A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM

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1 A THREE CRITICAL POINTS THEOREM AND ITS APPLICATIONS TO THE ORDINARY DIRICHLET PROBLEM DIEGO AVERNA AND GABRIELE BONANNO Abstact. The aim of this pape is twofold. On one hand we establish a thee citical points theoem fo functionals depending on a eal paamete λ Λ, which is diffeent fom the one poved by B.Riccei in [5] (Ach. Math. 75 (), -6) and gives an estimate of whee Λ can be located. On the othe hand, as an application of the pevious esult, we pove an existence theoem of thee classical solutions fo a two-point bounday value poblem which is independent fom the one by J.Hendeson and H.B.Thompson ([], J. Diffeential Equations 66 (), ). Specifically, an example is given whee the key assumption of [] fails. Nevetheless, the existence of thee solutions can still be deduced using ou theoem.. Intoduction Recently, B.Riccei established a vey inteesting thee citical points esult ([5], Theoem ), that we ecall in an equivalent fomulation (see [3], Theoem.3 and Remak.): Theoem A. Let X be a sepaable and eflexive eal Banach space, Φ : X IR a continuously Gâteaux diffeentiable and sequentially weakly lowe semicontinuous functional whose Gâteaux deivative admits a continuous invese on X, Ψ : X IR a continuously Gâteaux diffeentiable functional whose Gâteaux deivative is compact. Assume that: (i) lim (Φ(x) + λψ(x)) = + fo all λ [, + [; x + (ii) thee ae IR, x, x X such that: Φ(x ) < < Φ(x ), inf Ψ(x) > (Φ(x ) )Ψ(x ) + ( Φ(x ))Ψ(x ). x Φ (],]) Φ(x ) Φ(x ) Key wods and phases. citical points, thee solutions, two point bounday value poblem. Mathematics Subject Classification: 58E5, 34B5. This eseach was suppoted by 6% MURST. Typeset by L A TEX ε.

2 DIEGO AVERNA AND GABRIELE BONANNO Then, thee exists an open inteval Λ ], + [ and a positive eal numbe q such that, fo each λ Λ, the equation Φ (x) + λψ (x) = () has at least thee solutions in X whose noms ae less than q. Applications of Theoem A to nonlinea bounday value poblems have been given in [], [3], [4], [5], [6], [8], [], [5], (see also [3] fo the non smooth case), establishing multiplicity esults fo equations depending on a paamete λ. We note that Theoem A gives no estimate of whee Λ can be located in ], + [. Vey ecently, anothe thee citical point theoem was established (Theoem. of [7]), which povides an uppe bound fo Λ. The aim of this pape is to establish some theoems ensuing the existence of at least thee solutions fo the equation () fo each λ in an explicitly detemined inteval. The main esult of Section is Theoem.. Its poof is based on the vaiational pinciple of B.Riccei ([6]) (see also [7]) and on the mountain pass theoem as expessed by P.Pucci and J.Sein in [4]. The following is a paticula case of Theoem.. Theoem B. Let X be a eflexive eal Banach space, Φ, Ψ be as in Theoem A, and assume that (i) of Theoem A holds. Futhe put, fo each > inf X Φ, ϕ () := ϕ () := inf x Φ (],[) inf x Φ (],[) y Φ ([,+ [) Ψ(x) inf wψ Φ (],[), Φ(x) sup Ψ(x) Ψ(y) Φ(y) Φ(x), whee Φ (], [) w is the closue of Φ (], [) in the weak topology, and assume that (ii ) thee is IR such that: and inf X Φ <, ϕ () < ϕ (). Then, fo each λ ], ϕ () ϕ [ the equation () has at least thee solutions in X. () Howeve, ϕ () in Theoem B could be (see Theoem.). In this and simila cases, hee and in the sequel, we agee to ead as +. In Theoem B, the sepaability of X is not equied. Moeove, hypotheses (ii) and (ii ) in Theoems A and B espectively seem to be diffeent. Theoem B gives

