Research Article Multiplicity of Positive Solutions for a p-q-laplacian Type Equation with Critical Nonlinearities

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1 Abstact and Alied Analysis Volume 214, Aticle ID 82969, 9 ages htt://dx.doi.og/1.1155/214/82969 Reseach Aticle Multilicity of Positive Solutions fo a --Lalacian Tye Euation with Citical Nonlineaities Tsing-San Hsu and Huei-Li Lin Division of Natual Science, Cente fo Geneal Education, Chang Gung Univesity, Taoyuan 333, Taiwan Coesondence should be addessed to Tsing-San Hsu; tshsu@mail.cgu.edu.tw Received 25 Setembe 213; Acceted 9 Mach 214; Published 3 Ail 214 Academic Edito: Guozhen Lu Coyight 214 T.-S. Hsu and H.-L. Lin. This is an oen access aticle distibuted unde the Ceative Commons Attibution License, which emits unesticted use, distibution, and eoduction in any medium, ovided the oiginal wok is oely cited. We study the effect of the coefficient f(x) of the citical nonlineaity on the numbe of ositive solutions fo a --Lalacian euation. Unde suitable assumtions fo f(x) and g(x), we should ove that fo sufficiently small λ >, thee exist at least k ositive solutions of the following --Lalacian euation, Δ u Δ u=f(x) u 2 u+λg(x) u 2 u in, u=on, whee R N is a bounded smooth domain, N>, 1 < < N( 1)/(N 1) < max{, /( 1)} < <, =N/(N )is the citical Sobolev exonent, and Δ s u=div( u s 2 u is the s-lalacian of u. 1. Intoduction This ae is concened with the multilicity of ositive solutions to the following --Lalacian euation with citical nonlineaities: Δ u Δ u=f(x) u 2 u+λg(x) u 2 u in, u= on, (E λ ) whee R N is a bounded smooth domain with smooth bounday, 1 < < < N, λ >,andδ s u = div( u s 2 u) is the s-lalacian of u, and assume that (H1) 1 < < N( 1)/(N 1) < max{, /( 1)} < < = N/(N ); (H2) f and g ae ositive continuous functions in ; (H3) Thee exist k oints a 1,a 2,...,a k in such that f(a i ) ae stict local maxima satisfying f(a i )=maxf (x) =1 fo 1 i k, (1) x and fo some β > N/( 1), f(x) = f(a i )+O( x a i β ) as x a i unifomly in i. Poblem (E λ ) comes, fo examle, fom a geneal eaction-diffusion system u t = div [H (u) u] +c(x, u), (2) whee H(u) = u 2 + u 2. This system has a wide ange of alications in hysics and elated science such as biohysics, lasma hysics, and chemical eaction design. In such alications, the function u descibes a concentation, the fist tem on the ight-hand side of(2) coesonds to the diffusion with a diffusion coefficient H(u), wheeas the second one is the eaction and elates to souces and loss ocesses. Tyically, in chemical and biological alications, the eaction tem c(x, u) has a olynomial fom with esect to the concentation u. The stationay solution of (2) was studied by many authos; that is, many woks ae consideed the solutions of the following oblem: div [H (u) u] =c(x, u). (3) See [1 5] fodiffeentc(x, u). In the esent ae we ae concened with oblem (E λ ) in a bounded domain with c(x, u) = f(x) u 2 u + λg(x) u 2 u in (3). Recently, in [6], the authos obtain the existence of cat () ositive solutions

