In this paper, a fourth-order boundary value problem on the half-line is considered and existence of solutions is proved using a minimization principle and the mountain pass theorem

Briki, Mabrouk

Moussaoui, Toufik

O'Regan, Donal

Electronic journal of qualitative theory of differential equations

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Electronic Journal of Qualitative Theory of Differential Equations2016, No. 24, 1–11; doi: 10.14232/ejqtde.2016.1.24 http://www.math.u-szeged.hu/ejqtde/Existence of solutions for a fourth-order boundaryvalue problem on the half-line via critical point theoryMabrouk Briki1, Toufik MoussaouiB 1 and Donal O’Regan21Laboratory of Fixed Point Theory and Applications, École Normale Supérieure,Kouba, Algiers, Algeria2School of Mathematics, Statistics and Applied Mathematics, National University of Ireland,Galway, IrelandReceived 14 December 2015, appeared 2 May 2016Communicated by Alberto CabadaAbstract. In this paper, a fourth-order boundary value problem on the half-line isconsidered and existence of solutions is proved using a minimization principle and themountain pass theorem.Keywords: fourth-order BVPs, unbounded interval, critical point, minimization princi-ple, mountain-pass theorem.2010 Mathematics Subject Classification: 35A15, 35B38.1 IntroductionWe consider the existence of solutions for the following fourth-order boundary value problemset on the half-line u(4)(t)− u′′(t) + u(t) = f (t, u(t)), t ∈ [0,+∞),u(0) = u(+∞) = 0,u′′(0) = u′′(+∞) = 0,(1.1)where f ∈ C([0,+∞)×R,R).Many authors used critical point theory to establish the existence of solutions for fourth-order boundary value problems on bounded intervals (see for example [8,9,13]), but there areonly a few papers that consider the above problem on the half-line using critical point theory.We cite [5] where the authors consider the existence of solutions for a particular fourth-orderBVP on the half-line using critical point theory.We endow the following spaceH20(0,+∞) ={u ∈ L2(0,+∞), u′ ∈ L2(0,+∞), u′′ ∈ L2(0,+∞), u(0) = 0, u′(0) = 0}BCorresponding author. Email: moussaoui@ens-kouba.dz2 M. Briki, T. Moussaoui and D. O’Reganwith its natural norm‖u‖ =(∫ +∞0u′′ 2(t)dt +∫ +∞0u′ 2(t)dt +∫ +∞0u2(t)dt) 12.Note that if u ∈ H20(0,+∞), then u(+∞) = 0, u′(+∞) = 0, (see [3, Corollary 8.9]). Letp, q : [0,+∞) −→ (0,+∞) be two continuously differentiable and bounded functions withM1 = max(‖p‖L2 , ‖p′‖L2) < +∞, M2 = max(‖q‖L2 , ‖q′‖L2) < +∞.We also consider the following spacesCl,p[0,+∞) ={u ∈ C([0,+∞),R) : limt→+∞ p(t)u(t) exists}endowed with the norm‖u‖∞,p = supt∈[0,+∞)p(t)|u(t)|,andC1l,p,q[0,+∞) ={u ∈ C1([0,+∞),R) : limt→+∞ p(t)u(t), limt→+∞ q(t)u′(t) exist}endowed with the natural norm‖u‖∞,p,q = supt∈[0,+∞)p(t)|u(t)|+ supt∈[0,+∞)q(t)|u′(t)|.LetCl [0,+∞) ={u ∈ C([0,+∞),R) : limt→+∞ u(t) exists}endowed with the norm ‖u‖∞ = supt∈[0,+∞) |u(t)|.To prove that H20(0,+∞) embeds compactly in C1l,p,q[0,+∞), we need the followingCorduneanu compactness criterion.Lemma 1.1 ([4]). Let D ⊂ Cl([0,+∞),R) be a bounded set. Then D is relatively compact if thefollowing conditions hold:(a) D is equicontinuous on any compact sub-interval of R+, i.e.∀ J ⊂ [0,+∞) compact, ∀ ε > 0, ∃ δ > 0, ∀ t1, t2 ∈ J :|t1 − t2| < δ =⇒ |u(t1)− u(t2)| ≤ ε, ∀ u ∈ D;(b) D is equiconvergent at +∞ i.e.,∀ ε > 0, ∃ T = T(ε) > 0 such that∀ t : t ≥ T(ε) =⇒ |u(t)− u(+∞)| ≤ ε, ∀ u ∈ D.Similar reasoning as in [6] yields the following compactness criterion in the spaceC1l,p,q([0,+∞),R).Fourth-order BVP via critical point theory 3Lemma 1.2. Let D ⊂ C1l,p,q([0,+∞),R) be a bounded set. Then D is relatively compact if thefollowing conditions hold:(a) D is equicontinuous on any compact sub-interval of [0,+∞), i.e.∀ J ⊂ [0,+∞) compact, ∀ ε > 0, ∃ δ > 0, ∀ t1, t2 ∈ J :|t1 − t2| < δ =⇒ |p(t1)u(t1)− p(t2)u(t2)| ≤ ε, ∀ u ∈ D,|t1 − t2| < δ =⇒ |q(t1)u′(t1)− q(t2)u′(t2)| ≤ ε, ∀ u ∈ D;(b) D is equiconvergent at +∞ i.e.,∀ ε > 0, ∃ T = T(ε) > 0 such that∀ t : t ≥ T(ε) =⇒ |p(t)u(t)− (pu)(+∞)| ≤ ε, ∀ u ∈ D,∀ t : t ≥ T(ε) =⇒ |q(t)u′(t)− (qu′)(+∞)| ≤ ε, ∀ u ∈ D.Now we recall some essential facts from critical point theory (see [1, 2, 10]).Definition 1.3. Let X be a Banach space, Ω ⊂ X an open subset, and J : Ω −→ R a functional.We say that J is Gâteaux differentiable at u ∈ Ω if there exists A ∈ X∗ such thatlimt→0J(u + tv)− J(u)t= Av,for all v ∈ X. Now A, which is unique, is denoted by A = J′G(u).The mapping which sends to every u ∈ Ω the mapping J′G(u) is called the Gâteaux differ-ential of J and is denoted by J′G.We say that J ∈ C1 if J is Gâteaux differential on Ω and J′G is continuous at every u ∈ Ω.Definition 1.4. Let X be a Banach space. A functional J : Ω −→ R is called coercive if, forevery sequence (uk)k∈N ⊂ X,‖uk‖ → +∞ =⇒ |J(uk)| → +∞.Definition 1.5. Let X be a Banach space. A functional J : X −→ (−∞,+∞] is said to besequentially weakly lower semi-continuous (swlsc for short) ifJ(u) ≤ lim infn→+∞ J(un)as un ⇀ u in X, n→ ∞.Lemma 1.6 (Minimization principle [2]). Let X be a reflexive Banach space and J a functionaldefined on X such that(1) lim‖u‖→+∞ J(u) = +∞ (coercivity condition),(2) J is sequentially weakly lower semi-continuous.Then J is lower bounded on X and achieves its lower