The aim of this paper is to establish the existence of at least one non-trivial solution for Neumann quasi-linear problems. Our approach is based on variational methods

Pizzimenti, Pasquale F.

Sciammetta, Angela

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

LE MATEMATICHEVol. LXVI (2011) – Fasc. I, pp. 163–171doi: 10.4418/2011.66.1.16EXISTENCE RESULTS FOR A QUASI-LINEARDIFFERENTIAL PROBLEMPASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTAThe aim of this paper is to establish the existence of at least onenon-trivial solution for Neumann quasi-linear problems. Our approachis based on variational methods.1. IntroductionThe aim of this paper is to ensure the existence of at least one non-trivial solutionfor the following Neumann boundary value problem{ −u′′+uh(u′) = λα(x) f (u)h(u′)u′(a) = u′(b) = 0, (Nλ )where α : [a,b]→R is a positive continuous function, f :R→R and h :R→Rare continuous functions and λ is a positive real parameter.Existence and multiplicity of solutions for Neumann boundary value prob-lems have been investigated by several authors and, for an overview on thissubject, we refer to [1], [3] - [6], [8], [9], [11] - [14].The main result of this paper is Theorem 3.1, which generalizes [6, Theo-rem 3.1] to the case where the nonlinear term is not constant with respect to u′.Two relevant consequences of Theorem 3.1 (that is, Corollary 3.2 and TheoremEntrato in redazione: 20 dicembre 2010AMS 2010 Subject Classification: 34B15; 34B18.Keywords: Neumann problem, Positive solutions, Critical points.164 PASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTA3.3) are also pointed out. Here, as an example, we presented a special case ofour main result.Theorem 1.1. Let α : [a,b] → R be a nonnegative continuous function andf ,h : R→ R be continuous functions. Suppose that h is bounded and strictlypositive, and thatlimξ→0+f (ξ )ξ=+∞.Then, there exists λ ∗ such that, for each λ ∈ ]0,λ ∗[, the problem (Nλ ) admits atleast one positive classical solution.Our approach is based on a critical point theorem obtained in [2] (see The-orem 2.1).The paper is arranged as follows: in Section 2, we recall some basic defini-tions and our main tool, while Section 3 is devoted to our main results.2. Preliminaries and basic notationsOur main tool is the Ricceri variational principle [10, Theorem 2.5] as given in[2, Theorem 5.1] which is below recalled (see also [2, Proposition 2.1] and [7,Theorem 2.1]). First, given Φ,Ψ : X → R, putβ (r1,r2) = infv∈Φ−1(]r1,r2[)supu∈Φ−1(]r1,r2[)Ψ(u)−Ψ(v)r2−Φ(v) (1)andρ2(r1,r2) = supv∈Φ−1(]r1,r2[)Ψ(v)− supu∈Φ−1(]−∞,r1])Ψ(u)Φ(v)− r1 , (2)for all r1,r2 ∈ R, with r1 < r2.Theorem 2.1. ([2, Theorem 5.1]) Let X be a reflexive real Banach space; Φ :X → R be a sequentially weakly lower semicontinuous, coercive and continu-ously Gaˆteaux differentiable function whose Gaˆteaux derivative admits a con-tinuous inverse on X∗; Ψ : X → R be a continuously Gaˆteaux differentiablefunction whose Gaˆteaux derivative is compact. Put Iλ = Φ−λΨ and assumethat there are r1,r2 ∈ R, with r1 < r2, such thatβ (r1,r2)< ρ2(r1,r2), (3)where β and ρ2 are given by (1) and (2).Then, for each λ ∈]1ρ2(r1,r2),1β (r1,r2)[there is u0,λ ∈Φ−1(]r1,r2[) such thatIλ (u0,λ )≤ Iλ (u) for all u ∈Φ−1(]r1,r2[) and I′λ (u0,λ ) = 0.EXISTENCE RESULTS FOR