In this paper we prove the existence of infinitely many solutions for a mixed boundary value problem involving the one dimensional p-Laplacian. A result on the existence of three solutions is also established. The approach is based on multiple critical points theorems.<br /

Averna, D.

Buccellato, S.

Tornatore, E.

AVERNA, Diego

TORNATORE, Elisabetta

Buccellato,SM

Archivio istituzionale della ricerca - Università di Palermo

LE MATEMATICHEVol. LXVI (2011) – Fasc. I, pp. 93–104doi: 10.4418/2011.66.1.9ON AMIXED BOUNDARY VALUE PROBLEMINVOLVING THE p-LAPLACIAND. AVERNA - S. M. BUCCELLATO - E. TORNATOREIn this paper we prove the existence of infinitely many solutions fora mixed boundary value problem involving the one dimensional p-Lapla-cian. A result on the existence of three solutions is also established. Theapproach is based on multiple critical points theorems.1. IntroductionWe want to study the following mixed boundary value problem involving theone-dimensional p-Laplacian{ −(r|u′|p−2u′)′+ s|u|p−2u = λ f (t,u) in I =]a,b[u(a) = u′(b) = 0 (RDλ )where p > 1, λ is a positive parameter, f : [a,b]×R→ R is a L2-Carathe´odoryfunction and r,s ∈ L∞([a,b]) such thatr0 := ess inft∈[a,b]r(t)> 0, s0 := ess inft∈[a,b]s(t)≥ 0.In this paper we generalize the results obtained in [1] and [6] with p = 2.Our main tool to investigate the existence of infinitely many solutions formixed boundary value problems is the infinitely many critical points theoremdue to Ricceri ([7]). Here, we recall it as given in [3].Entrato in redazione: 2 dicembre 2010AMS 2010 Subject Classification: 34B15.Keywords: Critical points, Mixed boundary value problem, p-Laplacian.94 DIEGO AVERNA - STEFANIA M. BUCCELLATO - ELISABETTA TORNATORETheorem 1.1. (see [7, Theorem 2.5] and [3, Theorem 2.1]) Let X be a reflex-ive Banach space, Φ : X → R be a continuously Gaˆteaux differentiable, coer-cive and sequentially weakly lower semicontinuous functional, Ψ : X → R besequentially weakly upper semicontinuous and continuously Gaˆteaux differen-tiable functional.Put, for each r > infX Φϕ(r) := infu∈Φ−1(]−∞,r[)supv∈Φ−1(]−∞,r[)Ψ(v)−Ψ(u)r−Φ(u) , (1)γ := liminfr→+∞ ϕ(r), δ := liminfr→(infX Φ)+ϕ(r).One has(α) For every r > infX Φ and every λ ∈]0, 1ϕ(r)[, the restriction of the func-tional Φ− λΨ to Φ−1(]−∞,r[) admits a global minimum, which is acritical point (local minimum) of Φ−λΨ in X.(β ) If γ < ∞ then, for each λ ∈]0, 1γ [, the following alternative holds: either(β1) Φ−λΨ possesses a global minimum, or(β2) there is a sequence {un} of critical points (local minima) of Φ−λΨsuch that limn→+∞Φ(un) = +∞.(ζ ) If δ <+∞ then, for each λ ∈]0, 1δ [, the following alternative holds: either(ζ1) there is a global minimum ofΨwhich is a local minimum ofΦ−λΨ,or(ζ2) there is a sequence {un} of pairwise distinct critical points (localminima) of Φ− λΨ, with limn→+∞Φ(un) = infX Φ which weaklyconverges to a global minimum of Φ.Now, we recall a result which ensures the existence of three critical points;the result has been obtained in [5], it is a more precise version of theorem 3.2 of[4]Theorem 