Dynamic Systems and Applications 26 (2017) xx-xx. GRADIENT NONLINEAR ELLIPTIC SYSTEMS DRIVEN BY A (p, q)-laplacian OPERATOR

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1 Dynamic Sytem and Application xx-xx GRADIENT NONLINEAR ELLIPTIC SYSTEMS DRIVEN BY A p, -LAPLACIAN OPERATOR DIEGO AVERNA a, GABRIELE BONANNO b, AND ELISABETTA TORNATORE c a Dipatimento di Matematica e Infomatica, Univeità degli tudi di Palemo, Via Achiafi, Palemo, Italy. diego.avena@unipa.it b Depatment of Engineeing, Univeity of Meina, c.da Di Dio Sant Agata, Meina, Italy. bonanno@unime.it c Dipatimento di Matematica e Infomatica, Univeità degli tudi di Palemo, Via Achiafi, Palemo, Italy. elia.tonatoe@unipa.it Dedicated with geat eteem to Pofeo R. Agawal ABSTRACT. In thi pape, uing vaiational method and citical point theoem, we pove the exitence of multiple weak olution fo a gadient nonlinea Diichlet elliptic ytem diven by a p, -Laplacian opeato. AMS MOS Subject Claification. 35J35, 35J60, 35J92, 58E30. Keywod: Nonlinea elliptic ytem, citical point, vaiational method, Diichlet condition, multiple olution, p-laplacian. Ω.. INTRODUCTION Let Ω R N N 3 be a non-empty bounded open et with a mooth bounday In thi pape we tudy the following gadient nonlinea elliptic ytem with Diichlet condition p u + ax u p 2 u = λf u x, u, v in Ω,. v + bx v 2 v = λf v x, u, v in Ω, u = v = 0 on Ω whee, p, >, λ i a poitive eal paamete, by we denote the -Laplacian opeato defined by u = div u 2 u fo all u W, 0 Ω = p,. In the tatement of ytem. the eaction tem F : Ω R 2 R i a C - function uch Received Septembe 3, $5.00 c Dynamic Publihe, Inc.

2 2 D. AVERNA, G. BONANNO, AND E. TORNATORE that F x, 0, 0 = 0 fo evey x Ω, F u, F v denote the patial deivative of F epect on u and v epectively. We uppoe, moeove that a, b L Ω and.2 e inf x Ω ax = a 0 0, e inf x Ω bx = b 0 0. In ecent yea the exitence and tuctue of olution fo poblem diven by p- Laplacian ha found many inteet and diffeent appoache ha been developed. The vaiational method ae ued to obtain weak olution a citical point fo a uitable enegy function. Thi appoach i employed to deal with ytem of gadient type i.e. the nonlineaitie ae the gadient of a C functional. We efe the eade to 3] fo a complete oveview fo thi ubject and ],2], 3], 4], 8] 9],],4] and the efeence theein fo moe development. In 4] the autho tudied the exitence of citical point of functional thee point wee the olution of uailinea elliptic ytem involving p, -Laplacian with < p, < N. They conideed ubcitical gowth condition, and unde uitable condition on the lineaitie poved the exitence of non-tivial olution accoding vaiou cae: ublinea, upelinea and eonant cae. In 9] and 4] the autho get the exitence of thee olution fo a cla of uailinea elliptic ytem involving p, -Laplacian with p, > N. In ] the autho genealized the eult obtained in 4] to ytem involving p, p 2,, p n -Laplacian. Analogou tudie but fo diffeent bounday condition can be found in 2], 3] mixed bounday condition. It i woth noticing that in 4] pecie value of paamete λ ae not etablihed. The aim of pape i to detemine the exitence of multiple olution a the paamete λ > 0 vaie in an appopiate inteval. In thi wok, without loing geneality we uppoe that < p < N. The pape i aanged a follow. Fit we obtain the exitence of one non-zeo weak olution of ytem. without auming any aymptotic condition neithe at zeo no at infinity ee Theoem 3.. Next we pove the exitence of at leat two non zeo weak olution in which the Amboetti-Rabinowitz condition i ued ee Theoem 3.2. Finally, we peent a thee olution exitence eult unde an appopiate condition on the the nonlinea tem F ee Theoem 3.3. Moeove the cae in which F i autonomuo i peented and ome example ae given. 2. PRELIMINARIES In thi ection, we ecall definition and theoem ued in the pape. Let X, be a eal Banach pace and Φ, Ψ : X R be two Gâteaux diffeentiable functional and ], + ]. We ay that functional I = Φ Ψ atifie the Palai-Smale condition cut off uppe at in hot P S ] -condition if any euence {u n } in X uch that