3 A THREE CRITICAL POINTS THEOREM... 3 a lowe bound fo Λ, wheeas Theoem A assues the stability of the thee solutions with espect to λ, namely the unifom boundedness of noms of solutions. In Section 3, as an application of Theoem B and its consequences, we study the following odinay autonomuous Diichlet poblems { u = λf(u) (ADE) u() = u() =, and { u = f(u) u() = u() =, (AD) establishing the existence of thee classical solutions unde a suitable set of assumptions (see Theoem 3., Theoem 3. and Theoem 3.3). Multiple solutions to the above mentioned poblems have been obtained by seveal authos using diffeent techniques. We efe to [] and the efeences theein fo poblem (ADE) and to [], [], [] fo poblem (AD). In [] (see also [3], [7]), using citical points theoems and set-valued analysis aguments, a λ-unifom nom-boundedness of the thee solutions to poblem (ADE) was established unde assumptions which ae vey simila to ous (see Remak 3.). In the vey inteesting wok [], J.Hendeson and H.B.Thompson ensued the existence of at least thee solutions by using a method of lowe and uppe solutions. It is woth to note that thei key assumption, which we ecall in Remak 3.4, fails in examples whee, on the contay, we can apply ou Theoem 3. (see Example 3.). The main esult of Section 3 is Theoem 3.. Hee ae two paticula cases of it. Theoem C. Let f : IR IR be a nonnegative and bounded continuous function such that f(ξ)dξ < < f(ξ)dξ. Then, the poblem (AD) has at least thee classical solutions. Theoem D. Let f : IR IR be a continuous function with f() = and f(x) f(x) in a ight neighbouhood of, and such that lim ], + [ fo some q ], [. x + x q Then, thee exists a positive eal numbe λ such that, fo each λ > λ, the poblem (ADE) has at least two nontivial and nonnegative classical solutions.. Citical points theoems In this section we establish some thee citical points theoems fo a suitable class of functionals depending on a eal paamete λ. The main esult is Theoem.. As its consequences we obtain Theoem B given in Intoduction, and Theoem..

4 4 DIEGO AVERNA AND GABRIELE BONANNO Theoem.. Let X be a eflexive eal Banach space, and let Φ, Ψ : X IR be two sequentially weakly lowe semicontinuous functionals. Assume also that Φ is (stongly) continuous, satisfies Φ(x) = + and, fo each λ >, the functional lim x + Φ+λΨ is continuously Gâteaux diffeentiable, bounded below, and satisfies the Palais- Smale condition. Futhe, assume that thee exists > inf Φ such that, given ϕ and X ϕ as in Theoem B, ϕ () < ϕ (). Then, fo each λ ] ϕ (), ϕ () points. [, the functional Φ + λψ has at least thee citical Poof. Fix λ ], [ and conside the functional Ψ + Φ. Since > ϕ ϕ () ϕ () λ λ (), thanks to Theoem 5 of [7], the functional Ψ + Φ has a local minimum, say x λ, which lies in Φ (], [). Moeove, fom < ϕ λ () we have that fo evey x Φ (], [) thee exists y Φ ([, + [) such that Ψ(y) + Φ(y) < Ψ(x) + Φ(x); hence x λ λ is not a global minimum fo Ψ + Φ in X. λ On the othe hand, by Theoem 38.F of [8], Ψ + Φ admits a global minimum, λ say x, in X. Then, by Coollay of [4], the functional Ψ + Φ admits a thid citical point λ distinct fom x and x. Of couse, also the functional Φ+λΨ has the same thee distinct citical points. Now, we give the poof of Theoem B stated in the Intoduction. Poof of Theoem B. The compactness of Ψ implies that Ψ is sequentially weakly continuous ([8], Coollay 4.9). Moeove, Φ + λψ satisfies the Palais-Smale condition (see, fo instance, Example 38.5 of [8]) and is bounded below. Theoem.. Let X, Φ, Ψ be as in Theoem B and assume that (i) of Theoem A holds. Futhe, assume that thee ae IR, x, x X such that (j) Φ(x ) < < Φ(x ); (jj) inf Ψ = Ψ(x ) > Ψ(x ). Φ (],[) w Φ(x ) inf Φ Φ Then, fo each λ (],[), +, the functional Φ+λΨ has at least Ψ(x ) Ψ(x ) thee citical points. Poof. Thanks to ou assumptions, we have and ϕ () ϕ () = Ψ(x ) Ψ(x ) Φ(x ) inf Φ >. Φ (],[)