2 2 Abstact and Alied Analysis of oblem (E λ ) fo N > 2 and f(x) g(x) 1 when condition (H1) holds, whee cat () denotes the Lustenik- Schnielmann categoy of in itself. Secially, if =, (E λ ) canbeeducedtothefollowing ellitic oblems: Δ u=f(x) u 2 u+λg(x) u 2 u in, u= on. Afte the well-known esults of Bézis and Nienbeg [7], who studied (4) inthecaseof = = 2 and f(x) g(x) 1, a lot of oblems involving the citical gowth in bounded and unbounded domains have been consideed; see, fo examle, [8 1] and efeence theein. In aticula, the fist multilicity esult fo (4)hasbeenachievedbyReyin[11] in the semilinea case. Pecisely Rey oved that if N 5, = = 2, andf(x) g(x) 1, foλ small enough, oblem (4) has at least cat () solutions. Futhemoe, Alves and Ding in [12] obtained the existence of cat () ositive solutions to oblem (4) with 2, [, ),andf(x) g(x) 1. Finally, we mention that [13] studied(4) when1<<<n and f, g ae sign-changing and veified the existence of two ositive solutions fo λ (,λ ) fosomeositiveconstant λ. The main uose of this ae is to analyze the effect of the coefficient f(x) of the citical nonlineaity to ove the multilicity of ositive solutions of oblem (E λ ) fo small λ>.bythesimilaagumentin[14], we can constuct the k comact Palais-Smale seuences that ae suitably localized in coesondence of k maximum oints of f. Unde some assumtions (H1) (H3), we couldshow thatthee aeat least k ositive solutions of oblem (E λ ) fo sufficiently small λ>. This ae is oganized as follows. Fist of all, we study the agument of the Nehai manifold M λ. Next, we ove the existence of a ositive solution u M λ.finally,weshowthat the condition (H3) affects the numbe of ositive solution of (E λ );thatis,theeaeatleastk citical oints u i M λ of J λ such that J λ (u i )=α i λ ((PS)-value) fo 1 i k. The main esults of this ae ae given as follows. Theoem 1. Suose that (H1) (H3) hold; then oblem (E λ ) has a ositive solution u in W 1, () fo all λ>. Theoem 2. Suose that (H1) (H3) hold; then thee exists a λ >such that fo any λ (,λ ),oblem(e λ ) admits at least k ositive solutions in W 1, (). 2. Peliminaies In what follows, we denote by, the noms on W 1, () and L (),esectively;thatis, (4) u = ( 1/ 1/ u dx), u = ( u dx). (5) We denote the dual sace of W 1, () by W ().Setalso D 1, (R N ) :={u L (R N ): euied with the nom u L (R N ) fo i=1,2,...,n} x i (6) u = u (R 1/ dx). (7) N We will denote by S the best Sobolev constant as follows: S= u W 1, ()\{} u u. (8) It is well known that S is indeendent of and is neve achieved excet when = R N (see [15]). Thoughout this ae, we denote the Lebesgue measue of by and denote a ball centeed at a R N with adius by B (a) and also denote ositive constants (ossibly diffeent) by C, C i. O(ε t ) denotes O(ε t ) /ε t C, o(ε t ) denotes o(ε t ) /ε t as ε,ando n (1) denotes o n (1) as. Associated with (E λ ),weconsidetheenegyfunctional J λ in W 1, (),foeachu W 1, (), J λ (u) = 1 u + 1 u 1 f (x) u dx 1 λg (x) u dx. It is well known that J λ is of C 1 in W 1, () and the solutions of (E λ ) ae the citical oints of the enegy functional J λ (see [16]). We define the Nehai manifold whee (9) M λ := {u W 1, () \ {} : J λ (u),u =}, (1) J λ (u),u = u + u f (x) u dx λg (x) u dx=. (11) The Nehai manifold M λ contains all nontivial solutions of (E λ ). Note that J λ is not bounded fom below in W 1, ().Fom the following lemma, we have that J λ is bounded fom below on the Nehai manifold M λ. Lemma 3. Suose that 1<<<< and (H2) hold. Then fo any λ>, one has that J λ is bounded fom below on M λ.moeove,j λ (u) > fo all u M λ.