3 α {Iu n } i bounded, α 2 lim n + I u n X = 0, α 3 Φu n < n N, ha a convegent ubeuence. GRADIENT NONLINEAR ELLIPTIC SYSTEMS... 3 When = + the peviou definition coincide with the claical P S-condition, while if < uch condition i moe geneal than the claical one. We efe to 5] fo moe detail. We ay that functional I atifie the weak Palai-Smale condition W P S-condition if any bounded euence {u n } in X uch that α and α 2 hold, admit a convegent ubeuence. Ou main tool i a of local minimum theoem obtained in 5]. We ecall hee it veion peented in 6]. Theoem 2.. 6, Theoem 2.3]. Let X be a eal Banach pace, and let Φ, Ψ : X R be two continuouly Gâteaux diffeentiable functional uch that inf X Φ = Φ0 = Ψ0 = 0. Aume that thee exit R and ū X, with 0 < Φū <, uch that 2. and, fo each λ Λ := P S ] -condition. ] Φū Ψū, Then, fo each λ Λ := up u Φ ], Ψu up u Φ ], Ψu ] Φū, Ψū < Ψū Φū that I λ u λ I λ u fo all u Φ ]0, and I λ u λ = 0. the functional I λ = Φ λψ atified, thee i u up u Φ ], Ψu λ Φ ]0, uch Now, we alo ecall a ecent eult obtained in 8] that inue the exitence of at leat two non-zeo citical point fo diffeentiable functional. Theoem , Theoem 3.2]. Let X be a eal Banach pace and let Φ, Ψ : X R be two continuouly Gâteaux diffeentiable functional uch that inf X Φ = Φ0 = Ψ0 = 0. Aume that thee exit R and ū X, with 0 < Φū <, uch that and fo each λ Λ := up u Φ ],] Ψu < Ψū Φū, the functional I λ = Φ λψ atifie ] Φū Ψū up u Φ ],] Ψu PS-condition and it i unbounded fom below. Then, fo each λ Λ := ] Φū, Ψū up u Φ ],] Ψu non-zeo citical point u λ,, u λ,2 uch that I λ u λ, < 0 < I λ u λ,2., the functional I λ admit two

4 4 D. AVERNA, G. BONANNO, AND E. TORNATORE Finally we point out an othe eult which inue the exitence of at leat thee citical point. Theoem 2.3. ha been obtained in 6], it i a moe pecie veion of Theoem 3.2 of 7] and Theoem 3.6 of 0]. Theoem , Theoem 2.]. Let X be a eal Banach pace, Φ, Ψ : X R be two continuouly Gâteaux diffeentiable functional with Φ bounded fom below and Φ0 = Ψ0 = 0. Aume that thee exit R and ū X, with 0 < < Φū, uch that i up u Φ ],] Ψu ii fo each λ Λ := < Ψū Φū ] Φū, Ψū up u Φ ],] Ψu bounded fom below and atifie PS-condition. the functional I λ = Φ λψ i Then, fo each λ Λ, the functional I λ = Φ λψ ha at leat thee ditinct citical point in X. Thoughout in the pape we uppoe that the following condition hold H thee exit two non negative contant a, a 2 and two contant, pn N p and, N N uch that F tx, t, t 2 a + a 2 t fo evey x, t Ω R 2. F t2 x, t, t 2 a + a 2 t 2 Clealy, fom H follow t 2.2 F x, t, t 2 a t + t 2 + a 2 In fact, thee exit 0 < θ < uch that + t 2 fo evey x, t Ω R 2. by uing H, we have F x, t 2 i= F x, t = F x, t F x, 0 = F x, θt t 0 F t i x, θtt i dθ a 0 + a 2 θt t + a + θt 2 t 2 ]dθ a t + t 2 + a t 2. + t 2 We conide the Sobolev pace X = W,p 0 Ω W, 0 Ω endowed with the nom fo all u, v X, whee u, v := u W,p 0 Ω + v W, 0 Ω u W,p 0 Ω Ω := ux p + ax ux p dx p,