bound at some point u0.Definition 1.7. Let X be a real Banach space, J ∈ C1(X,R). If any sequence (un) ⊂ X forwhich (J(un)) is bounded in R and J′(un) −→ 0 as n → +∞ in X′ possesses a convergentsubsequence, then we say that J satisfies the Palais–Smale condition (PS condition for brevity).4 M. Briki, T. Moussaoui and D. O’ReganLemma 1.8 (Mountain Pass Theorem, [11, Theorem 2.2], [12, Theorem 3.1]). Let X be a Banachspace, and let J ∈ C1(X,R) satisfy J(0) = 0. Assume that J satisfies the (PS) condition and thereexist positive numbers ρ and α such that(1) J(u) ≥ α if ‖u‖ = ρ,(2) there exists u0 ∈ X such that ‖u0‖ > ρ and J(u0) < α.Then there exists a critical point. It is characterized byJ′(u) = 0, J(u) = infγ∈Γmaxt∈[0,1]J(γ(t)),whereΓ = {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = u0}.1.1 Variational settingTake v ∈ H20(0,+∞), and multiply the equation in Problem (1.1) by v and integrate over(0,+∞), so we get∫ +∞0(u(4)(t)− u′′(t) + u(t))v(t)dt = ∫ +∞0f (t, u(t))v(t)dt.Hence ∫ +∞0(u′′(t)v′′(t) + u′(t)v′(t) + u(t)v(t))dt =∫ +∞0f (t, u(t))v(t)dt.This leads to the natural concept of a weak solution for Problem (1.1).Definition 1.9. We say that a function u ∈ H20(0,+∞) is a weak solution of Problem (1.1) if∫ +∞0(u′′(t)v′′(t) + u′(t)v′(t) + u(t)v(t))dt =∫ +∞0f (t, u(t))v(t)dt,for all v ∈ H20(0,+∞).In order to study Problem (1.1), we consider the functional J : H20(0,+∞) −→ R definedbyJ(u) =12‖u‖2 −∫ +∞0F(t, u(t))dt,whereF(t, u) =∫ u0f (t, s)ds.2 Some embedding resultsWe begin this section by proving some continuous and compact embeddings. Here p and q(and M1, M2) are as in Section 1.Lemma 2.1. H20(0,+∞) embeds continuously in C1l,p,q[0,+∞).Fourth-order BVP via critical point theory 5Proof. For u ∈ H20(0,+∞), we have|p(t)u(t)| = |p(+∞)u(+∞)− p(t)u(t)|=∣∣∣∣∫ +∞t (pu)′(s)ds∣∣∣∣≤∣∣∣∣∫ +∞t p′(s)u(s)ds∣∣∣∣+ ∣∣∣∣∫ +∞t p(s)u′(s)ds∣∣∣∣≤(∫ +∞0p′2(s)ds) 12(∫ +∞0u2(s)ds) 12+(∫ +∞0p2(s)ds) 12(∫ +∞0u′2(s)ds) 12≤ max(‖p′‖L2 , ‖p‖L2)‖u‖≤ M1‖u‖,and|q(t)u′(t)| = |q(+∞)u′(+∞)− q(t)u′(t)|=∣∣∣∣∫ +∞t (qu′)′(s)ds∣∣∣∣≤∣∣∣∣∫ +∞t q′(s)u′(s)ds∣∣∣∣+ ∣∣∣∣∫ +∞t q(s)u′′(s)ds∣∣∣∣≤(∫ +∞0q′2(s)ds) 12(∫ +∞0u′ 2(s)ds) 12+(∫ +∞0q2(s)ds) 12(∫ +∞0u′′2(s)ds) 12≤ max(‖q′‖L2 , ‖q‖L2)‖u‖≤ M2‖u‖.Hence ‖u‖∞,p,q ≤ M‖u‖, with M = max(M1, M2).The following compactness embedding is an important result.Lemma 2.2. The embedding H20(0,+∞) ↪→ C1l,p,q[0,+∞) is compact.Proof. Let D ⊂ H20(0,+∞) be a bounded set. Then it is bounded in C1l,p,q[0,+∞) by Lemma 2.1.Let R > 0 be such that for all u ∈ D, ‖u‖ ≤ R. We will apply Lemma 1.2.