3 Abstact and Alied Analysis 3 Poof. Fo u M λ,(1)leadsto Then fo u M λ, Define J λ (u) =( 1 1 ) u +(1 1 ) u +( 1 1 ) f (x) u dx>. (12) φ λ (u),u = u + u f (x) u dx λg (x) u dx =( ) λg (x) u dx ( ) u ( ) u α λ := u M λ J λ (u). (13) Now we show that J λ ossesses the mountain-ass (MP, in shot) geomety. =( ) u +( ) u +( ) f (x) u dx<. (17) Lemma 4. Suose 1<<<< and (H2) holds. Then fo any λ>, one has that (i) thee exist ositive numbes R and d such that J λ (u) d fo u =R; (ii) thee exists u W 1, () such that u > R and J λ (u) <. Poof. (i) By (8), the Hölde ineuality, and the Sobolev embedding theoem, we have that J λ (u) 1 u 1 f (x) u dx Lemma 5. Suose that 1<<<< and (H2) holds. If u M λ satisfies then u is a solution of (E λ ). J λ (u )= min u M λ J λ (u) =α λ, (18) Poof. By (17), φ λ (u), u < fo u M λ.sincej λ (u ) = min u Mλ J λ (u), by the Lagange multilie theoem, thee is τ R such that J λ (u )=τφ λ (u ) in W (). This imlies that = J λ (u ),u =τ φ λ (u ),u. (19) 1 λg (x) u dx 1 u 1 S / u 1 λ g ( )/ S / u. (14) It then follows that τ=and J λ (u )=in W ().Thus,u is a nontivial solution of (E λ ) and J λ (u )=α λ. Lemma 6. Suose that 1 < < < < and (H2) holds. Fo each u W 1, ()\{}, thee exists a uniue ositive numbe t u such that t u u M λ and J λ (t u u) = su t J λ (tu) fo any λ>. Poof. Fo fixed u W 1, ()\{}, conside Hence, thee exist ositive R and d such that J λ (u) d fo u = R. (ii) Fo any u W 1, ()\{},fom J λ (tu) = t u + t u t f (x) u dx t λg (x) u dx, (15) we have lim t J λ (tu) =.Fofixedsomeu W 1, () \ {}, thee exist t>such that tu >Rand J λ (tu) <.Let u=tu. Define φ λ (u) := J λ (u),u. (16) h (t) =J λ (tu) = t u + t u t f (x) u dx t λg (x) u dx. (2) Since h() =,lim t h(t) =,bylemma 4(i),thenitis easy to see that thee exists a uniue ositive numbe t u such that su t h(t) is achieved at t u.thismeansthath (t u )=; that is, t u u M λ. We will denote by α λ the MP level: α λ := u W 1, ()\{} suj λ (tu). t Thenwehavethefollowingesult. (21) Lemma 7. Suose that 1<<<< and (H2) holds, then α λ = α λ fo any λ>.

4 4 Abstact and Alied Analysis Poof. By Lemma 6,wehave α λ = u W 1, ()\{} suj λ (tu) = t u M λ J λ (u) =α λ. t u u M λ J λ (t u u) (22) On the othe hand, fo u M λ,bylemma 6,wehavet u =1 and J λ (u) = su t J λ (tu).hence, Poof. Let {u n } W 1, () be a (PS) c -seuence fo J λ which satisfies J λ (u n ) =c+o n (1), J λ (u n) =o n (1) in W (). (26) Then c+s n + t n u n J λ (u n ) 1 J λ (u n),u n α λ = u M λ J λ (u) = u W 1, ()\{} su u M λ t J λ (tu) suj λ (tu) = α λ. t (23) =( 1 1 ) u n +(1 1 ) u n +( 1 1 ) f (x) u n dx (27) Now the desied esult follows fom (22)and(23). Remak 8. By Lemma 7 and the definition, it is aaent that α λ1 α λ2 if λ 1 λ 2 ;thatis,α λ is noninceasing in λ. Moeove, by Lemma 4(i), fo any λ >, thee exists a d=d(λ ), elated to the MP geomety, such that <d α λ α λ [, λ ]. (24) Hee α is the MP level associated to the functional J (u) = 1 u + 1 u 1 f (x) u dx. (25) 3. (PS) c -Condition in W 1, () fo J λ Fist, we define the Palais-Smale (denote by (PS)) seuence, (PS)-value, and (PS)-conditions in W 1, () fo J λ. Definition 9. (i) Fo c R,aseuence{u n } is a (PS) c -seuence in W 1, () fo J λ if J λ (u n )=c+o n (1) and J λ (u n)=o n (1) stongly in W () as. (ii) c Ris a (PS)-value in W 1, () fo J λ if thee exists a (PS) c -seuence in W 1, () fo J λ. (iii) J λ satisfies the (PS) c -condition in W 1, () if evey (PS) c -seuence {u n } in W 1, () fo J λ contains a convegent subseuence. Alying Ekeland s vaiational incile and using the same agument as in Cao and Zhou [17] o Taantello [18], we have the following lemma. Lemma 1. Suose that 1<<<< and (H2) holds. Then fo any λ>, thee exists a (PS) αλ -seuence {u n } in M λ fo J λ. To ove the existence