5 GRADIENT NONLINEAR ELLIPTIC SYSTEMS... 5 v W, 0 Ω Ω := vx + bx vx dx, that ae, taking into account.2, euivalent to uual one. A function u, v X i aid a weak olution to ytem. if ux p 2 ux w xdx + vx 2 ux w 2 x]dx Ω + ax ux p 2 uxw x + bx vx 2 vxw 2 x]dx Ω λ F u x, ux, vxw x + F v x, ux, vxw 2 x]dx = 0 Ω fo evey w, w 2 X. Now, conide < h < N and put h = hn. Denote by Γ the Gamma function N h defined by Γ = + 0 z e z dz, > 0. Fom the Sobolev embedding theoem, fo evey u W,h 0 Ω thee exit a contant cn, h R + uch that 2.3 u L h Ω cn, h u W,h 0 Ω the bet contant that appea in 2.3 i ee 5]. cn, h = π 2 N h h N h h Γ + N 2 ΓN Γ N h Γ + N N h Fixing, h in vitue of Sobolev embedding theoem, fo evey u W,h 0 Ω, thee exit a poitive contant c,h uch that 2.4 u L Ω c,h u W,h 0 Ω and, in vitue of Rellich theoem the embedding i compact. By uing Hölde ineuality, we have 2.5 c,h µω h h cn, h whee µω denote the Lebegue meaue of the et Ω. Now, we put 2.6 c, = max{c,p, c, }, c, = max{c,p, c, }, whee the contat and ae given by H. Moeove, let 2.7 D := up ditx, Ω. x Ω N

6 6 D. AVERNA, G. BONANNO, AND E. TORNATORE Simple calculation how that thee i x 0 Ω uch that Bx 0, D Ω. Finally, we et 2.8 κ = π N 2 D N Γ + N 2, { 2.9 σ = 2p max + a D p 2 N, { 2.0 τ = 2 p min + a D p 2 N 0, } + b D 2 N. } + b D 2 N 0. In ode to tudy poblem., we will ue the functional Φ, Ψ : X R defined by putting 2. Φu, v := p u p + W,p 0 Ω v, Ψu, v := W, 0 Ω fo evey u, v X and put I λ = Φ λψ fo λ > 0. Ω F x, ux, vxdx Clealy, Φ i a coecive, weakly euentially lowe emicontinuou, continuouly Gâteaux diffeentiable and it deivative at point u, v X i defined by Φ u, vw, w 2 = Ω ux p 2 ux w x + ax ux p 2 uxw x]dx + Ω vx 2 vx w 2 x + bx vx 2 vxw 2 x]dx. fo evey w, w 2 X. Moeove, Ψ i well defined, weakly euentially uppe emicontinuou continuouly Gâteaux diffeentiable with compact deivative and it deivative at point u, v X i defined by Ψ u, vw, w 2 = F u x, ux, vxw x + F v x, ux, vxw 2 x]dx, Ω fo evey w, w 2 X. A citical point fo the functional I λ := Φ λψ i any u, v X uch that Φ u, vw, w 2 λψ u, vw, w 2 = 0 w, w 2 X. Hence, the citical point fo functional I λ olution to ytem.. := Φ λψ ae exactly the weak We have the following eult Lemma 2.4. Fix λ > 0 the functional I λ = Φ λψ atifie the WPS-condition.