5 A THREE CRITICAL POINTS THEOREM... 5 Thus, the conclusion follows by Theoem B. 3. Applications to the odinay Diichlet poblem In this section, we apply Theoem. and its consequences to the Diichlet poblems (ADE) and (AD). Let us assume f : IR IR continuous and put g(ξ) := ξ f(t) dt, The main esult of this section is the following ξ IR. Theoem 3.. Assume that thee exist fou positive constants c, d, a, s, with c < d and s <, such that: (k) max < g(d) + g(t)dt max d ; c 4 d (kk) g(ξ) a( + ξ s ) fo all ξ IR. Then, fo each 8d λ g(d) + d g(t)dt max (ADE) admits at least thee classical solutions. g(ξ), c max g(ξ), the poblem Poof. Let X be the Sobolev space W, ([, ]) endowed with the nom u := ( u (t) dt ) /. Fo each u X, put: Φ(u) := u, Ψ(u) := g(u(t))dt. It is well known that the citical points in X of the functional Φ + λψ ae pecisely the classical solutions of poblem (ADE). So, ou end is to apply Theoem B to Φ and Ψ. Clealy, Φ and Ψ ae as in Theoem A. Futhemoe, thanks to (kk) and to Hölde inequality, we have lim (Φ(u) + λψ(u)) = + u + fo all λ [, + [. In ode to pove (ii ) of Theoem B, we claim that: fo each >, and ϕ () max g(ξ) ξ (C)

6 6 DIEGO AVERNA AND GABRIELE BONANNO ϕ () ξ y (C) fo each > and evey y X such that y and g(y(t))dt max g(ξ). ξ In fact, fo >, taking into account that Φ (], [) w = Φ (], ]), we have ϕ () sup x g(x(t))dt Thus, since max x(t) x fo evey x X, we obtain t [,] sup x g(x(t))dt. max g(ξ) ξ. So, (C) is poved. Moeove, fo each > and each y X such that y, we have ϕ () inf x < g(y(t))dt g(x(t))dt y, x thus, since max x(t) x fo evey x X, we obtain t [,] inf x < g(y(t))dt g(x(t))dt y inf x x < ξ, y x fom which, being < y x y fo evey x X such that x <, and unde futhe condition g(y(t))dt max g(ξ), ξ we can wite inf x < So, (C) is also poved. ξ y x ξ. y

7 A THREE CRITICAL POINTS THEOREM... 7 Now, in ode to pove (ii ) of Theoem B, taking into account (C) and (C), it suffices to find > and y X such that y, max g(ξ) ξ < To this end, we define 4dt if t [, [ 4 y(t) := d if t [, 3] 4 4 g(y(t))dt max g(ξ), and ξ ξ. (3.) y 4d( t) if t ] 3 4, ] and := c. Clealy, y X and y = 8d. Hence, since c < d, we have y >. Moeove, we have so that g(y(t))dt = g(d) + g(t)dt, d ξ = y g(d) + d g(t)dt max g(ξ) 8d, hence hypothesis (k) gives (3.) and g(y(t))dt > max g(ξ). ξ Thus, the conclusion follows by Theoem B, taking into account that, witing (C) and (C) with the y(t) and defined above, ϕ () 4d g(d) + d g(t)dt max g(ξ) and ϕ () c max g(ξ). Remak 3.. In Theoem 3. instead of hypothesis (k) we can also use the following less geneal, but simple: (k ) g(ξ)dξ ; (k ) g(ξ) c < 6 g(d), fo evey ξ [ c, c]. d