of ositive solutions, we claim that J λ satisfies the (PS) c -condition in W 1, () fo c (, (1/N)S N/ ). Lemma 11. Suose that 1<<<<, f =1,and (H2) holds. Then fo any λ>, J λ satisfies the (PS) c -condition in W 1, () fo all c (, (1/N)S N/ ). u n, whee s n =o n (1), t n =o n (1),as.Itfollowsthat{u n } is bounded in W 1, (). Thus, thee exist a subseuence still denoted by {u n } and u W 1, () such that u n u weakly in W 1, (), u n u stongly in L s () 1 s <, u n u a.e. in. (28) Futhemoe, we have that J λ (u) = in W (). Byg being continuous on,we get λ g (x) u n dx=λ g (x) u dx+o n (1). (29) Let V n =u n u.thenbyf being ositive continuous on and Bézis-Lieb lemma (see [19]), we obtain f (x) V n dx V n = u n u +o n (1), V n = u n u +o n (1), = f (x) u n dx f (x) u dx + o n (1). (3) Fom (26) (3), we can deduce that 1 V n + 1 V n 1 f (x) V n dx =c J λ (u) +o n (1), (31) V n + V n = f (x) V n dx + o n (1). (32) Without loss of geneality, we may assume that V n =a+o n (1), V n =b+o n (1). (33)

5 Abstact and Alied Analysis 5 So (32)and f =1imly that V n dx f (x) V n dx=a+b+o n (1). (34) BytheSobolevineualityand(33)and(34), we have V n S( V n dx) / and a S(a+b) / Sa /. (35) If a>,then(35)imliesthata S N/,combinedwith(31), (33) (35)andLemma 3, 1<<<,as;weget c= a + b a+b +J λ (u) 1 N a 1 N SN/, (36) which is a contadiction. So, we have a=; J λ satisfies the (PS) c -condition in W 1, () fo all c (, (1/N)S N/ ). 4. Existence of k Positive Solutions In this section, we fist give some eliminay notations and useful lemmas. Choose >small enough such that B (a i ) and B (a i ) B (a j )= fo i =j, i,j=1,2,...,k. Define Q i (u) = φ i (x) u dx, u dx φ i (x) = min {1, x ai }, 1 i k. Then we have the following seaation esult. (37) Lemma 12. If Q i (u) /3 and Q j (u) /3 fo u W 1, () \ {},theni=j. Poof. Fo any u W 1, () \ {} satisfying Q i (u) /3 (1 i k),weget 3 u dx φ i (x) u dx which imlies that φ i (x) u dx \B (a i ) u dx, \B (a i ) (38) u dx 3 u dx, \B (a i ) 1 i k. (39) Hence, fom (39), we obtain 2 u dx 3( u dx + u dx) \B (a i ) \B (a j ) which is a contadiction. 3 u dx if i =j, (4) Fo i=1,2,...,k,weset and define α i λ N i λ := {u M λ :Q i (u) < 3 } N i λ := {u M λ :Q i (u) = 3 }, := J λ (u), N i λ α i λ (41) := J λ (u). (42) N i λ Now let us assume that (H1) (H3) hold. Fom conditions (H2) and (H3), wecanchooseaρ (, /2) small enough and thee exist some ositive constants γ 1,γ 2 such that fo 1 i k,wehave B 2ρ (a i ), 1 i k f (x) f(ai ) γ 1 x ai β g (x) γ 2 x B 2ρ (a i ), x B 2ρ (a i ), 1 i k (43) fo some β > N/( 1). Foi {1,2,...,k}and ε>,we define u ai ε (x) = η i (x), (ε + x ai /( 1) ) (N )/ V ai ε (x) =ε(n )/2 u ai ε (x), (44) whee η i C (B 2ρ(a i )) is a function such that η i (x) 1 and η i (x) 1 on B ρ (a i ). Then we obtain the following estimates (see [2]): t uai ε dx K 1 ε (N( 1) t(n ))/ +O(1), { = K 1 ln ε +O(1), { O (1), { t uai ε dx K 2 ε (t+n( 1) tn)/ +O(1), { = K 2 ln ε +O(1), { O (1), { t > N( 1), t = N( 1), t < N( 1), t > N( 1), t = N( 1), t < N( 1). (45) (46)

