MULTIPLICITY OF POSITIVE SOLUTIONS FOR SECOND-ORDER DIFFERENTIAL INCLUSION SYSTEMS DEPENDING ON TWO PARAMETERS

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1 Electronic Journl of Differentil Equtions, Vol ), No. 9, pp ISSN: URL: or ftp ejde.mth.txstte.edu MULTIPLICITY OF POSITIVE SOLUTIONS FOR SECOND-ORDER DIFFERENTIAL INCLUSION SYSTEMS DEPENDING ON TWO PARAMETERS ZIQING YUAN, LIHONG HUANG, CHUNYI ZENG Abstrct. We consider the two-point boundry-vlue system u i + u i λ ui F u 1,..., u n) + µ ui Gu 1,..., u n), u i ) = u i b) = 0 u i 0, 1 i n. Applying version of nonsmooth three criticl points theorem, we show the existence of t lest three positive solutions. 1. Introduction In the previous decdes, there hs been lot of interest in sclr periodic problems driven by the one dimension p-lplcin. Some results cn be found in [3, 18, 5, 4] nd references therein. We mention the works by Guo [1], Pino et l [3], Fbry nd Fyd [9] nd Dng nd Oppenheimer [7]. The uthors used degree theory nd ssumed tht the right-hnd side nonlinerity ft, ζ) is jointly continuous in t T = [, b] nd ζ R. Their conditions on f re lso symptotic nd there is no interction between the nonlinerity nd the Fučik spectrum of the one-dimensionl p-lplcin. Especilly, Heidrkhni nd Yu [13] considered the existence of t lest three solutions for clss of two-point boundry-vlue systems of the form u i + u i = λ ui F u 1,..., u n ) + µ ui Gu 1,..., u n ), u i) = u 1.1) ib) = 0 for 1 i n, where F, G : [, b] R n R re C 1 -functionls with respect to u 1,..., u n ) R n for.e. x [, b]. From the bove results, nturl question rises: wht will hppen when the potentil functions F nd G re not differentible in 1.1)? This is the min point of interest in our pper. Here, we extend the min results in [13] to clss of perturbed Motrenu-Pngiotopoulos functionls [19], which rises some essentil difficulties. The presence of non-differentible function probbly leds to no solution of 1.1) in generl. Therefore to overcome this difficulty, setting f i = ui F u 1,..., u n ) nd g i = ui Gu 1,..., u n ), we consider such functions f i 010 Mthemtics Subject Clssifiction. 49J40, 35R70, 35L85. Key words nd phrses. Neumnn problem; differentil inclusion system; loclly Lipschitz; nonsmooth criticl point. c 015 Texs Stte University. Submitted Februry 0, 015. Published November 30,

2 Z. YUAN, L. HUANG, C. ZENG EJDE-015/9 nd g i, which re loclly essentilly bounded mesurble nd we fill the discontinuity gps of f i nd g i, replcing f i nd g i by intervls [f i u 1,..., u n ), f + i u 1,..., u n )] nd [g i u 1,..., u n ), g + i u 1,..., u n )], where f i u 1,..., u n ) = lim ess inf u δ 0 + i ui <δ ui F u 1,..., u i,..., u n ), f + i u 1,..., u n ) = lim ess sup δ 0 + u i ui <δ ui F u 1,..., u i,..., u n ), g i u 1,..., u n ) = lim ess inf u δ 0 + i ui <δ ui Gu 1,..., u i,..., u n ), g + i u 1,..., u n ) = lim ess sup δ 0 + u i ui <δ ui Gu 1,..., u i,..., u n ). Then f i u 1,..., u n ), g i u 1,..., u n ) re lower semi-continuous, nd f + i u 1,..., u n ), g + i u 1,..., u n ) re upper semi-continuous. So insted of 1.1) we consider the following second-order Neumnn inclusion systems on bounded intervl [, b] in R < b) with nonsmooth potentils hemivritionl inequlity): u i + u i λ ui F u 1,..., u n ) + µ ui Gu 1,..., u n ), u i) = u ib) = 0, u i 0 for 1 i n, 1.) where λ, µ re two positive prmeters, F, G : R n R re mesurble nd loclly Lipschitz functions. We denote by ui F