1.2. (see [5, Theorem 3.6] ) Let X be a reflexive real Banach space,Φ : X → R be a continuously Gaˆteaux differentiable, coercive and sequentiallyweakly lower semicontinuous functional whose Gaˆteaux derivative admits acontinuous inverse on X∗, Ψ : X → R be a continuously Gaˆteaux differentiablefunctional whose Gaˆteaux derivative is compact. Assume thatΦ(0) =Ψ(0) = 0and that there exist r ∈ R and u¯ ∈ X, with 0 < r <Φ(u¯), such thatON A MIXED BOUNDARY VALUE PROBLEM INVOLVING THE p-LAPLACIAN 95(a1)supΦ(u)≤r Ψ(u)r <Ψ(u¯)Φ(u¯) ;(a2) for each λ ∈Λr :=]Φ(u¯)Ψ(u¯) ,rsupΦ(u)≤r Ψ(u)[the functional Φ−λΨ is coercive.Then, for each λ ∈ Λr, the functional Φ−λΨ has at least three distinct criticalpoints in X.2. PreliminariesNow, consider problem (RDλ ).We recall that a function f : [a,b]×R→R is a L2-Carathe´odory function if(·) t→ f (t,x) is measurable for every x ∈ R;(·) x→ f (t,x) is continuous for every t ∈ [a,b];(·) for every ρ > 0, sup|x|≤ρ | f (t,x)| ∈ L2([a,b]).PutF(t,x) :=∫ x0f (t,ξ )dξ ∀(t,x) ∈ [a,b]×R,k :=2(p+1)r02p(p+1)||r||∞+(p+2)(b−a)p||s||∞ , (2)where||r||∞ := esssupt∈[a,b]r(t), ||s||∞ := esssupt∈[a,b]s(t)A := liminfξ→+∞∫ ba max|x|≤ξ F(t,x)dtξ p, B := limsupξ→+∞∫ ba+b2F(t,ξ )dtξ p, (3)λ1 :=r0p(b−a)p−1kB , λ2 :=r0p(b−a)p−1A . (4)where we suppose λ1 = 0 if B = ∞, and λ2 =+∞ if A = 0.Denote by X the Sobolev space {u ∈W 1,p([a,b]), u(a) = 0} endowed withthe following norm||u|| :=(∫ bar(t)|u′(t)|pdt+∫ bas(t)|u(t)|pdt) 1p.We observe that the norm || · || is equivalent to the usual one.96 DIEGO AVERNA - STEFANIA M. BUCCELLATO - ELISABETTA TORNATOREA function u : [a,b]→ R is said a generalized solution to problem (RDλ ) ifu ∈C1([a,b]), r|u′|p−2u′ ∈ AC([a,b]), u(a) = u′(b) = 0, and−(r(t)|u′(t)|p−2u′(t))′+ s(t)|u(t)|p−2u(t) = λ f (t,u)for almost every t ∈ I =]a,b[.A function u ∈ X is said a weak solution of problem (RDλ ) if∫ bar(t)|u′(t)|p−2u′(t)v′(t)dt+∫ bas(t)|u(t)|p−2u(t)v(t)dt == λ∫ baf (t,u(t))v(t)dt ∀v ∈ X .Standard methods show that generalized solutions to problem (RDλ ) coincideswith weak ones when f is a L2-Caratheodory function.It is well known that (X , || · ||) is compactly embedded in (C0([a,b]), || · ||∞)and one has||u||∞ ≤ p√(b−a)p−1r0||u|| ∀u ∈ X . (5)In order to study problem (RDλ ), we will use the functionals Φ,Ψ : X →Rdefined by puttingΦ(u) :=1p||u||p, Ψ(u) :=∫ baF(t,u(t))dt ∀u ∈ X , (6)Φ is continuous and convex, hence it is weakly sequentially lower semicontinu-ous. Moreover Φ is continuously Gaˆteaux differentiable and its Gaˆteaux deriva-tive admits a continuous inverse. On the other hand, Ψ is Gaˆteaux differentiablewith compact derivative and one hasΦ′(u)(v) =∫ bar(t)|u′(t)|p−2u′(t)v′(t)dt+∫ bas(t)|u(t)|p−2u(t)v(t)dt,Ψ′(u)(v) =∫ baf (t,u(t))v(t)dt ∀v ∈ X ,moreoverΦ(0) =Ψ(0) = 0.A critical point for the functional Φ−λΨ is any u ∈ X such thatΦ′(u)(v)−λΨ′(u)(v) = 0 ∀v ∈ X .We can observe that each critical point for functional Φ−λΨ is a general-ized solution for problem (RDλ ).ON A MIXED BOUNDARY VALUE PROBLEM INVOLVING THE p-LAPLACIAN 973. Main resultsOur main results are the following theoremsTheorem 3.1. Assume