o some i= 1,2. Now we ake =u2 2 u1as es unc ion in he equa ion sa is ied by u1and =u2in ha sa is ied by u2. Thus, sub ac ing bo h equali ies we ha e ha : 0 = ZΩ A(x, u1)∇u1· ∇ µu2 2 u1¶−ZΩ A(x, u2)∇u2· ∇u2 =−ZΩ A(x, u1)µu2 u1 ∇u1− ∇u2¶·µu2 u1 ∇u1− ∇u2¶ −ZΩ (A(x, u2)−A(x, u1)) ∇u2· ∇u2<0. This con adic ion gi es he uniqueness. u 3. The case o unbounded ma ices A In his sec ion, we s udy (Pλ) when Ais no necessa ily bounded and does no sa is y (A3). We p o e i s ly ha e e y solu ion o (Pλ) is bounded. Mo e p ecisely we ha e Lemma 3.1. Le A(x, s)sa is y (A1)and u∈H1 0(Ω) be a solu ion o (Pλ), hen u∈E. Mo eo e , he e exis posi i e cons an s c1, c2, γ1, γ2such ha kukγ1 0≤c1+c2kukγ2.(3.1) P oo . Once we know ha u∈L∞(Ω), and kukγ1 ∞≤c1+c2kukγ2 o some posi i e cons an s c1, c2, γ1, γ2, hen he esul ollows di ec ly om he De Gio gi-S ampacchia Theo em. Le us p o e he L∞(Ω)-es ima e. We conside o e e y k∈R+ he unc ion Gk:R+ 0→R+ 0gi en by Gk(s) = (0 0 ≤s≤k, s−k s > k. 6J. Ca mona and A. Su´a ez Thus, we can ake =Gk(u) as es unc ion in he weak equa ion sa is ied by uand using (A1) we ha e αk∇Gk(u)k2 2≤ZΩ A(x, u)∇u∇Gk(u)≤λZ Ωk uGk(u),(3.2) whe e Ωk≡ {x∈Ω : u(x)> k}. Using he Sobole and H¨olde inequali ies, in he case N > 2, by (3.2) we yield, o u∈L (Ω) wi h > 2∗ 2∗−1, and some posi i e cons an c, kGk(u)k2 2∗≤ckuk kGk(u)k2∗(meas Ωk)(1−1/ −1/2∗).(3.3) Taking in o accoun ha , o e e y h > k,Gk(u)≥h−kin Ωh, (3.3) implies ha (h−k)(meas Ωh)1/2∗≤ckuk (meas Ωk)(1−1/ −1/2∗), o equi alen ly meas Ωh≤ckuk2∗ (meas Ωk)2∗−1−2∗/ (h−k)2∗.(3.4) We can now apply he S ampacchia Lemma ([8, Lemma 4.1]) o deduce ha : i) i u∈L (Ω) wi h > N 2, hen u∈L∞(Ω) and kuk∞≤ckuk , ii) i u∈L (Ω) wi h =N 2, hen u∈L (Ω) o ∈[1,∞) and kuk ≤c+c0kuk , iii) i u∈L (Ω) wi h < N 2, hen u∈L (Ω) o =2∗ (2−2∗) +2∗−δand δ > 0 a bi a ily small. Mo eo e , kuk ≤c+c0kuk +δ . Since u∈L2∗(Ω) and 2∗>2∗ 2∗−1, we can a gue as be o e o 0= 2∗. Thus, i 2∗>N 2 we conclude by i em i). In he case 2∗=N 2we use i em ii) in o de o ake 1>N 2and conclude again by i em i). Finally, in he case 2∗<N 2we can ake 1=2∗ 0 (2 −2∗) 0+ 2∗−δ1> 0. As be o e, i 1≥N 2we easily conclude. In o he case we ake 2=2∗ 1 (2 −2∗) 1+ 2∗−δ2. By an i e a i e a gumen we conclude a e a ini e numbe o s eps. Indeed, in o he case, we ha e ha nis bounded, whe e nis de ined ecu en ly by    0= 2∗ n+1 =2∗ n (2 −2∗) n+ 2∗−δn+1. Non-bounded quasi-linea ope a o 7 whe e limn→∞ δn= 0. Mo eo e , nis non dec easing and so i con e ges o ∈(2∗,N 2] ha sa is ies =2∗ (2 −2∗) + 2∗, ha is, 2∗= (2 −2∗) + 2∗, which implies ha = 0 and his is a con adic ion. Obse e ha he es ima e (3.1) ollows, a e his ini e numbe o s eps, om es ima es in i ems i)-iii), and he Sobole embedding. Finally, in he case N= 2 we can choose > q q−2 o any q > 2 and a gue as be o e wi h 2∗ eplaced by q. In his case we inish by i em i). u Along his sec ion, we assume, ins ead o (A2), ha o each s0∈R+ he e exis s β(s0) such ha |A(x, s)| ≤ β(s0),(˜ A2) o (x, s)∈Ω×[0, s0]. We conside he unca ed p oblems (−di (A(x, Tn(u))∇u) = λu, x ∈Ω, u= 0, x ∈∂Ω,(Pλ,n) being Tn(s) he map de ined, o each n∈N, by Tn(s) = (s0≤s≤n, n s > n. By Theo em 2.2, he e exis Σnunbounded maximal con inua o posi i e solu ions such ha (λ0,0) ∈Σn o each n∈N. Now, we can p o e Theo em 3.2. Suppose ha Asa is ies (A1,4)and (˜ A2). Then, he e exis s an unbounded con inuum Σ⊂ S such ha (λ0,0) ∈Σ. P oo . Fi s ly, we deno e by Σn k he connec ed componen o Σk∩(R×Bn(0)) con- aining (λ0,0). We claim ha Σn k= Σn n o k≥n. (3.5) Indeed, i k≥nand (λ, u)∈Σn k hen uis solu ion o (Pλ,n). Thus, Σn kis a closed and connec ed subse o cl{(λ, u)∈R×E:uis solu ion non- i ial o (Pλ,n)} con aining (λ0,0). So, Σn k⊂Σn, whence we deduce ha Σn k⊂Σn n. We can eason simila ly and ob ain ha Σn n⊂Σk∩(R×Bn(0)), and so i ollows (3.5). So, we ge Σn n= lim kΣn k. The e o e, o each n∈Nwe ha e a con inuum Σn n⊂cl{(λ, u)∈R×E:uis a non- i ial solu ion o (Pλ)} 8J. Ca mona and A. Su´a ez con aining (λ0,0) and i (λ, u)∈Σn n hen kuk0≤n. Now, we a e going o p o e ha Σn n⊂Σn+1 n+1 o each n∈N.