(a) D is equicontinuous on every compact interval of [0,+∞). Let u ∈ D and t1, t2 ∈ J ⊂[0,+∞) where J is a compact sub-interval. Using the Cauchy–Schwarz inequality, we have|p(t1)u(t1)− p(t2)u(t2)| =∣∣∣∣∫ t1t2 (pu)′(s)ds∣∣∣∣=∣∣∣∣∫ t1t2 (p′(s)u(s) + u′(s)p(s)) ds∣∣∣∣≤(∫ t1t2p′2(s)ds) 12(∫ t1t2u2(s)ds) 12+(∫ t1t2p2(s)ds) 12(∫ t1t2u′2(s)ds) 12≤ max[(∫ t1t2p′2(s)ds) 12,(∫ t1t2p2(s)ds) 12]‖u‖≤ R max[(∫ t1t2p′2(s)ds) 12,(∫ t1t2p2(s)ds) 12]−→ 0,6 M. Briki, T. Moussaoui and D. O’Reganas |t1 − t2| → 0, and|q(t1)u′(t1)− q(t2)u′(t2)| =∣∣∣∣∫ t1t2 (qu′)′(s)ds∣∣∣∣=∣∣∣∣∫ t1t2 (q′(s)u′(s) + q(s)u′′(s)) ds∣∣∣∣≤(∫ t1t2q′2(s)ds) 12(∫ t1t2u′2(s)ds) 12+(∫ t1t2q2(s)ds) 12(∫ t1t2u′′2(s)ds) 12≤ max[(∫ t1t2q′2(s)ds) 12,(∫ t1t2q2(s)ds) 12]‖u‖≤ R max[(∫ t1t2q′2(s)ds) 12,(∫ t1t2q2(s)ds) 12]−→ 0,as |t1 − t2| → 0.(b) D is equiconvergent at +∞. For t ∈ [0,+∞) and u ∈ D, using the fact that (pu)(+∞) =0, (qu′)(+∞) = 0 (note that u(∞) = 0, u′(∞) = 0 and p, q are bounded) and using the Cauchy–Schwarz inequality, we have|(pu)(t)− (pu)(+∞)| =∣∣∣∣∫ +∞t (pu)′(s)ds∣∣∣∣=∣∣∣∣∫ +∞t (p′(s)u(s) + u′(s)p(s)) ds∣∣∣∣≤ max[(∫ +∞tp′2(s)ds) 12,(∫ +∞tp2(s)ds) 12]‖u‖≤ R max[(∫ +∞tp′2(s)ds) 12,(∫ +∞tp2(s)ds) 12]−→ 0,as t→ +∞, and|(qu′)(t)− (qu′)(+∞)| =∣∣∣∣∫ +∞t (qu′)′(s)ds∣∣∣∣=∣∣∣∣∫ +∞t (q′(s)u′(s) + q(s)u′′(s)) ds∣∣∣∣≤ max[(∫ +∞tq′ 2(s)ds) 12,(∫ +∞tq2(s)ds) 12]‖u‖≤ R max[(∫ +∞tq′ 2(s)ds) 12,(∫ +∞tq2(s)ds) 12]−→ 0,as t→ +∞.Corollary 2.3. C1l,p,q[0,+∞) embeds continuously in Cl,p[0,+∞) .Corollary 2.4. The embedding H20(0,+∞) ↪→ Cl,p[0,+∞) is continuous and compact.Fourth-order BVP via critical point theory 73 Existence resultsHere p (and M1) are as in Section 1.Theorem 3.1. Assume that F satisfy the following conditions.(F1) There exist two constants 1 < α < β < 2 and two functions a, b with ap α ∈ L1([0,+∞), [0,+∞)),bp β ∈ L1([0,+∞), [0,+∞)) such that|F(t, x)| ≤ a(t)|x|α, ∀(t, x) ∈ [0,+∞)×R, |x| ≤ 1and|F(t, x)| ≤ b(t)|x|β, ∀(t, x) ∈ [0,+∞)×R, |x| > 1.(F2) There exist an open bounded set I ⊂ [0,+∞) and two constants η > 0 and 0 < γ < 2 such thatF(t, x) ≥ η|x|γ, ∀(t, x) ∈ I ×R, |x| ≤ 1.Then Problem (1.1) has at least one nontrivial weak solution.Proof.Claim 1. We first show that J is well defined.LetΩ1 = {t ≥ 0, |u(t)| ≤ 1}, Ω2 = {t ≥ 0, |u(t)| > 1}.Given u ∈ H20(0,+∞), it follows from (F1) and Corollary 2.4 that∫ +∞0|F(t, u(t))|dt =∫Ω1|F(t, u(t))|dt +∫Ω2|F(t, u(t))|dt≤∫Ω1a(t)|u(t)|αdt +∫Ω2b(t)|u(t)|βdt≤∫Ω1a(t)pα(t)|p(t)u(t)|αdt +∫Ω2b(t)pβ(t)|p(t)u(t)|βdt≤∣∣∣∣ ap α∣∣∣∣L1‖u‖α∞,p +∣∣∣∣ bp β∣∣∣∣L1‖u‖β∞,p≤ Mα1∣∣∣∣ ap α∣∣∣∣L1‖u‖α + Mβ1∣∣∣∣ bp β∣∣∣∣L1‖u‖β.Thus|J(u)| ≤ 12‖u‖2 + Mα1∣∣∣∣ ap α∣∣∣∣L1‖u‖α + Mβ1∣∣∣∣ bp β∣∣∣∣L1‖u‖β < +∞.Claim 2. J is coercive.From (F1) and Corollary 2.4, we haveJ(u) =12‖u‖2 −∫Ω1F(t, u(t))dt−∫Ω2F(t, u(t))dt≥ 12‖u‖2 −Mα1∣∣∣∣ ap α∣∣∣∣L1‖u‖α −Mβ1∣∣∣∣ bp β∣∣∣∣L1‖u‖β.