u 1,..., u n ) 1 i n) the prtil generlized grdient of F u 1,..., u n ) with respect to u i 1 i n), nd by ui Gu 1,..., u n ) 1 i n) the prtil generlized grdient of Gu 1,..., u n ) to u i 1 i n). Hemivritionl inequlity is new type of vritionl expressions which rises in problems of engineering nd mechnics, when one dels with nonsmooth nd nonconvex energy functionls. For severl concrete pplictions, we refer the reder to the monogrphs of [11, 19, 0, 1, ] nd references [8, 15, 10] nd therein. More precisely, Innizzoto [14] estblished nonsmooth three criticl points theorem nd give two pplictions for vritionl-hemivritionl inequlities depending on two prmeters. Mrno nd Motrenu [17] obtined nonsmooth version of Ricceri s theorem nd used the theorem to discuss the existence of solutions to the following discontinuous vritionl-hemivritionl inequlity problem ux) vx) ux)) dx Ω λ [J x, ux); vx) ux)) + µk) x, ux); vx) ux))] dx. Ω Kristály et l [16] generlized result of Ricceri concerning the existence of three criticl points of certin nonsmooth functionl, lso gve two pplictions, both in the theory of differentil inclusions. Our pproch is bsed on the nonsmooth criticl point for non-differentil functions due to Chng [5] nd the nonsmooth three criticl points which ws proved by Innizzotto [14]. Compred with the results in [14, 16, 17], our frmework presents new nontrivil difficulties. In prticulr, the presence of set-vlued rection terms G nd F require completely different devices in order to verify the pproprite conditions.

3 EJDE-015/9 MULTIPLICITY OF POSITIVE SOLUTIONS 3 We sy tht u = u 1,..., u n ) W 1, [, b])) n is wek solution of 1.) if the following conditions re stisfied + λ + µ u ix)v ix) dx + u i x)v i x) dx F 0 u i u 1 x),..., u n x); v i x)) dx G 0 u i u 1 x),..., u n x); v i x)) dx 0 1.3) for 1 i n nd ll v = v 1,..., v n ) W 1, [, b])) n. Moreover we ssume tht the nonsmooth potentil functions F nd G stisfy the following ssumptions: A1) F nd G re regulr on R n in the sense of Clrke [6]); A) There exists k 1 > 0 nd 1 > 0 such tht ω ω n k 1 u u n ) + 1 for ll u 1,..., u n ) R n nd ll ω i ui F u 1,..., u n ) 1 i n); A3) There exists k > 0 nd > 0 such tht ξ ξ n k u u n )+ for ll u 1,..., u n ) R n nd ll ξ i ui Gu 1,..., u n ) 1 i n). Our min results re the following: Theorem 1.1. Assume tht A1) A3) re stisfied nd there exist n + 3 positive constnts d, e, rη i, γ i, for 1 i n, such tht d + e < b, 0 < k 1 < M1 ηi c γi, nr nd where c = deb 4 5 d + e)) d + e), nd A4) F ζ 1,..., ζ n ) 0 for ech ζ i [0, deγ i ] 1 i n) nd F 0,..., 0) = 0; A5) n M 1 = η i c n b d + e))f deγ 1,..., deγ n ) γ i b ) mx F ζ 1,..., ζ n ) > 0, ζ 1,...,ζ n) A 1 where A 1 = {ζ 1,..., ζ n ) ζ i de b ηi }; then, there exist λ, λ 0, ν], 0 < ν < 1 nk 1, λ < λ, µ 1 > 0 nd σ 1 > 0 such tht for every λ [λ, λ ] nd µ 0, µ 1 ), system 1.) hs t lest three positive solutions in W 1, [, b])) n whose norms re less thn σ 1. If n = 1, then system 1.) turns into u + u λ u F u) + µ u Gu), u ) = u b) = 0, u 0. From Theorem 1.1, we hve the following result. Corollry 1.. Assume tht the following conditions re stisfied: A1 ) F nd G re regulr on R; 1.4)