7 GRADIENT NONLINEAR ELLIPTIC SYSTEMS... 7 Poof. Fixed λ > 0, we claim that the functional I λ = Φ λψ atifie the WPS- condition. Fo thi end, let {u n, v n } be a bounded euence in X uch that I λ u n, v n i bounded and I λ u n, v n ω u n, ω 2 v n ε n ω u n, ω 2 v n fo all ω, ω 2 X and whee ε n 0 +. Hence, taking a ubeuence if neceay, we have u n, v n u, v in X, u n u in L α Ω fo all α, p v n v in L β Ω fo all β, Fom the peviou elation, witten with ω = u and ω 2 = v we infe 2.2 Φ u n, v n u u n, v v n λψ u n, v n u u n, v v n ε n u u n, v v n. We obeve that Φ u n, v n u u n, v v n = u n p W,p 0 Ω v n W, 0 Ω + Ω u nx p 2 u n x ux + ax u n x p 2 u n xux]dx + Ω v nx p 2 v n x ux + bx v n x p 2 v n xvx]dx and, beaing in mind that fo all a, b R and p >, one ha a p b p p a p + p b p 2.3 Φ u n, v n u u n, v v n p u p W,p 0 Ω + v W, 0 Ω p u n p W,p 0 Ω v n W, 0 Ω. Moeove, by uing H we have Ψ u n, v n u u n, v v n a un u L Ω + v n v L Ω whee α = +a 2 u n u L p Ω n u L α Ω + v n v L Ω n v L β Ω p and β = p +, hence obeving that α < + p and β <, we obtain 2.4 lim n + Ψ u n, v n u u n, v v n = 0. Fom 2.2 and 2.3 we obtain ε n u u n, v v n + p u n p W,p 0 Ω + v n W, 0 Ω p u p W,p 0 Ω + v W, 0 Ω λψ u n, v n u u n, v v n,

8 8 D. AVERNA, G. BONANNO, AND E. TORNATORE fom thi, taking into account 2.4 we have lim up n + p u n p + W,p 0 Ω v n W, 0 Ω p u p + W,p 0 Ω v W, 0 Ω thu, ince X i unifomly convex, Popoition III. 30 of 2] enue that {u n, v n } convege to u, v in X. Hence ou claim i poved. 3. MAIN RESULTS By uing the notation of Section 2 we have ou main eult Theoem 3.. We uppoe that H hold and aume that i F x, t 0 fo evey x, t Ω R 2 + whee R 2 + = {t = t, t 2 R 2 : t i 0 i =, 2}; i 2 thee exit two poitive contant and δ with δ p + δ < κσ, uch that inf x Ω F x, δ, δ δ p + δ > 2N σ a c, p p p + + a 2 c, p + whee a, a 2, and ae given by H and κ, σ ae given by 2.8 and 2.9. Then, fo each λ 2N σδ p +δ, inf x Ω F x,δ,δ a c, p + +a 2 c, ytem. ha at leat one non-zeo weak olution. Poof. Ou goal i to apply Theoem 2.. opeato defined in Taking into account 2.2, it follow that Ψu, v = F x, ux, vxdx Ω a u L Ω + v L Ω + a 2 u L Ω p p p + ], the Conide the Sobolev pace X and the + v L Ω. Let ]0, +, then fo evey u, v X uch that Φu, v <, by uing 2.4 and 2.6 we get 3.2 Ψu, v a c, p p + + a2 c, p p Hence, fom 3.2, we have 3.3 up u,v Φ ], Ψu, v + a c, p p p + p p +a 2 c, p +.,

9 fo evey > 0. GRADIENT NONLINEAR ELLIPTIC SYSTEMS... 9 Now, we chooe the function ū, ū defined by putting 0 if x Ω \ Bx 0, D 3.4 ūx = 2δ D D N j= x j x j0 2 if x Bx 0, D \ Bx 0, D, 2 δ if x Bx 0, D Clealy ū, ū X and by uing 2.8, 2.9 and 2.0 we have δ p + δ κτ < Φū, ū < δp + δ κσ. p In vitue of 3.5 and beaing in mind that δ p + δ <, we obtain κσ 0 < Φū, ū < and by uing i we have 3.6 Ψū, ū = F x, ūx, ūxdx F x, δ, δdx k Ω Bx 0, D 2 2 inf F x, δ, δ. N x Ω Hence, by 3.5 and 3.6, one ha 3.7 Ψū, ū Φū, ū inf x Ω F x, δ, δ. 2 N σ δ p + δ By uing 3.3, 3.7 and taking into account i 2, we get up u,v Φ ], Ψu,v a c, p p p + + a 2 c, p + < 2 N σδ p +δ inf x Ω F x, δ, δ Ψū,ū Φū,ū. Moeove, let be 2 > 0 and {u n, v n } a euence in X uch that α 3 hold, ince Φ i coecive we have that {u n, v n } i bounded. Then by uing Lemma 2.. we obtain that WPS-condition implie P S 2] -condition. Theefoe, all the aumption of Theoem 2. ae atified. So, fo each λ ] Φū,ū Ψū,ū, 2N σδ p +δ, inf x Ω F x,δ,δ a c, p + +a 2 c, up u,v Φ ], Ψu,v p + the functional I λ ha at leat one non-zeo citical point that i weak olution of ytem..