(3.6) Indeed, obse e ha Σn n= Σn n+1 ⊂Σn+1 ∩(R×Bn(0)) ⊂Σn+1 ∩(R×Bn+1(0)), so, since Σn+1 n+1 is he connec ed componen o Σn+1 ∩(R×Bn+1(0)) con aining (λ0,0) and Σn nis a connec ed o such subse con aining i , (3.6) ollows. Finally, we show ha he se Σ = ∞ [ n=1 Σn n sa is ies he heo em. Fi s ly, obse e ha since Σnis unbounded, Σ is also unbounded. Indeed, since P ojRΣnis bounded, so he e exis s a connec ed subse o Σn∩(R×Bn(0)) con aining (λ0,0) and in e sec ing wi h R×∂Bn(0) o each n∈N; i.e., o each n∈N he e exis s (λn, un)∈Σn n, wi h kunk0=n. On he o he hand, since Σn nis connec ed and (λ0,0) ∈Σn n o each n∈N, i ollows ha Σ is connec ed. Finally, we will p o e ha Σ is closed. Le (λ, u)∈Σ. Since Σ is connec ed, he e exis s a connec ed and bounded se Σ0⊂Σ con aining (λ0,0) and (λ, u). Thus, he e exis s n∈Nsuch ha Σ0⊂cl{(λ, u)∈R×E:kuk0≤n, u is non- i ial solu ion o (Pλ,n)}. In pa icula , Σ0⊂Σn∩(R×Bn(0)) whence Σ0⊂Σn nand so, (λ, u)∈Σn n⊂Σ. u Rema k 3.3. 1. We would like o poin ou ha he abo e esul is ue e en in he case ha he limi o A(x, s) does no exis as s→ ∞. 2. In he case Abounded in some subse o Ω, hen we can conclude ha P ojRΣ is bounded. Indeed, assume ha |A(x, s)| ≤ γi x∈B, whe e Bis a ball such ha B⊂Ω, hen using he mono ony o he p incipal eigen alue wi h espec o he domain, we ob ain λ=λ1(A(x, u)) ≤λB 1(A(x, u)) ≤λB 1(γI) = γλB 1(I). 3. In his case we can ob ain a simila esul o he main one in [2]. Indeed, o each > 0 he e exis s λ >0 and u ∈H1 0(Ω) solu ion o (Pλ) wi h kuk0= . In he nex esul we show ha when A(x, s) ends o in ini y as s→ ∞ in he sense o (A∞), hen he bi u ca ion a in ini y disappea s, in some sense λ∞→+∞when A(x, s) ends o in ini y. Non-bounded quasi-linea ope a o 9 Theo em 3.4. Assume ha Asa is ies (A4),(˜ A2)and (A∞). Then, he e exis s a con inuum Σ⊂ S such ha (λ0,0) ∈Σ. Mo eo e , he in e al (λ0,+∞)⊂P ojRΣand lim λ→+∞ (λ, uλ)∈Σ kuλk0= +∞. P oo . The exis ence o he con inuum unbounded Σ bi u ca ing om (λ0,0) ollows by Theo em 3.2. Since λ=λ1(A(x, u)) ≥λ1(αI) = αλ1(I), he e do no exis posi i e solu ions o λsmall. So, i su ices o p o e ha i is no possible bi u ca ion om in ini y. In o de o do ha we obse e ha p oblem (Pλ) can be w i en as (−di (B(x, u)g(u)∇u) = λu, x ∈Ω, u= 0, x ∈∂Ω,(Pλ) whe e gis gi en by hypo hesis (A∞) and B(x, u) := A(x, u) g(u). Mo eo e , i we pe o m he change o a iable w= ˜g(u) = Zu 0 g( )d , p oblem (Pλ) is equi alen o (−di (C(x, w)∇w) = λ (w), x ∈Ω, w= 0, x ∈∂Ω,(Qλ) whe e C(x, w) := B(x, ˜g−1(w)) and (w) := ˜g−1(w). Now we a gue by con adic ion, and assume ha he e exis s a sequence o solu ions (λn, un) o (Pλn) such ha λn→λ > 0 and kunk0→ ∞. Then, by (3.1) we ha e ha kunk→∞and aking wn= ˜g(un), i is clea ha kwnk0→ ∞. In addi ion, since (A∞) implies ha α2kunk2≤ kwnk2, we also ha e ha kwnk → ∞. Fo he no malized sequence zn:= wn kwnkwe know he exis ence o z∈H1 0(Ω), such ha zn→zs ongly in L2(Ω), and a.e. in Ω. and so, aking wn/kwnk2as a es unc ion in (Qλn), we ob ain ha α≤ZΩ C(x, wn)∇zn· ∇zn=λnZΩ (wn) kwnkzn.(3.7) Now, aking in o accoun ha (s) s→0 as s→ ∞,