4 4 Z. YUAN, L. HUANG, C. ZENG EJDE-015/9 A ) There exists k 3 > 0 nd 3 > 0 such tht ω k 3 u + 3 for ll u R nd ll ω u F u); A3 ) There exists k 4 > 0 nd 4 > 0 such tht ξ k 4 u + 4 for ll u R nd ll ξ u Gu); nd there exist five positive constnts d, e, r, η, γ such tht d + e < b, k 3 < M r nd η cγ, where c = deb 4 5 d + e)) d + e), nd A4 ) F u) 0 for ll u [0, deγ] nd F 0) = 0; A5 ) M = η cγ b d + e))f deγ) b ) mx u A F u) > 0, where A = {u η de b )1/ u η de b )1/ }; then, there exist λ, λ 0, ν], 0 < ν < 1 nk 3, λ < λ, µ 1 > 0 nd σ 1 > 0 such tht for every λ [λ, λ ] nd µ 0, µ 1 ), system 1.4) hs t lest three positive solutions in W 1, [, b]) whose norms re less thn σ 1. Next, we give n exmple tht illustrte Theorem 1.1 nd Corollry 1.). Set e 9900 ue u3 u 10, u u 1, F u) = u 00 e u4 0 < u < 10, Gu) = u 0 < u < 1, 0 u 0, 0 u 0, nd choose = 0, b = 1, d = 0.5, e = 0.5, η = 0.5, γ = 16. Then it is esy to check tht F u) nd Gu) stisfy the ssumptions in Theorem Preliminries In this section we stte some definitions nd lemms, which will be used in this rticle. First of ll, we give some definitions: X, ) denotes rel) Bnch spce nd X, ) its topologicl dul. While x n x respectively, x n x) in X mens the sequence {x n } converges strongly respectively, wekly) in X. Definition.1. A function ϕ: X R is loclly Lipschitz if for every u X there exist neighborhood U of u nd L > 0 such tht for every ν, ω U, ϕν) ϕω) L ν ω. If ϕ is loclly Lipschitz on bounded sets, then clerly it is loclly Lipschitz. Definition.. Let ϕ : X R be loclly Lipschitz functionl nd u, ν X, the generlized derivtive of ϕ in u long the direction ν, is ϕ 0 ϕω + τν) ϕω) u; ν) = lim sup. ω u,τ 0 τ + It is esy to see tht the function ν ϕ 0 u; ν) is subliner, continuous nd so is the support function of nonempty, convex nd w compct set ϕu) X, If ϕ C 1 X), then ϕu) = {ϕ u)}. ϕu) = {u X : u, ν X ϕ 0 u; ν)}. Clerly, these definitions extend those of the Gâteux directionl derivtive nd grdient.

5 EJDE-015/9 MULTIPLICITY OF POSITIVE SOLUTIONS 5 Definition.3. A mpping A : X X is of type S) +, for every sequence {u n } such tht u n u X nd one hs u n u. lim sup Au n ), u n u 0, n Definition.4. Let ϕ : X R be loclly Lipschitz functionl nd X : X R {+ } be proper, convex, lower semicontinuous l.s.c.) functionl whose restriction to the set domx ) = {u X : X u) < + } is continuous, then ϕ + X is Motrenu-Pngiotopouls functionl. In most pplictions, C is nonempty, closed, convex subset of X; the indictor of C is the function X C : X R {+ } defined by { 0 if u C, X C = + if u C. It is esy to see tht X C is proper, convex nd l.s.c., while its restriction to domx ) = C is the constnt 0. Definition.5. Let ϕ + X be Motrenu-Pngiotopouls functionl, u X. Then, u is criticl point of ϕ + X if for every v X ϕ u; v u) + X v) X u) 0. The next propositions will be used lter. Proposition.6 [6]). Let h : X R be loclly Lipschitz on X. Then i) h u; v) = mx{ ω, v X : ω hu)} for ll u, v X, ii) Lebourg s men vlue theorem) Let u nd v be two points in X, then there exists point ζ in the open segment between u nd v nd ω ζ hζ) such tht hu) hv) = ω ζ, u v X. We sy tht h is regulr t u X in the sense of Clrke [6]) if for ll z X the usul one-sided directionl derivtive h hu + tz) hu) u; z) = lim t 0 + t exists nd h u; z) = h u; z). Moreover, we sy tht h is regulr on X, if it is regulr in every point u X. Proposition.7. Let h : X R be loclly Lipschitz function which is regulr t u 1,..., u n ) X, then i) hu 1,..., u n ) u1 hu 1,..., u n )... un hu 1,..., u n ), where ui hu 1,..., u n ) denotes the prtil generlized grdient of hu 1,..., u i,..., u n ) to u i for 1 i n. ii) h u 1,..., u n ; v 1,..., v n ) h 1u 1,..., u n ; v 1 ) h nu 1,..., u n ; v n ) for ll v 1,..., v n ) X.