10 0 D. AVERNA, G. BONANNO, AND E. TORNATORE The following eult, in which Amboetti-Rabinowitz condition i alo ued, enue the exitence at leat two non-zeo weak olution. Theoem 3.2. We uppoe that H hold. Aume that j F x, t 0 fo evey x, t Ω R 2 + whee R 2 + = {t = t, t 2 R 2 : t i 0 i =, 2}; j 2 thee ae two poitive contant and δ with uch that inf x Ω F x, δ, δ δ p + δ > 2N σ δ p + δ < κσ, a c, p p p + + a 2 c, p + whee a, a 2, and ae given by H and κ, σ ae given by 2.8 and 2.9, and that thee ae two poitive contant µ > p and R uch that AR 0 < µf x, t t t F x, t fo all x Ω and t > R. Then, fo each λ 2N σδ p +δ p p, inf x Ω F x,δ,δ a c, p + +a 2 c, the ytem. ha at leat two non-zeo weak olution. Poof. Ou goal i to apply Theoem 2.2. p + ], Conide the Sobolev pace X and the opeato defined in 2. taking into account that the egolaity aumption on Φ and Ψ ae atified. Aguing a in the poof of Theoem 3., put ū, ū a in 3.4, by uing i, j 2, 3.5 and beaing in mind that δ p + δ >, we obtain κσ and Fix λ 2N σδ p +δ 0 < Φū, ū < up u,v Φ ],] Ψu, v, inf x Ω F x,δ,δ a c, p + +a 2 c, < Ψū, ū Φū, ū. p + tandad computation, thee i a poitive contant C uch that 3.8 F x, t C t µ x Ω, t > R. Fom 3.8 it follow that I λ i unbounded fom below., fom AR, by

11 GRADIENT NONLINEAR ELLIPTIC SYSTEMS... Now, by uing Lemma 2.. to veify PS-condition it i enough to pove that any euence of Palai-Smale i bounded. To thi end, taking into account AR one ha 3.9 µi λ u n, v n I λ u n, v n X u n, v n µi λ u n, v n I λ u n, v n u n, v n = µφu n, v n λµψu n, v n Φ u n, v n u n, v n + λψ u n, v n u n, v n = µ p u n p W,p 0 Ω + µ v n W, 0 Ω λ Ω µf x, u nx, v n x F u x, u n x, v n xu n x + F v x, u n x, v n xv n x µ p u n p W,p 0 Ω + µ v n W, 0 Ω + C. whee C i a contant. If {u n, v n } i not bounded fom 3.9 we have a contadiction. Theefoe, all condition of Theoem 2.2 ae atified, then the ytem., fo each λ 2N σδ p +δ, inf x Ω F x,δ,δ a c, p + +a 2 c, leat two non-zeo weak olution. p +, admit at Now, we point out the following eult on the exitence of at leat thee weak olution. Theoem 3.3. We uppoe that H hold and aume that j F x, t 0 fo evey x, t Ω R 2 + whee R 2 + = {t = t, t 2 R 2 : t i 0 i =, 2}; h 2 thee exit thee poitive contant α, β and b with α < p and β < uch that F x, t, t 2 b + t α + t 2 β fo almot evey x Ω and fo evey t, t 2 R 2 +; h 3 thee exit two poitive contant and δ with uch that inf x Ω F x, δ, δ δ p + δ > 2N σ δ p + δ > p κτ, a c, p p p + + a 2 c, p + whee a, a 2, and ae given by H and κ, σ ae given by 2.8 and 2.9. p p ]