that(h1)∫ a+b2a F(t,ξ )dt ≥ 0 ∀ξ ≥ 0,(h2) liminfξ→+∞∫ ba max|x|≤ξ F(t,x)dtξ p< k limsupξ→+∞∫ ba+b2F(t,ξ )dtξ pwhere k is given by (2).Then, for each λ ∈]λ1,λ2[, where λ1, λ2 are given by (4), the problem (RDλ )has a sequence of weak solutions which is unbounded in X.Proof. Our goal is to apply Theorem 1.1. Consider the Sobolev space X and theoperators defined in (6). Pick λ ∈]λ1,λ2[.By using (3), let {cn} be a real sequence such that limn→+∞ cn =+∞ andlimn→+∞∫ ba max|ξ |≤cn F(t,ξ )dtc2n= A.Put rn = r0p(b−a)p−1 cpn for all n ∈ N, taking into account (5), one has ||v||∞ ≤ cnfor all v ∈ X such that ||v||p ≤ prn.Hence, for all n ∈ N, one hasϕ(rn) = infu∈Φ−1(]−∞,rn[)supv∈Φ−1(]−∞,rn[)Ψ(v)−Ψ(u)rn−Φ(u)≤ sup||v||p<prn∫ ba F(t,v(t))dtrn≤∫ ba max|ξ |≤cn F(t,ξ )dtrn=p(b−a)p−1r0∫ ba max|ξ |≤cn F(t,ξ )dtcpntherefore, since from (h2) one has A < ∞, we obtainγ := liminfn→∞ ϕ(rn)≤p(b−a)p−1r0A < ∞.Now, we claim that the functional Iλ =Φ−λΨ is unbounded from below.98 DIEGO AVERNA - STEFANIA M. BUCCELLATO - ELISABETTA TORNATOREBy using (3), let {dn} be a real sequence such that limn→∞ dn =+∞ andlimn→+∞∫ ba+b2F(t,dn)dtdpn= B. (7)For all n ∈ N defineωn(t) :={ 2dnb−a(t−a) if t ∈ [a, a+b2 [dn if t ∈ [ a+b2 ,b].Clearly ωn ∈ X and||ωn||p ≤ dpn2(p+1)(b−a)p−1 (2p(p+1)||r||∞+(p+2)(b−a)p||s||∞) (8)thereforeΦ(ωn)−λΨ(ωn) = 1p ||ωn||p−λ∫ baF(t,ωn(t))dt (9)≤ dpn2(p+1)(b−a)p−1 (2p(p+1)||r||∞+(p+2)(b−a)p||s||∞)−λ∫ baF(t,ωn(t))dt.Taking into account (h1), we have∫ baF(t,ωn(t))dt ≥∫ ba+b2F(t,dn)dt. (10)Then, for all n ∈ NΦ(ωn)−λΨ(ωn)≤ dpn2(p+1)(b−a)p−1 (2p(p+1)||r||∞+ (11)(p+2)(b−a)p||s||∞)−λ∫ ba+b2F(t,dn)dt =dpn r0p(b−a)p−1k −λ∫ ba+b2F(t,dn)dt.Now, if B < ∞, we fix ε ∈]r0pλ (b−a)p−1kB ,1[, from (7) there exists νε ∈ N suchthat ∫ ba+b2F(t,dn)dt > εBdpn ∀n > νεthereforeΦ(ωn)−λΨ(ωn)≤[r0p(b−a)p−1k −λεB]dpn ∀n > νεON A MIXED BOUNDARY VALUE PROBLEM INVOLVING THE p-LAPLACIAN 99by the choice of ε , one haslimn→∞[Φ(ωn)−λΨ(ωn)] =−∞.On the other hand, if B =+∞, we fixM >r0pλ (b−a)p−1k ,from (7) there exists νM ∈ N such that∫ ba+b2F(t,dn)dt > Mdpn ∀n > νMthereforeΦ(ωn)−λΨ(ωn)≤[r0p(b−a)p−1k −λM]dpn ∀n > νMby the choice of M, one haslimn→∞[Φ(ωn)−λΨ(ωn)] =−∞.Hence, our claim is proved.Since all assumptions of Theorem 1.1 are verified, the functional Iλ = Φ−λΨadmits a sequence {un} of critical points such that limn→∞ ||un|| = +∞ and theconclusion is achieved.Now, we point out the following consequence of Theorem 3.1Corollary 3.2. Let f : R→ R be a nonnegative continuous function,s ∈C0([a,b]), r ∈C1([a,b]) and put F(x) = ∫ x0 f (ξ )dξ ∀x ∈ R. Assume thatliminfξ→+∞F(ξ )ξ p<k2limsupξ→+∞F(ξ )ξ p.Then, for eachλ ∈ 2r0pk(b−a)p limsupξ→+∞ F(ξ )ξ p,r0p(b−a)p liminfξ→+∞ F(ξ )ξ pthe problem { −(r|u′|p−2u′)′+ s|u|p−2u = λ f (u) in I =]a,b[u(a) = u′(b) = 0possesses a sequence of pairwise distinct classical solutions.100 DIEGO AVERNA - STEFANIA M. BUCCELLATO - ELISABETTA TORNATORERemark 3.3. In Theorem 3.1 we can replace ξ → +∞ by ξ → 0+, applyingin the proof part (ζ ) of Theorem 1.1 instead of (β ). In this case a sequence ofpairwise distinct solutions to the problem (RDλ ) which converges uniformly tozero is obtained.Example 3.4. Putan :=2n!