6 6 Z. YUAN, L. HUANG, C. ZENG EJDE-015/9 Proof. For the proof of i), see [6, Proposition.3.15]. From Proposition.6 i), it follows tht there exists ω hu, v) such tht h u; v) = ω, v X. From i) we hve ω = ω 1,..., ω n ), where ω i ui hu 1,..., u n ) 1 i n), nd using the definition of the generlized grdient, we derive h u; v) = ω 1, v 1 W 1, [,b]) ω n, v n W 1, [,b]) h 1u 1,..., u n ; v 1 ) h nu 1,..., u n ; v n ). The following theorems re the min tools for proving our min results. Theorem.8 see [14]). Let X, ) be reflexive Bnch spce, Λ R n intervl, C nonempty, closed, convex subset of X, N C 1 X, R) sequentilly wekly l.s.c. functionl, bounded on ny bounded subset of X, such tht N is of type S) +, Γ : X R is loclly Lipschitz functionl with compct grdient, nd ρ 1 R. Assume lso tht the following conditions hold: i) sup λ Λ inf u C [N u) + λρ 1 Γu))] < inf u C sup λ Λ [N u) + λρ 1 Γu))]; ii) lim u + [N u) λγu)] = + for every λ Λ. Then there exist λ, λ Λ λ < λ ) nd σ 1 > 0 such tht for every λ [λ, λ ] nd every loclly Lipschitz functionl G : X R with compct grdient, there exists µ 1 > 0 such tht for every µ 0, µ 1 ), the functionl N λγ µg + X C hs t lest three criticl points whose norms re less thn σ 1. Theorem.9 []). Let X be nonempty set nd Φ, Ψ two rel functions on X. Assume tht Φu) 0 for every u X nd there exists u 0 X such tht Φu 0 ) = Ψu 0 ) = 0. Further, ssume tht there exist u 1 X, r > 0 such tht Φu 1 ) > r nd sup Ψu) < r Ψu 1) Φu)<r Φu 1 ). Then, for every h > 1 nd for every ρ R stisfying one hs where sup Ψu) + r Ψu1) Φu sup 1) Φu)<r Ψu) < ρ < r Ψu 1) Φu)<r h Φu 1 ), sup inf [Φu) + λ Ψu) + ρ)] < inf λ R u X sup u X λ [0,ν] hr ν = r Ψu1) Φu sup 1) Φu)<rΦu)). 3. Proof of min results Let X = W 1, [, b])) n equipped with the norm n u 1,..., u n ) = u i ) 1/, [Φu) + λ Ψu) + ρ))], where u i = u i x) + u i x) ) dx) 1/ for 1 i n, nd we introduce the functionls Φ, Ψ : X R for ech u = u 1,..., u n ) X, s follows Φu) = 1 u i, Ψu) = F u 1 x),..., u n x)) dx

7 EJDE-015/9 MULTIPLICITY OF POSITIVE SOLUTIONS 7 nd Ju) = Gu 1 x),..., u n x)) dx. Let C = {u X : ux) 0 for every x [, b]}, then for ll λ, µ > 0 nd u X, ϕu) = Φu) λψu) µju) + X C u). The next lemm displys some properties of Φ. Lemm 3.1. Φ C 1 X, R) nd its grdient, defined for u, v X by Φ u), v = ux) vx) dx is of type S) +, where u = u 1,..., u n ) nd v = v 1,..., v n ). The proof of the bove lemm is similr to the one in Chbrowski [4, Section.]. We omit it here. Next we consider some properties of Ψ. Lemm 3.. If A1) A) re stisfied, then Ψu) : X R is loclly Lipschitz function with compct grdient. Moreover, Ψ u 1,..., u n ; v 1,..., v n ) for ll u 1,..., u n ), v 1,..., v n ) X. F u 1,..., u n ; v 1,..., v n ) dx, 3.1) Proof. First, let u = u 1,..., u n ), v = v 1,..., v n ) X be fixed elements. Using the regulrity of F nd Lebourg s men vlue theorem see Proposition.6) we derive ω F ζ 1,..., ζ n ) such tht F u) F v) = ω, u v, where ζ 1,..., ζ n ) is in the open line segment between u 1,..., u n ) nd v 1,..., v n ). Using Proposition.7, there exist ω i ζi F ζ 1,..., ζ n ) 1 i n), such tht F u 1,..., u n ) F v 1,..., v n ) = ω 1 u 1 v 1 ) ω n u n v n ). 3.) From A1) nd 3.), we obtin F u 1,..., u n ) F v 1,..., v n ) ω ω n ) u 1 v u n v n ) [k 1 u u n + v v n ) + 1 ] u 1 v u n v n ). 3.3) Using 3.3), Hölder s inequlity nd the fct the embedding W 1, [, b]) L [, b]) is continuous, we derive Ψu 1,..., u n ) Ψv 1,..., v n ) nk 1 c 1 n n u i + v i + m 1 ) u 1 v u n v n ) u i + v i + m 1 ) u 1 v u n v n ) for some c 1 > 0, m 1 > 0 nd denotes the L norm. From this reltion it follows tht Ψ is loclly Lipschitz on X.