12 2 D. AVERNA, G. BONANNO, AND E. TORNATORE Then, fo each λ 2N σδ p +δ, inf x Ω F x,δ,δ a c, p + +a 2 c, ytem. ha at leat thee weak olution. p +, the Poof. Ou goal i to apply Theoem 2.3. Conide the Sobolev pace X and the opeato defined in 2. taking into account that the egolaity aumption on Φ and Ψ ae atified, ou aim i to veify i and ii. Aguing a in the poof of Theoem 3., put ū, ū a in 3.4, by uing 3.5 and beaing in mind that δ p + δ > p κτ, we obtain Φū, ū > > 0. Theefoe, the aumption i of Theoem 2.3 i atified. We pove that the functional I λ = Φ λψ i coecive fo all poitive paamete, in fact by uing condition h 2 we have I λ u, v = Φu, v λψu, v = p u p + W,p 0 Ω v λ F x, ux, vxdx W, 0 Ω Ω p u p λbc α W,p 0 Ω,µΩ p α p u α W,p 0 Ω + v λbc β W,,µΩ β v β 0 Ω W, 0 Ω λbµω. We obeve that the functional I λ = Φ λψ i bounded fom below becaue it i coecive and weakly euentially lowe emicontinuou. Now, by uing Lemma 2.. to veify PS-condition it i enough to obeve that ince the functional I λ = Φ λψ i coecive any euence of Palai-Smale i bounded. Then alo condition ii hold. Hence all the aumption of Theoem 2.3 ae atified. So, fo each λ 2N σδ p +δ, inf x Ω F x,δ,δ a c, p + +a 2 c, p +, the functional I λ ha at leat thee ditinct citical point that ae weak olution of ytem.. Now, we point out the cae when F doe not depend on x Ω, we conide the following ytem p u + ax u p 2 u = λf u u, v in Ω, 3.0 v + bx v 2 v = λf v u, v in Ω, u = v = 0 on Ω we have the following eult.

13 GRADIENT NONLINEAR ELLIPTIC SYSTEMS... 3 Coollay 3.4. Let F : R 2 R be a nonnegative and C -function atifying H and aume that F t, t lim = +. t 0 + t Then, thee i λ > 0 uch that, fo each λ ]0, λ, the poblem 3.0 admit at leat one non-zeo weak olution. Poof. Fix λ = a c, p p + + a 2 c, +. whee the contant a, a 2, c, and c, ae given by condition H and 2.6. By uing 2.8 and 2.9 and taking into account that F t, t lim t 0 + t = +. we obtain that fo each λ ]0, λ thee exit h > 0 uch that F t,t t t < h. Now, conide 0 < δ < min{ h, 2κσ } we have > 2N σ λ fo each F δ,δ δ p +δ > 2N σ λ > 2N σ λ δ p + δ < κσ Then, by chooing = all aumption of Theoem 3. ae atified and the poof i complete. Coollay 3.5. Let F : R 2 R be a nonnegative and C -function atifying H, AR and aume that F t, t lim = +. t 0 + t Then, thee i λ > 0 uch that, fo each λ ]0, λ, the poblem 3.0 admit at leat two non-zeo weak olution. Poof. λ = a c, p p + + a 2 c, +. whee the contant a, a 2, c, and c, ae given by condition H and 2.6. The concluion follow aguing a in the poof of Coollay 3. taking into account Theoem 3.2. Now, we peent ome example that illutate ou eult.