(n+2)!−14(n+1)!, bn :=2n!(n+2)!+14(n+1)!for every n ∈ N, and define the nonnegative continuous function f : R→ R asfollowsf (ξ ) ={32(n+1)!2[(n+1)!p−n!p]pi√116(n+1)!2 −(ξ − n!(n+2)2)2 if ξ ∈⋃n∈N[an,bn]0 otherwise.By a simple computation, we obtain liminfξ→+∞F(ξ )ξ p= 0, limsupξ→+∞F(ξ )ξ p= 2p,so0 = liminfξ→+∞F(ξ )ξ p<2(p+1)e(2p(p+1)+(p+2))=(p+1)2pe(2p(p+1)+(p+2))limsupξ→+∞F(ξ )ξ p.Hence, from Corollary 3.2, for each λ > 2p(p+1)+(p+2)2p p(p+1) , the problem{ −(e−t |u′|p−2u′)′+ e−t |u|p−2u = λ f (u) in ]0,1[u(0) = u′(1) = 0has a sequence of pairwise distinct generalized solutions.Now we point out the following result of three generalized solutions.Theorem 3.5. Assume that there exist three positive constants c, d and ν withc < d, ν < p and a function µ ∈ L1([a,b]) such that(i1)∫ a+b2a F(t,ξ )dt > 0 ∀ξ ∈ [0,d],(i2)∫ ba max|ξ |≤c F(t,ξ )dtcp < k∫ ba+b2F(t,d)dtdpwhere k is given by (2),(i3) F(t,ξ )≤ µ(t)(1+ |ξ |ν)∀t ∈ [a,b] ∀ξ ∈ R.ON A MIXED BOUNDARY VALUE PROBLEM INVOLVING THE p-LAPLACIAN 101Then, for eachλ ∈ ] r0dppk(b−a)p−1 ∫ ba+b2F(t,d)dt,r0cpp(b−a)p−1 ∫ ba max|ξ |≤c F(t,ξ )dt[the problem (RDλ ) has at least three generalized solutions.Proof. Our goal is to apply Theorem 1.2. Consider the Sobolev space X and theoperators defined in (6).Now, we claim that (i2) ensures (a2) of Theorem 1.2. In fact, setr = r0cpp(b−a)p−1 and consider the function u¯ ∈ X defined by puttingu¯(t) :={ 2db−a(t−a) if t ∈ [a, a+b2 [d if t ∈ [ a+b2 ,b].(12)We observe thatpΦ(u¯) =2pdp(b−a)p∫ a+b2ar(t)dt+2pdp(b−a)p∫ a+b2a(t−a)2s(t)dt+dp∫ ba+b2s(t)dtfrom 0 < c < d by using the previous relation and (2) we have0 < r <Φ(u¯)<r0dppk(b−a)p−1 .In virtue of (i1) we haveΨ(u¯)≥∫ ba+b2F(t,d)dt.Therefore, one hasΨ(u¯)Φ(u¯)≥ p(b−a)p−1kr0dp∫ ba+b2F(t,d)dt. (13)From (5) if Φ(u)≤ r, we have maxt |u(t)| ≤ c thereforesupΦ(u)≤rΨ(u)≤∫ bamax|ξ |≤cF(t,ξ )dt. (14)Hence, owing to (13), (14) and (i2) condition (a1) of Theorem 1.1 is verified.We prove that the operator Φ−λΨ is coercive, in fact, for each u ∈ X , byusing (i3) one hasΦ(u)−λΨ(u) = 1p||u||p−λ∫ baF(t,u(t))dt ≥102 DIEGO AVERNA - STEFANIA M. BUCCELLATO - ELISABETTA TORNATORE1p||u||p−λ∫ baµ(t)(1+ |u(t)|ν)dt ≥1p||u||p−λ∫ baµ(t)dt−λ∫ baµ(t)|u(t)|νdtand, by using (5)∫ ba|µ(t)||u(t)|νdt ≤ ||u||ν∞∫ ba|µ(t)|dt ≤((b−a)p−1r0) νp||u||ν ||µ||1,we haveΦ(u)−λΨ(u)≥ 1p||u||p−λ ||µ||1−λ((b−a)p−1r0) νp||u||ν ||µ||1 (15)hence condition (a2) of Theorem 1.2 is verified. All assumptions of Theorem1.2 are satisfied and the proof is complete.Now, we point out the following consequence of Theorem 3.2.Corollary 3.6. Let f : R→ R be a nonnegative continuous function and r ∈C1([a,b]), s ∈C0([a,b]) . Assume that there exist positive constants a, c, d andν , with c < d and ν < p, such that(i1’)F(c)cp ≤ k2 F(d)dpwhere k is given by (2),(i2’) F(ξ )≤ a(1+ |ξ |ν) ∀ξ ∈ R.Then, for each λ ∈]2r0dppk(b−a)pF(d) ,r0cpp(b−a)pF(c)[the problem{ −(r|u′|p−2u′)′+ s|u|p−2u = λ f (u) in I =]a,b[u(a) = u′(b) = 0has at least three classical solutions.Example 3.7. The problem{ −(|u′|u′)′+(2+t3 )|u|u = λ t2[2e−t2u17(9−u2)+1] in I =]0,1[u(0) = u′(1) = 0admits at least three classical solutions for each λ ∈]128e47(218+ e4),3e2(e+1)[.In fact, if we choose, for example, c = 1 and d = 2, hypotheses of Theorem3.1 are satisfied.ON A MIXED BOUNDARY VALUE PROBLEM INVOLVING THE p-LAPLACIAN 103Remark 3.8. In ([2]), has been studied a mixed boundary problem involvingthe one dimensional p-Laplacian of type{ −(|u′|p−2u′)′+ |u|p−2u = λ f (t,u) in I =]a,b[u(a) = u′(b) = 0.We observe that our equation gives back that case with r = s = 1.REFERENCES[1] D. Averna - N. Giovannelli - E. Tornatore, Existence of three solutions for a mixedboundary value problem with the Sturm-Liouville equation, preprint.[2] D. Averna - R. Salvati, Three solutions for a mixed boundary value problem in-volving the one dimensional p-Laplacian, J. Math. Appl. 298 (2004), 245–283.[3] G. Bonanno - G. Molica Bisci, Infinitely many solutions for a boundary valueproblem with discontinuous nonlinearities, Bound. Value Probl. (2009), 1–20.[4] G. Bonanno - P. Candito, Non-differentiable functionals and applications to el-liptic problems with discontinuous nonlinearities, J. Differential Equations 244(2008), 3031–3059.[5] G. Bonanno - S. A. Marano, On the structure of the critical set of non-diffe-rentiable functions with a weak compactness condition, Applicable Analysis 89(1) (2010), 1–10.[6] G. Bonanno - E. Tornatore, Infinitely many solutions for a mixed boundary valueproblem, Ann. Polon. Math. 99 (2010), 285–293.[7] B. Ricceri, A general variational principle and some of its applications, J. Com-put. Appl. Math. 113 (2000), 401–410.[8] E. Zeidler, Nonlinear functional analysis and its applications Vol. III, Springer,Berlin, 1990.104 DIEGO AVERNA - STEFANIA M. BUCCELLATO - ELISABETTA TORNATOREDIEGO AVERNADipartimento di Matematica ed InformaticaFacolta` di ScienzeUniversita` degli Studi di PalermoVia Archirafi 23, 90143 - Palermo (Italy)e-mail: averna@math.unipa.itSTEFANIA M. BUCCELLATODipartimento di Metodi e Modelli MatematiciFacolta` di IngegneriaUniversita` degli Studi di PalermoViale delle Scienze, 90100 - Palermo (Italy)e-mail: stefania maria.buccellato@unipa.itELISABETTA TORNATOREDipartimento di Metodi e Modelli MatematiciFacolta` di IngegneriaUniversita` degli Studi di PalermoViale delle Scienze, 90100 - Palermo (Italy)e-mail: elisa@math.unipa.it

On a mixed boundary value problem involving the p-Laplacian

In this paper we prove the existence of infinitely many solutions for a mixed boundary value problem involving the one dimensional p-Laplacian. A result on the existence of three solutions is also established. The approach is based on multiple critical points theorems

Averna, Diego

Buccellato, Stefania Maria

Tornatore, Elisabetta

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

On a mixed boundary value problem involving the  p-Laplacian

Diego Averna

Stefania Maria Buccellato

Elisabetta Tornatore

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On a mixed boundary value problem involving the p-Laplacian

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