8 8 Z. YUAN, L. HUANG, C. ZENG EJDE-015/9 Now choose u = u 1,..., u n ), h = h 1,..., h n ) X, since F u 1,..., u n ) is continuous, F u 1,..., u n ; h 1,..., h n ) cn be expressed s the upper limit of F u th 1,..., u 0 n + th n ) F u 0 1,..., u 0 n), t where t 0 + tking rtionl vlues nd u 0 1,..., u 0 n) u 1,..., u n ) tking vlues in countble dense subset of X. Therefore, the mp x F u 1 x),..., u n x); h 1 x),..., h n x)) is lso mesurble. By A), the mp x F u 1 x),..., u n x); h 1 x),..., h n x)) belongs to L 1 [, b]). Since X is seprble, there exist functions u k 1,..., u k n) X nd numbers t k 0 + such tht u k 1,..., u k n) u 1,..., u n ) in X nd Ψ u 1,..., u n ; h 1,..., h n ) = We define g n : [, b] R {+ } by Ψu k 1 + t k h 1,..., u k n + t k h n ) Ψu k lim 1,..., u k n). k + t k g k x) = F uk 1 + t k h 1,..., u k n + t k h n ) F u k 1,..., u k n) + k 1 h h n ) t k u k u k n + u k 1 + t k h u k n + t k h n + ) 1. k 1 Then the function g k is mesurble nd nonnegtive see 3.3)). From Ftou s Lemm, we hve I = lim sup k + Let L k = B k + g k, where [ g k x)] dx lim sup[ g k x)] dx = H. k + B k x) = F uk 1 + t k h 1,..., u k n + t k h n ) F u k 1,..., u k n). t k From the Lebesgue dominted convergence theorem, we obtin lim sup k + L k x) dx = k 1 h 1 x) h n x) ) Hence, we derive I = lim sup k + u 1 x) u n x) + 1 k 1 ) Ψu k 1 + t k h 1,..., u k n + t k h n ) Ψu k 1,..., u k n) t k lim k + = Ψ u 1,..., u n ; h 1,..., h n ) k 1 h 1 x) h n x) ) u 1 x) u n x) + ) 1 dx. k 1 dx. L k dx Now, we obtin the estimtes H H B H L, where H B = lim sup k + B k x) dx nd H L = lim inf k + L k x) dx. Since u k 1x),..., u k nx)) u 1 x),..., u n x)).e. in [, b] nd t k 0 +, we derive H L = k 1 h 1 x) h n x) ) u 1 x) u n x) + ) 1 dx. k 1

9 EJDE-015/9 MULTIPLICITY OF POSITIVE SOLUTIONS 9 On the other hnd, H B = = lim sup k + F u k 1 + t k h 1,..., u k n + t k h n ) F u k 1,..., u k n) t k F u th 1,..., u 0 n + th n ) F u 0 lim sup 1,..., u 0 n) u 0 1,...,u0 n ) u1,...,un), t t 0+ F u 1,..., u n ; h 1,..., h n ) dx, which implies 3.1). At lst, we prove tht Ψ is compct. Let {u k } k 1 be sequence in X, where u k = u k 1,..., u k n), such tht u k M nd choose ω k Ψu k ), where ω k = ω k 1,..., ω k n), k 1, k N nd M > 0. From A1), for every v = v 1,..., v n ) X, we obtin ω k, v ω k