(3.1)Now since 0 < α < β < 2, then (3.1) implies thatlim‖u‖→+∞J(u) = +∞.8 M. Briki, T. Moussaoui and D. O’ReganConsequently, J is coercive.Claim 3. J is sequentially weakly lower semi-continuous.Let (un) be a sequence in H20(0,+∞) such that un ⇀ u as n −→ +∞ in H20(0,+∞). Thenthere exists a constant A > 0 such that ‖un‖ ≤ A, for all n ≥ 0 and ‖u‖ ≤ A. Now (seeCorollary 2.4) (p(t)un(t)) converges to (p(t)u(t)) as n −→ +∞ for t ∈ [0,+∞). Since F iscontinuous, we have F(t, un(t)) −→ F(t, u(t)) as n −→ +∞, and using (F1) we have|F(t, un(t))| ≤ a(t)|un(t)| α + b(t)|un(t)| β≤ a(t)p α(t)|p(t)un(t)| α + b(t)p β(t) |p(t)un(t)|β≤ a(t)p α(t)‖un‖ α∞,p +b(t)p β(t)‖un‖ β∞,p≤ a(t)p α(t)M α1 ‖un‖ α +b(t)p β(t)Mβ1‖un‖ β≤ a(t)p α(t)M α1 Aα +b(t)p β(t)Mβ1 Aβ,so from the Lebesgue Dominated Convergence Theorem we havelimn→+∞∫ +∞0F(t, un(t))dt =∫ +∞0F(t, u(t))dt.The norm in the reflexive Banach space is sequentially weakly lower semi-continuous, solim infn→+∞ ‖un‖ ≥ ‖u‖.Thus one haslim infn→+∞ J(un) = lim infn→+∞(12‖un‖2 −∫ +∞0F(t, un(t))dt)≥ 12‖u‖2 −∫ +∞0F(t, u(t))dt = J(u).Then, J is sequentially weakly lower semi-continuous.From Lemma 1.6, J has a minimum point u0 which is a critical point of J.Claim 4. We show that u0 6= 0.Let u1 ∈ H20(0,+∞) \ {0} and |u1(t)| ≤ 1, for all t ∈ I. Then from (F2), we haveJ(su1) =s22‖u1‖2 −∫ +∞0F(t, su1(t))dt≤ s22‖u1‖2 −∫Iη|su1(t)|γdt≤ s22‖u1‖2 − s γη∫I|u1(t)|γdt, 0 < s < 1.Since 0 < γ < 2, it follows that J(su1) < 0 for s > 0 small enough. Hence J(u0) < 0, andtherefore u0 is a nontrivial critical point of J.Finally, it is easy to see that under (F1), the functional J is Gâteaux differentiable and theGâteaux derivative at a point u ∈ X is(J′(u), v) =∫ +∞0(u′′(t)v′′(t) + u′(t)v′(t) + u(t)v(t))dt−∫ +∞0f (t, u(t))v(t)dt, (3.2)Fourth-order BVP via critical point theory 9for all v ∈ H20(0,+∞). Therefore u is a weak solution of Problem (1.1).Theorem 3.2. Assume that f satisfies the following assumptions.(F3) There exist nonnegative functions ϕ, g such that g ∈ C(R, [0,+∞)) with| f (t, x)| ≤ ϕ(t)g(x), for all t ∈ [0,+∞) and all x ∈ R,and for any constant R > 0 there exists a nonnegative function ψR with ϕψR ∈ L1(0,+∞) andsup{g(yp(t)): y ∈ [−R, R]}≤ ψR(t) for a.e. t ≥ 0.(F4)1a(t)F(t,1p(t)x)= o(|x|2) as x −→ 0uniformly in t ∈ [0,+∞) for some function a ∈ L1(0,+∞) ∩ C[0,+∞).