8 8 DIEGO AVERNA AND GABRIELE BONANNO In fact, taking into account that < c < d, using (k ) and (k ) we obtain g(d) < g(d) max g(ξ) g(d) + d 6 d 4 d d 4 g(t)dt max g(ξ) d thus, using again (k ), hypothesis (k) of Theoem 3. is fulfilled. We obseve that the assumptions (k ) and (k ) ae vey simila to those of Theoem of [] (see Remak of []) and Theoem 3. of [7]. Hee we have a pecise estimate of the inteval of paametes fo which the poblem has at least thee solutions, while in those theoems the unifom boundedness of the noms of the solutions with espect to λ is obtained. Remak 3.. In Theoem 3. the assumption (kk), togethe with (k), ensues the thid solution and cannot be dopped as the function f(u) = e u shows (see [9]). Also the assumption (k) cannot be dopped as the function f(u) = shows (see also Remak 3.4). We now give a simple example of application of Theoem 3.. Example 3.. It is simple to veify that the function g(u) = e u u + 3(u +) 5 3 3, 5 5 besides (kk) of Theoem 3., satisfies (k ) and (k ) of Remak 3. by choosing, fo instance, c = and d = ; moeove, we have ] 8, [ g(d) + d 8d c g(t)dt max g(ξ), max g(ξ) Theefoe, thanks to Theoem 3., fo each λ ], [, the poblem 8 { u = λ ( ) e u u ( u) + (u 3 + ) u() = u() =, admits at least thee non tivial classical solutions. An immediate consequence of Theoem 3. is the following Theoem 3.. Assume that thee exist fou positive constants c, d, a, s, with c < d and s <, such that: (k ) (kk) max g(ξ) c < < 4 g(d) + d g(ξ) a( + ξ s ) fo all ξ IR. g(t)dt max g(ξ) d ; Then, the poblem (AD) admits at least thee classical solutions.

9 A THREE CRITICAL POINTS THEOREM... 9 Poof. It is clea that Theoem 3. can be used. So, it is enough to obseve that, owing to (k ), we have 8d c g(d) + d g(t)dt max g(ξ), max g(ξ). Remak 3.3. On the basis of Remak 3., in Theoem 3. instead of hypothesis (k ) we can use the following simple: (k ) g(ξ)dξ, (k ) g(ξ) c < < 6 g(d), fo evey ξ [ c, c]. d Poof of Theoem C. Taking into account Remak 3.3, we can choose c =, d = and apply Theoem 3.. Remak 3.4. Poblem (AD) has been studied, fo instance, in [], [] and []. The key assumption in [] is (see (iii) in Theoem of []) (HT) thee exist b > and < e < such that f(y) b fo evey e( e) y [b, b(e+) ], 4e and the authos give an example (see Remak 7 of []) whee (HT) fails and the poblem has only the tivial solution. The following example shows a poblem that admits at least two positive classical solutions even if the assumption (HT) is not veified. Example 3.. Let h : IR IR be the function defined as follows if ξ ], ] 5 ξ 4 if ξ ] h(ξ) :=, ] 5 ξ + 36 if ξ ], 9] 5 if ξ ] 9, + [. 5 By choosing, fo instance, c = and d =, it is simple to veify all the assumptions 5 of Theoem 3.. So, taking into account that h is nonnegative and vanishes at, fom the maximum pinciple the poblem { u = h(u) u() = u() =, admits at least two positive classical solutions. On the othe hand, the assumption (HT) fails, as it is simple to see.

10 DIEGO AVERNA AND GABRIELE BONANNO As application of Theoem. we give the following Theoem 3.3. Assume that thee exist fou positive constants c, d, a, s, with c < d and s <, such that: (k ) g(ξ)dξ > ; (k ) max g(ξ) = ; (kk) g(ξ) a( + ξ s ) fo all ξ IR. ] d 3 Then, fo each λ [, g(t)dt, + the poblem (ADE) admits at least two nontivial and nonnegative classical solutions. Poof. Since assumption (k ) implies that f() =, it is not estictive to suppose that f() = fo x <. Clealy, the solutions of the poblem (ADE) with such an f ae nonnegative and they ae also solutions of the poblem (ADE) with the oiginal one. Now, let X, Φ, Ψ be as in poof of Theoem 3., and define dt if t [, ] x (t) := d( t) if t ], ], x (t) := fo evey t [, ], and := c. Clealy, we have Φ(x ) =, Ψ(x ) =, Φ(x ) = d and Ψ(x ) = d g(t)dt. Since c < d, one has that Φ(x ) < < Φ(x ). Moeove, taking into account that max x fo evey x X, we have Ψ(x) max g(ξ) = fo evey t [,] x X such that Φ(x). Then, inf ), and, thanks Φ (],[) w Φ (],]) to (k ), Ψ(x ) < Ψ(x ). Hence, using Theoem., since Φ(x ) inf Φ (],[) Ψ(x ) Ψ(x ) = Φ(x ) d 3 we have the conclusion. Ψ(x ) g(t)dt, Poof of Theoem D. As in the poof of Theoem 3.3, we can suppose f(x) = fo x <. Clealy, thee exists c > such that max g(t) =. Moeove, since t c f(x) lim x + x q ], + [, thee exists d > c such that g(ξ)dξ >, and thee exists a > such that g(ξ) a(+ ξ +q ) fo all ξ IR. Theefoe, we can use Theoem 3.3 to each the conclusion.