14 4 D. AVERNA, G. BONANNO, AND E. TORNATORE Example 3.6. Let Ω be an open ball of adiu one in R 6. Conide the function F : R 2 R defined by F t, t 2 = log + t 2 + t 2 2. We obeve that 2t F t t, t 2 = + t 2 + t 2 2 2t 2 F t2 t, t 2 = + t 2 + t 2 2 then, chooing = 3, p = 4, = = 2 a = 0 and a 2 = 2 the condition H hold. We obeve Then by uing Coollay 3., put F t, t lim t 0 + t 3 λ = 0, 46 = +. λ ]0, λ the following ytem 4 u + u 3 u = λf u u, v in Ω, 3 v + v 2 v = λf v u, v in Ω, u = v = 0 on Ω admit at leat one non-zeo weak olution in X = W,4 0 Ω W,3 0 Ω. Example 3.7. Let Ω be an open ball of adiu one in R 6. Conide the function F : R 2 R defined by t 8 + t 2 + t 4 + t 4 2e t 2 +t2 2 t, t 2 0, 0 F t, t 2 = 0 t, t 2 = 0, 0 We obeve that + 2 2t 2 8 i + t4 +t4 2 t t 2 i e t 2 +t t2 2 t, t 2 0, 0 F ti t, t 2 = t 8, t 2 = 0, 0 then, chooing p = = 3, = = 4, a = 3 and a 2 = 6 the condition H hold. Moeove, chooe µ = 4 and R = we have 0 < 4F t, t 2 t F t t, t 2 + t 2 F t2 t, t 2 fo evey t, t 2 R 2 with t, t 2 >. We obeve F t, t lim t 0 + t 3 = +.

15 GRADIENT NONLINEAR ELLIPTIC SYSTEMS... 5 Then by uing Coollay 3.2, put λ = 0.06, λ ]0, λ the following ytem 3 u + u 3 u = λf u u, v in Ω, 3 v + v 2 v = λf v u, v in Ω, u = v = 0 on Ω admit at leat two non-zeo weak olution in X = W,3 0 Ω W,3 0 Ω. Acknowledgment. The autho ae membe of the Guppo Nazionale pe l Analii Matematica, la Pobabilità e le loo Applicazioni GNAMPA of the Itituto Nazionale di Alta Matematica INdAM. REFERENCES ] G.A. Afouzi and S. Heidakhani, Exitence of thee olution fo a cla of Diichlet uailinea euation involving p,..., p m -Laplacian, Nonlinea Anal., 70:35 43, ] D. Avena and E. Tonatoe, Infinitely many weak olution fo a mixed bounday value ytem with p,..., p m -Laplacian, Electon. J. Qual. Theoy Diffe. Eu. 57: 8, ] D. Avena and E. Tonatoe, Odinay p,..., p m -Laplacian ytem with mixed bounday value condition, Nonlinea Anal.: Real Wold Appl., 28:20 3, ] L. Boccado and D. G. Defigueiedo, Some emak on a ytem of uailinea elliptic euation, NoDEA Nonlinea Diffeential Euation Appl., 9: , ] G. Bonanno, A citical point theoem via the Ekeland vaiational pinciple, Nonlinea Anal., 75: , ] G. Bonanno, Relation between the mountain pa theoem and local minima, Adv. Nonlinea Anal., : , ] G. Bonanno and P. Candito, Non-diffeentiable functional and application to elliptic poblem with dicontinuou nonlineaitie, J. Diffeential Euation 244: , ] G. Bonanno and G. D Aguì, Two non-zeo olution fo elliptic Diichlet poblem, Z. Anal. Anwend. 35: , ] G. Bonanno, S. Heidakhani and D. O Regan, Multiple oltion fo a cla of Diichlet uailinea elliptic ytem diven by a p, -laplacian opeato, Dynam. Sytem Appl., 20:89 99, 20. 0] G. Bonanno and S. A. Maano, On the tuctue of the citical et of non-diffeentiable function with a weak compactne condition, Appl. Anal., 89: 0, 200. ] G. Bonanno and E. Tonatoe, Exitence and multiplicity of olution fo nonlinea elliptic Diichlet ytem, Electon. J. Diffeential Euation, 83:, ] H. Bézi, Analye fonctionelle - théoie et application, Maon, Pai, ] D. G. Defigueiedo, Nonlinea Elliptic Sytem, An. Acad. Bail. Ciênc., 72: , ] C. Li and C.-L. Tang, Thee olution fo a cla of uailinea elliptic ytem involving the p, Laplacian, Nonlinea Anal., 69: , ] G. Talenti, Bet contant in Sobolev ineuality, Ann. Mat. Pua Appl. 0: , ] E. Zeidle, Nonlinea functional analyi and it application, vol. III, Spinge, Belin, 990.

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

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