(F5) There exists a positive function c1 and a nonnegative function c2 with c1, c2 ∈ L1(0,∞), andµ > 2 such that(a) F(t, x) ≥ c1(t)|x|µ − c2(t), for t ≥ 0, ∀ x ∈ R \ {0},(b) µF(t, x) ≤ x f (t, x), for t ≥ 0, ∀ x ∈ R.Then Problem (1.1) has at least one nontrivial weak solution.Proof. We have J(0) = 0.Claim 1. J satisfies the (PS) condition.Assume that (un)n∈N ⊂ H20(0,+∞) is a sequence such that (J(un))n∈N is bounded andJ′(un) −→ 0 as n −→ +∞. Then there exists a constant d > 0 such that|J(un)| ≤ d, ‖J′(un)‖E′ ≤ dµ, ∀n ∈N.From (F5)(b) we have2d + 2d‖un‖ ≥ 2J(un)− 2µ(J′(un), un)≥(1− 2µ)‖un‖2 + 2[∫ +∞0(1µun(t) f (t, un(t))− F(t, un(t)))dt]≥(1− 2µ)‖un‖2.Since µ > 2, then (un)n∈N is bounded in H20(0,+∞).Now, we show that (un) converges strongly to some u in H20(0,+∞). Since (un) is boundedin H20(0,+∞), there exists a subsequence of (un) still denoted by (un) such that (un) convergesweakly to some u in H20(0,+∞). There exists a constant c > 0 such that ‖un‖ ≤ c. Now (seeCorollary 2.4) (p(t)un(t)) converges to p(t)u(t) on [0,+∞). We have f (t, un(t)) −→ f (t, u(t))and| f (t, un(t))| =∣∣∣∣ f (t, 1p(t) p(t)un(t))∣∣∣∣≤ ϕ(t)g(1p(t)p(t)un(t))≤ ϕ(t)ψcM1(t),10 M. Briki, T. Moussaoui and D. O’Reganand using the Lebesgue Dominated Convergence Theorem, we havelimn→+∞∫ +∞0( f (t, un(t))− f (t, u(t))) (un(t)− u(t)) dt = 0. (3.3)Since limn→+∞ J′(un) = 0 and (un) converges weakly to some u, we havelimn→+∞〈J′(un)− J′(u), un − u〉 = 0. (3.4)It follows from (3.2) that(J′(un)− J′(u), un − u) = ‖un − u‖2 −∫ +∞0( f (t, un(t))− f (t, u(t)))(un(t)− u(t))dt.Hence limn→+∞ ‖un − u‖ = 0. Thus (un) converges strongly to u in H20(0,+∞), so J satis-fies the (PS) condition.Claim 2. J satisfies assumption (1) of Lemma 1.8.Let 0 < ε < 1|a|L1 M21. From (F4), there exists 0 < δ < 1 such that∣∣∣ 1a(t)F(t,1p(t)x)∣∣∣ ≤ ε2|x|2, for t ∈ [0,+∞) and |x| ≤ δ.Using Corollary 2.4, we have∫ +∞0|F(t, u(t))dt| =∫ +∞0∣∣∣∣F(t, 1p(t) p(t)u(t))dt∣∣∣∣≤∫ +∞0ε2|a(t)|p2(t)|u(t)|2dt≤ ε2M21|a|L1‖u‖2,whenever ‖u‖∞,p ≤ δ.Let 0 < ρ ≤ δM1 and α = 12 (1− ε|a|L1 M21)ρ2. Then for ‖u‖ = ρ (note ‖u‖∞,p ≤ δ), we haveJ(u) =12‖u‖2 −∫ +∞0F(t, u(t))dt≥ 12(1− ε|a|L1 M21)‖u‖2 = α,so assumption (1) in Lemma 1.8 is satisfied.Claim 3. J satisfies assumption (2) of Lemma 1.8.By (F5)(a) we have for some v0 ∈ H20(0,+∞), v0 6= 0,J(ξv0) =12ξ2‖v0‖2 −∫ +∞0F(t, ξv0(t))dt≤ 12ξ2‖v0‖2 − |ξ|µ∫ +∞0c1(t)|v0(t)|µdt +∫ +∞0c2(t)dt.Now since µ > 2, then for u0 = ξv0, J(u0) ≤ 0, as ξ → +∞, so assumption (2) in Lemma 1.8is satisfied. From Lemma 1.8, J possesses a critical point which is a nontrivial weak solutionof Problem (1.1).Fourth-order BVP via critical point theory 11As an example of the above theorem, take f (t, x) = 52 exp(−t)|x|12 x. To see this takec1(t) = exp(−t), c2(t) = 0,µ =52, a(t) =1(1+ t)2, p(t) =11+ t,ϕ(t) =52e−t, g(x) = |x| 32 