11 A THREE CRITICAL POINTS THEOREM... Refeences [] R.I.Avey, J.Hendeson, Thee symmetic positive solutions fo a second-ode bounday value poblem, Appl. Math. Lettes 3 (), -7. [] G.Bonanno, Existence of thee solutions fo a two point bounday value poblem, Appl. Math. Lettes 3 (), [3] G.Bonanno, A minimax inequality and its applications to odinay diffeential equations, J. Math. Anal. Appl. 7 (), -9. [4] G.Bonanno, Multiple solutions fo a Neumann bounday value poblem, J. Nonlinea Convex Anal., 4 (3), to appea. [5] G.Bonanno, P.Candito, Thee solutions to a Neumann poblem fo elliptic equations involving the p-laplacian, Ach. Math. (Basel), 8 (3), [6] G.Bonanno, R.Livea, Multiplicity theoems fo the Diichlet poblem involving the p-laplacian, Nonlinea Anal., 54 (3), -7. [7] G.Bonanno, Some emaks on a thee citical points theoem, Nonlinea Anal., 54 (3), [8] P.Candito, Existence of thee solutions fo a nonautonomous two point bounday value poblem, J. Math. Anal. Appl. 5 (), [9] I.M.Gelfand, Some poblems in the theoy of quasilinea equations, Ame. Math. Soc. Tanslations 9 (963), [] J.Hendeson, H.B.Thompson, Existence of multiple solutions fo second ode bounday value poblems, J. Diffeential Equations 66 (), [] J.Hendeson, H.B.Thompson, Multiple symmetic positive solutions fo a second ode bounday value poblem, Poc. Ame. Math. Soc. 8 (), [] R.Livea, Existence of thee solutions fo a quasilinea two point bounday value poblem, Ach. Math. (Basel), 79 (), [3] S.A.Maano, D.Moteanu, On a thee citical points theoem fo non-diffeentiable functions and applications to nonlinea bounday value poblems, Nonlinea Anal. 48 (), [4] P.Pucci, J.Sein, A mountain pass theoem, J. Diffeential Equations 6 (985), [5] B.Riccei, On a thee citical points theoem, Ach. Math. (Basel) 75 (), -6. [6] B.Riccei, A geneal vaiational pinciple and some of its applications, J. Comput. Appl. Math. 3 (), 4-4. [7] B.Riccei, On a classical existence theoem fo nonlinea elliptic equations, in Expeimental, constuctive and nonlinea analysis, M.Théa ed., 75-78, CMS Conf. Poc. 7, Canad. Math. Soc.,. [8] E.Zeidle, Nonlinea functional analysis and its applications, Vol. III. Belin-Heidelbeg-New Yok 985. (D.Avena) Dipatimento di Matematica ed Applicazioni, Facoltà di Ingegneia, Univesità di Palemo, Viale delle Scienze, 98 Palemo (Italy) addess: avena@unipa.it (G.Bonanno) Dipatimento di Infomatica, Matematica, Elettonica e Taspoti, Facoltà di Ingegneia, Univesità di Reggio Calabia, Via Gaziella (Feo di Vito), 89 Reggio Calabia (Italy) addess: bonanno@ing.unic.it

SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS

Fixed Point Theoy, Volume 5, No. 1, 2004, 71-80 http://www.math.ubbcluj.o/ nodeacj/sfptcj.htm SOME SOLVABILITY THEOREMS FOR NONLINEAR EQUATIONS G. ISAC 1 AND C. AVRAMESCU 2 1 Depatment of Mathematics Royal