x) vx) dx k 1 u k u k n + ) 1 v v n ) dx k 1 [ ) 1/ k 1 u k u k n ) 1 dx + b ) 1/] k 1 ) 1/ v v n ) dx = k 1 [ ) 1/ u k u k n + u k 1 u k u k n 1 u k n ) dx + 1 b ) 1/] ) 1/ v h n + v 1 v v n 1 v n ) dx k 1 k 1 [n ) 1/ u k u k n 1 ) dx + b ) 1/] k 1 ) 1/ v v n ) dx c 1 u k + 1 b ) 1/) v k 1 c 1 M + 1 b ) 1/) v, k 1 where c 1 is positive constnt. Hence ω k c 1 M + 1 k 1 b ) 1/ = c. This mens tht {ω k } is bounded. Pssing to subsequence ω k ω X, where ω = ω 1,..., ω n ). We need to prove tht the convergence is strong. We proceed by contrdiction. Suppose tht there exists ε > 0, such tht for every k N ω k ω > ε, where ω k = ω k 1,..., ω k n). Tht is for ll k N, there exists v k = v k 1,..., v k n) B0, 1)... B0, 1) such tht dx dx ω k ω, v k > ε. 3.4)

10 10 Z. YUAN, L. HUANG, C. ZENG EJDE-015/9 Since {v k } k 1 is bounded, pssing to subsequence, v n v = v 0 1,..., v 0 n) X nd v k v 0, so for k big enough, ω k ω, v < ε 3, ω, vk v < ε 3, vk v < ε 3c, this implies ω k ω, v k = ω k ω, v + ω k, v k v ω, v k v ε 3 + ε 3 + c v k v < ε 3 + ε 3 + ε 3 = ε, which contrdicts 3.4). The proof is complete. Anlogously, we deduce the properties of the function J. Lemm 3.3. If A1) nd A3) re stisfied, then J : X R is loclly Lipschitz function with compct grdient nd J u 1,..., u n ; v 1,..., v n ) for ll u 1,..., u n ), v 1,..., v n ) X. G u 1,..., u n ; v 1,..., v n ) dx Now, we re in position to estblish the following proposition. Lemm 3.4. If A1) A3) re stisfied, then, for every λ, µ > 0, ϕ : X R {+ } is Motrenu-Pngiotopoulos function nd the criticl points u 1,..., u n ) belong to X of ϕ is wek solution of 1.1). Proof. From Lemms the function I = Φ λψ µj is loclly Lipschitz; furthermore, C is closed convex subset of X nd C ; thus ϕ is Motrenu- Pngiotopoulos function. Since u 1,..., u n ) X is criticl point of ϕ, then u C nd for ll v = v 1,..., v n ) C we hve 0 I u 1,..., u n ; v 1,..., v n ) = u iv i + u i v i ) dx + λ Ψ) u 1,..., u n ; v 1,..., v n ) + µ J) u 1,..., u n ; v 1,..., v n ) u iv i + u i v i ) dx + λ F u 1,..., u n ; v 1,..., v n ) dx + µ From Proposition.7 ii), we hve 0 u iv i + u i v i ) dx + λ + λ + µ G u 1,..., u n ; v 1,..., v n ) dx. F u n u 1,..., u n ; v n ) dx G u 1 u 1,..., u n ; v 1 ) dx µ F u 1 u 1,..., u n ; v 1 ) dx +... G u n u 1,..., u n ; v n ) dx.