and ψR(t) = (1+ t) 32 R 32 .References[1] A. Ambrosetti, G. Prodi, A primer of nonlinear analysis, Cambridge University Press,Cambridge, 1995. MR1336591[2] M. Badiale, E. Serra, Semilinear elliptic equations for beginners. Existence results via thevariational approach, Universitext, Springer, London, 2011. MR2722059; url[3] H. Brézis, Functional analysis, Sobolev spaces and partial differential equations,Springer, New York, 2010. MR2759829[4] C. Corduneanu, Integral equations and stability of feedback systems, Academic Press, NewYork, 1973. MR0358245[5] R. Enguiça, A. Gavioli, L. Sanchez, Solutions of second-order and fourth-order ODEson the half-line, Nonlinear Anal. 73(2010), 2968–2979. MR2678658; url[6] O. Frites, T. Moussaoui, D. O’Regan, Existence of solutions for a variational inequalityon the half-line, B. Iran. Math. Soc., accepted.[7] O. Kavian, Introduction à la théorie des points critiques et applications aux problèmes elliptiques(in French), Springer-Verlag, Paris, 1993. MR1276944[8] F. Li, Q. Zhang, Z. Liang, Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear Anal. 62(2005), 803–816. MR2153213; url[9] X. L. Liu, W. T. Li, Existence and multiplicity of solutions for fourth-order boundary valueproblems with three parameters, Math. Comput. Modelling 46(2007), 525–534. MR2329456;url[10] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Springer-Verlag, NewYork, 1989. MR982267; url[11] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differ-ential equations, in: CBMS Regional Conference Series in Mathematics, Vol. 65, Published forthe Conference Board of the Mathematical Sciences, Washington, DC; by the AmericanMathematical Society, Providence, RI, 1986. MR845785; url[12] M. Struwe, Variational methods. Applications to nonlinear partial differential equations andHamiltonian systems, Springer-Verlag, Berlin, 1996. MR1411681; url[13] Y. Yang, J. Zhang, Existence of infinitely many mountain pass solutions for some fourth-order boundary value problems with a parameter, Nonlinear Anal. 71(2009), 6135–6143.MR2566519; url

Existence of solutions for a fourth-order  boundary value problem on the half-line via critical point theory

Mabrouk Briki

Toufik Moussaoui

Donal O'Regan

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Existence of solutions for a fourth-order boundary value problem on the half-line via critical point theory

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Existence of solutions for a fourth-order boundary value problem on the half-line via critical point theory

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