A QUASI-LINEAR DIFFERENTIAL PROBLEM 165Let X be the Sobolev space W 1,2([a,b]) endowed with the norm‖u‖ :=(∫ ba|u′(x)|2dx+∫ ba|u(x)|2dx) 12.Throughout the sequel, f : R→ R is a continuous function, h : R→ R is apositive continuous function, α : [a,b]→ R is a sommable function and λ is apositive real parameter. PutF(t) =∫ t0f (ξ )dξ , for all t ∈ R,F1(x, t) =∫ t0α(x) f (ξ )dξ = α(x)F(t), for all (x, t) ∈ [a,b]×R,and, putH(y) =∫ y0(∫ σ01h(τ)dτ)dσ , for all y ∈ R.We recall that u : [a,b]→ R is called weak solution of Problem (Nλ ) if u ∈W 1,2([a,b]) and∫ baH ′(u′(x))v′(x)dx+∫ bau(x)v(x)dx = λ∫ baα(x) f (u(x))v(x)dx,for all v ∈W 1,2([a,b]).We also recall that a weak solution is a generalized solution, that is,u ∈C1([a,b]), u′ ∈ AC([a,b]), −u′′(x)+u(x)h(u′(x)) = λα(x) f (u(x))h(u′(x)),for a.e. x ∈ [a,b], and u′(a) = u′(b) = 0.Moreover, if α is continuous, each weak solution is a classical solution, that is,u ∈C2([a,b]), −u′′(x)+u(x)h(u′(x)) = λα(x) f (u(x))h(u′(x)) for all x ∈ [a,b],and u′(a) = u′(b) = 0. Finally, putγ =(max{2(b−a); 2b−a}) 12,we recall the following inequality which we use in the sequelmaxx∈[a,b]|u(x)| ≤ γ‖u‖, (4)for all u ∈ X and for all x ∈ [a,b].166 PASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTA3. Main ResultsIn this Section, we establish existence results for the Neumann boundary valueproblem (Nλ ).Given two positive constants m, M, with m≤M, putδ1 =(min{1M(b−a) ;1b−a}) 12, δ2 =(max{1m(b−a) ;1b−a}) 12.Moreover, given three nonnegative constants c1, c2, d, with δ1c1 < γd < δ2c2,puta(c2,d) :=max|t|≤c2F(t)−F(d)δ 22 c22− γ2d2andb(c1,d) :=F(d)−max|t|≤c1F(t)γ2d2−δ 21 c21.We give our main result.Theorem 3.1. Let α : [a,b]→R be a nonnegative function and let f ,h :R→Rbe continuous functions. Assume that there exist two positive constants m, M,such that(i) m≤ h(y)≤M, for all y ∈ R,and, assume that there exist three nonnegative constants c1, c2, d, with δ1c1 <γd < δ2c2, such thata(c2,d)< b(c1,d). (5)Then, for each λ ∈]b−a2γ2‖α‖1b(c1,d) ,b−a2γ2‖α‖1a(c2,d)[, the problem (Nλ ) ad-mits at least one weak solution u, such thatc1γ< ‖u‖< c2γ.Proof. PutΦ(u) :=12∫ ba|u(x)|2dx+∫ baH(u′(x))dx,Ψ(u) :=∫ baF1(x,u(x))dx,for all u ∈ X .It is well known that Φ and Ψ satisfy all regularity assumptions requested inEXISTENCE RESULTS FOR A QUASI-LINEAR DIFFERENTIAL PROBLEM 167Theorem 2.1 and that the critical points in X of the functional Φ−λΨ are ex-actly the weak solutions of the problem (Nλ ). By using (i), one hasmin{12M;12}‖u‖2 ≤Φ(u)≤max{12m;12}‖u‖2,for every u ∈ X . Our aim is to apply Theorem 2.1. To this end, putr1 =b−a2δ 21γ2c21, r2 =b−a2δ 22γ2c22andu0(x) = d, for all x ∈ [a,b].Clearly, u0 ∈ X and one hasΦ(u0) =12∫ ba|u0|2dx+∫ baH(u′0)dx =12d2(b−a),Ψ(u0) =∫ baF1(x,u0(x))dx = ‖α‖1F(d),where‖α‖1 :=∫ ba|α(x)|dx.From δ1c1 < γd < δ2c2, one has r1 <Φ(u0)< r2. Moreover, for all u ∈ X suchthat Φ(u)< r2, taking (4) into account, one has|u(x)|< c2, for all x ∈ [a,b],and ∫ baF1(x,u(x))dx≤∫ bamax|t|≤c2F1(x, t)dx = ‖α‖1 max|t|≤c2 F(t).Thereforesupu∈Φ−1(]−∞,r2[)Ψ(u)≤ ‖α‖1 max|t|≤c2 F(t).Arguing as before, we obtainsupu∈Φ−1(]−∞,r1])Ψ(u)≤ ‖α‖1 max|t|≤c1 F(t).Therefore, one hasβ (r1,r2)≤supu∈Φ−1(]−∞,r2[)Ψ(u)−Ψ(u0)r2−Φ(u0) ≤≤ 2γ2‖α‖1b−amax|t|≤c2F(t)−F(d)δ 22 c22− γ2d2=2γ2‖α‖1b−a a(c2,d). (6)168 PASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTAOn the other hand, one hasρ2(r1,r2)≥Ψ(u0)− supu∈Φ−1(]−∞,r1])Ψ(u)Φ(u0)− r1 ≥≥ 2γ2‖α‖1b−aF(d)−max|t|≤c1F(t)γ2d2−δ 21 c21=2γ2‖α‖1b−a b(c1,d). (7)Hence, from (5) one hasβ (r1,r2)< ρ2(r1,r2).Therefore, owing to Theorem 2.1, for eachλ ∈]b−a2γ2‖α‖1b(c1,d) ,b−a2γ2‖α‖1a(c2,d)[,Φ−λΨ admits at least one critical point u¯ such thatr1 <Φ(u¯)< r2,that is c1γ< ‖u¯‖< c2γ.Hence, the proof is complete.Now, we point out the following consequence of Theorem 3.1.Corollary 3.2. Let α : [a,b]→R be a nonnegative function, h :R→]0,+∞[ andf : R→ [0,+∞[ be continuous functions. Assume that (i) holds and that thereexist two positive constants c,d, with c >γδ2d, such thatF(c)c2<(δ2γ)2 F(d)d2. (8)Then, for each λ ∈]b−a2‖α‖1d2F(d),b−a2‖α‖1(δ2γ)2 c2F(c)[, the problem (Nλ ) ad-mits at least one nontrivial weak solution u such that ‖u‖< cγ.Proof. Our aim is to apply Theorem 3.1. To this end, we pick c1 = 0 and c2 = c.From (8) one hasa(c2,d) =max|t|≤cF(t)−F(d)δ 22 c2− γ2d2≤F(c)−(γ2d2δ 22 c2F(c))δ 22 c2− γ2d2=F(c)δ 22 c2.EXISTENCE RESULTS FOR A QUASI-LINEAR DIFFERENTIAL PROBLEM 169On the other hand, one has b(c1,d) =F(d)γ2d2. Hence, owing to (8), Theorem 3.1ensures the conclusion.Now, we point out the following relevant consequence of Corollary 3.2.Theorem 3.3. Let α : [a,b]→R be a nonnegative function and f ,h :R→R becontinuous functions. Assume that (i) holds. Assume thatlimξ→0+f (ξ )ξ=+∞, (9)and put λ ∗ =b−a2‖α‖1(δ2γ)2supc>0c2F(c). Then, for each λ ∈ ]0,λ ∗[, the problem(Nλ ) admits at least one positive weak solution.Proof. Fix λ ∈ ]0,λ ∗[. Then, there is c > 0 such that λ < b−a2‖α‖1(δ2γ)2 c2F(c).From (9) there is d <δ2γc such that2‖α‖1b−aF(d)d2>1λ. Hence, Corollary 3.2ensures the conclusion.Remark 3.4. Taking (9) into account, fix ρ > 0 such that f (ξ ) > 0 for allξ ∈ ]0,ρ[. Then, put λ¯ = b−a2‖α‖1(δ2γ)2supc∈]0,ρ[c2F(c). Clearly, λ¯ ≤ λ ∗. Now,fixed λ ∈ ]0, λ¯[ and arguing as in the proof of Theorem 3.3, there are c ∈ ]0,ρ[and d <δ2γc such thatb−a2‖α‖1d2F(d)< λ <b−a2‖α‖1(δ2γ)2 c2F(c). Hence, Corol-lary 3.2 ensures that, for each λ ∈ ]0, λ¯[, the problem (Nλ ) admits at least onepositive weak solution uλ such that|uλ (x)|<ργ,for all x ∈ [a,b].170 PASQUALE F. PIZZIMENTI - ANGELA SCIAMMETTA4. AcknowledgementsThe authors wish to thank Professor Gabriele Bonanno for his remarks and sug-gestions during the writing of this paper.REFERENCES[1] D. Averna - G. 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Zhilong, Existence of positive solutions of superlinear second-order Neumannboundary value problem, Nonlinear Anal. 72 (2010), 3216–3221.EXISTENCE RESULTS FOR A QUASI-LINEAR DIFFERENTIAL PROBLEM 171PASQUALE F. PIZZIMENTIDepartment of MathematicsUniversity of Messina98166 - Messina (Italy)e-mail: ppizzimenti@unime.itANGELA SCIAMMETTADepartment of MathematicsUniversity of Messina98166 - Messina (Italy)e-mail: asciammetta@unime.it

Existence results for a quasi-linear differential problem

The aim of this paper is to establish the existence of at least one non-trivial solution for Neumann quasi-linear problems. Our approach is based on variational methods.<br /

Pasquale F. Pizzimenti

Angela Sciammetta

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Existence results for a quasi-linear differential problem

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