11 EJDE-015/9 MULTIPLICITY OF POSITIVE SOLUTIONS 11 Tking v 1 =... = v i 1 = v i+1 =... = v n = 0 in the bove inequlity for 1 i n, then we led to 1.3), i.e., u 1,..., u n ) is wek solution of 1.). Proof of Theorem 1.1. We pply Theorem.8 to prove this theorem. For this purpose, it is esy to see tht X is reflexive Bnch spce. We put Λ = 0, ν], where 0 < ν < 1 nk 1. The functionl Φ C 1 X, R) is continuous nd convex, hence wekly l.s.c. nd obviously bounded on ny bounded subset of X. Moreover, Φ is of type S + ) Lemm 3.1) nd Ψ is loclly Lipschitz function with compct grdient Lemm 3.). We only need to test conditions i) nd ii) in Theorem.8. We first check condition i). Let vx) = v 1 x),..., v n x)) such tht for 1 i n, eγ i d x ) if x < + d, v i x) = deγ i if + d x b e, 3.5) dγ i e b x) if b e x b. It is obvious tht v X. From simple computtion, we hve Set r = de v ix) + v i x) ) dx = d e b 4d + e) ) + 4 ) 5 3 ded + e) γi. 3.6) n η i, since n η i c n γ i, from 3.6), we obtin Φv 1,..., v n ) = dec γ i > de η i = r. Let u 0 = 0,..., 0), u 1 = v 1,..., v n ). From A4), we hve Ψu 0 ) = 0, nd Φu 0 ) = 0. From [1], we obtin ) 1/ ui mx u ix) x [,b] b for ll u i W 1, [, b]), 1 i n. Hence sup x [,b] u i x) b u i x) for ll u = u 1,..., u n ) X. From 3.7), for ech r > 0, Φ 1, r)) = {u = u 1,..., u n ) X : Φu) < r} = { u i u = u 1,..., u n ) X : < r } { u = u 1,..., u n ) X : u i < 4r b for ll x [, b]}. 3.7) 3.8) By A5), we hve n η i c n b d + e))f deγ 1,..., deγ n ) > b ) mx F ζ 1,..., ζ n ). γ i ζ 1,...,ζ n) A 1 3.9)

12 1 Z. YUAN, L. HUANG, C. ZENG EJDE-015/9 From A4), A5), 3.5), 3.8) nd 3.9), for ll u = u 1,..., u n ) X, we hve sup Ψu) sup u Φ 1 P,r)) n uix) 4r b Note tht 0 < k 1 < M1 nr. Fix 1 < h < F u 1 x),..., u n x)) dx sup F ζ 1,..., ζ n ) dx ζ 1,...,ζ n) A 1 n < η i c n b d + e))f deγ 1,..., deγ n ) γ i r e +d F deγ 1,..., deγ n ) dx Φv 1,..., v n ) r F v 1,..., v n ) dx = rψv) Φv 1,..., v n ) Φv). M1 nk 1r nd ρ such tht sup Φu) + r Ψv) Φv) sup Φu)<r Ψu) < ρ < r Ψv) Φu)<r h Φv). From Theorem.9, we obtin sup inf [Φu) + λρ Ψu))] < inf λ R u X sup u X λ Λ [Φu) + λρ Ψu))]. 3.10) Next, we test condition ii). From A), Lebourg s men vlue theorem nd Proposition.7, there exist ω i ζi F ζ 1,..., ζ n ), where ζ i is between the segment u i nd 0 for 1 i n, such tht F u 1,..., u n ) = F u 1,..., u n ) F 0,..., 0) = ω 1 u ω n u n ω ω n ) u u n ) k 1 u u n ) + 1 u u n ) = k 1 u u n + u 1 u u n 1 u n ) + 1 u u n ) nk 1 u u n ) + 1 u u n ). By 3.11), we cn find two positive constnts K nd τ stisfying K nk 1 nd F ζ 1,..., ζ n ) K ζi + τ 3.11) for ll ζ 1,..., ζ n ) R n. Let u = u 1,..., u n ) X, then it is esy to see tht F u 1,..., u n ) K u i x) + τ.e. x [, b]. 3.1) Choosing λ 0, ν], from 3.1), we obtin u i Φu) λψu) = λ F u 1 x),..., u n x)) dx u i λk u i x) dx λb )τ

13 EJDE-015/9 MULTIPLICITY OF POSITIVE SOLUTIONS 13 u i λk 1 ) n λk u i b τ. nk 1 Since 0 < K nk 1 nd 0 < ν < 1 nk 1, we hve lim Φu) λψu)) = +, u + then we hve tested condition ii). The proof is complete. u i x) dx b nk 1 τ Acknowledgements. This reserch ws prtly supported by the Ntionl Nturl Science Foundtion of Chin ). The uthors would like to thnk the editor nd the reviewer for his/her vluble comments nd constructive suggestions, which help to improve the presenttion of this pper. References [1] G. Anello, G. Gordro; Positive infinitely mny rbitrrily smll solutions for the Dirichlet Problem involving the p-lplcin, Proc. Roy. Soc. Edinburgh Sect. A 13 00) [] G. Bonnno; Some remrks on three criticl points theorem, Nonliner Anl ) [3] A. Brdji; A theoreticl nlysis of new second order finite volume pproximtion bsed on low-order scheme using generl dmissible sptil meshes for the one dimensionl wve eqution, J. Mth. Anl. Appl ) [4] J. Chbrowski; Vritionl Methods for Potentil Opertor Equtions, De Gruyter, [5] K. Chng; Vritionl methods for nondifferentible functionls nd their pplictions to prtil differentil equtions, J. Mth. Anl. Appl ) [6] F. Clrke; Nonsmooth Anlysis nd Optimztion, Wiley New York, [7] H. Dng, S. Oppenheimer; Existence nd uniqueness results for some nonliner boundry vlue problems, J. Mth. Anl. Appl ), [8] Z. Denkowski, L. Gsiński, N. Ppgeorgiou; Existence nd multiplicity of solutions for semiliner hemivritionl inequlities t resonnce, Nonliner Anl ) [9] C. Fbry, D. Fgyd; Periodic solutions of second order differentil equtions with p- lplcin nd symmetric nonlinerities, Rend. Istit. Mt. Univ. Trieste 4 199) [10] M. Filippks, L. Gsiński, N. Ppgeorgiou; On the existence of positive solutions for hemivritionl inequlities driven by the p-lplcin, J. Globl Optim ) [11] L. Gsiński, N. Ppgeorgiou; Nonsmooth Criticl Point Theory nd Nonliner Boundry Vlue Problems, Chpmn nd Hll/CRC Press, Boc Rton, FL, 005. [1] Z. Guo; Boundry vlue problems of clss of qusiliner ordinry differentil equtions, Diff. Integ. Eqns ), [13] S. Heidrkhni, Y. Yu; Multiplicity results for clss of grdient systems depending on two prmeters, Nonliner Anl ) [14] A. Innizzotto; Three criticl points for perturbed nonsmooth functionls nd pplictions, Nonliner Anl ) [15] A. Innizzotto, N. Ppgeorgiou; Existence of three nontrivil solutions for nonliner Neumnn hemivritionl inequlities, Nonliner Anl ) [16] A. Kristály, W. Mrzntowicz, C. Vrg; A non-smooth three criticl points theorem with pplictions in differentil inclusions, J. Globl Optim ) [17] S. Mrno, D. Motrenu; On three criticl points theorem for nondifferentible functions nd pplictions to nonliner boundry vlue problems, Nonliner Anl ) [18] R. Milson, F. Vliquette; Point equivlence of second-order ODEs: Mximl invrint clssifiction order, J. Symbolic Computtion ) [19] D. Motrenu, P. Pngiotopoulos; Minimx Theorems nd Qulittive Properties of the Solutions of Hemivrittionl Inequlities, Kluwer Acdemic Publishers, Dordrecht, 1999.

14 14 Z. YUAN, L. HUANG, C. ZENG EJDE-015/9 [0] D. Motrenu, V. Rǎdulescu; Vritionl nd Non-Vritionl Methods in Nonliner Anlysis nd Boundry Vlue Problems, Kluwer Acdemic Publisher, Boston, 003. [1] Z. Nniewicz, P. Pngiotopoulos; Mthemticl Theory of Hemivritionl Inequlities nd Applictions, Mrcel Dekker, New York, [] P. Pngiotopoulos; Hemivritionl Inequlities, Applictions in Mechnics nd Engineering, Springer-Verlg, Berlin, [3] M. Pino, R. Mnsevich, A. Muru; Existence nd multiplicity of solutions with prescribled period for second-order OFR, Nonliner Anl ) [4] M. Schechter; Periodic nonutonomous second order dynmicl systems, J. Differentil Equtions 3 006) [5] Z. Wng, J. Xio; On periodic solutions of subqudrtic second order non-utonomous Hmiltonin systems, Appl. Mth. Lett ) Ziqing Yun College of Mthemtics nd Econometrics, Hunn University, Chngsh, Hunn 41008, Chin E-mil ddress: junjyun@sin.com Lihong Hung corresponding uthor) College of Mthemtics nd Econometrics, Hunn University, Chngsh, Hunn 41008, Chin E-mil ddress: lhhung@hnu.edu.cn Chunyi Zeng Deprtment of Foundtionl Eduction, Southwest University for Ntionlities, Chengdu, Sichun , Chin E-mil ddress: ykbzcy@163.com