6 6 Abstact and Alied Analysis Fom (43) (46) [13, Lemma 4.2] and conditions (H2)-(H3), we can deduce the following estimates: Vai ε Vai ε dx=k2 +O(ε (N )/ ), Vai ε dx=k2 +O(ε (N )/2 ), f (x) Vai ε (47) dx=k / 3 +O(ε N/ ), (48) dx = K1 ε (( 1)/)(N ((N )/)) +O(ε (N )/2 ), (49) whee K 1, K 2,andK 3 ae ositive constants indeendent of ε,ands=k 2 /K 3 is the best Sobolev constant given in (8). Next, we will investigate the effect of the coefficient f(x) to find some Palais-Smale seuences which ae used to ove Theoem 2. Lemma 13. If (H1) (H3) hold, then fo any i {1,2,...,k} and any λ>, thee exists a ε >such that fo ε (,ε ) one has suj λ (tv ai ε ) < 1 t N SN/ unifomly in i. (5) In aticula, <α λ α i λ <(1/N)SN/ fo all λ>. Poof. By Lemma 6, thee exists a t i ε >such that ti ε Vai ε M λ. Futhemoe, Set h (t) =J λ (tv ai ε )=t t Vai ε dx + t Vai ε f (x) Vai ε dx t λg (x) Vai ε dx. (54) Since h() =, lim t + h(t) =, then thee exists a t ε such that su t J λ (tv ai ε )=J λ(t ε V ai ε ) hold, and then t ε satisfies =h (t ε )=t 1 ε t 1 ε then we have Vai ε Vai ε +t 1 ε Vai ε f (x) Vai ε +t ε Vai ε >t ε dx t 1 ε λg (x) Vai ε f (x) Vai ε dx; (55) dx. (56) Fom (47) and(48), fixing any ε 2 > small enough, thee exists T 1 >such that Also,fom (55), we obtain t ε T 1 fo any ε (,ε 2 ). (57) Vai ε Q i (t i ε Vai ε )= φ i (x) Vai ε dx dx ε φ i (a i +εy) η i (a i + εy) V (y) dy = ε η i (a i + εy) V (y) dy (51) Vai ε <t ε f (x) Vai ε dx + t ε g λ Vai ε dx. (58) Fom (47) (49) and(58), thee exist ε 3 >and T 2 >such that φ i (a i )= as ε, t ε T 2 fo any ε (, ε 3 ). (59) whee ε = {x : εx + a i } and V(y) = 1/(1 + y /( 1) ) (N )/. Hence, thee exists an ε 1 >small enough such that fo any ε (,ε 1 ),wehave Q i (t i ε Vai ε )< 3, (52) Let ε 4 = min{ε 1,ε 2,ε 3 }>;then <T 2 t ε T 1 ε (, ε 4 ), (6) whee T 1 and T 2 ae indeendent of ε.fom[13,lemma4.2] and conditions (H2)-(H3),we also have which imlies t i ε Vai ε N i λ fo any ε (,ε 1),andthen <α λ α i λ J λ (t i ε Vai ε ) su J λ (tt i ε Vai ε )=su J λ (tv ai ε ). t t (53) su ( t t Vai ε t = 1 N SN/ +O(ε (N )/ ). f (x) Vai ε dx) (61)

7 Abstact and Alied Analysis 7 By (43), (46) (49), (6), and (61), fo ε (,ε 4 ),weobtain h(t ε ) su ( t t + t ε Vai ε Vai ε t t f (x) Vai ε dx) λg (x) Vai ε dx 1 N SN/ +O(ε (N )/ )+ T 1 T 2 λγ 2 Vai ε dx 1 N SN/ +C 1 ε (N )/ Vai ε +C 2 ε (N )/2 C 3 ε (/( 1))(N ((N )/)), (62) whee C 1,C 2,C 3 ae ositive constants indeendent of ε. Since 1 < < N( 1)/(N 1) < max{, /( 1)} < <,weobtainthat N > (N ) 2 > 1 (N N ) ; (63) then thee exists an ε (,ε 3 ) such that h(t ε ) = su t J λ (tv ai ε ) < (1/N)SN/ unifomly in i fo all ε (,ε ). Moeove, fom (53), we have <α λ α i λ <(1/N)SN/ fo all 1 i kand λ>.thiscomletestheoof. Poof of Theoem 1. Fom Lemmas 5, 1, 11, and13, weget fo all λ > that thee exists a u such that J λ (u ) = and J λ (u ) = α λ.setu + = max{u, }. Relacethe tems f(x) u dx and g(x) u dx of the functional J λ by f(x)u + dx and g(x)u + dx, esectively. It then follows that u is a nonnegative solution of (E λ ). Alying the maximum incile, (E λ ) admits at least one ositive solution u in W 1, (). By studying the agument as in [21, Theoem III 3.1] and [22], we obtain the following lemma. Lemma 14. Let {u n } W 1, () be a nonnegative function seuence with u n =1and u n S. Then thee exists a seuence (y n,σ n ) R + such that V n (x) := σ (N )/ n u n (σ n x+y n ) (64) contains a convegent subseuence denoted again by {V n } such that V n V in D 1, (R N ), (65) whee V(x) > in R N. Moeove, we have σ n, (1/σ n ) dist(y n, ),andy n y as. Lemma 15. Suosethat(H2)and(H3)hold.Thenfoany i {1,2,...,k}, thee exists λ i >such that α i λ > 1 N SN/ λ (, λ i ). (66) Poof. Fix i {1,2,...,k}. Assume the contay; that is, thee then exists a seuence {λ n } with λ n + as n such that α i λ n c (1/N)S N/. Conseuently, thee exists a seuence {u n } N i λ n such that, as, u n + u n = f (x) u n dx +λ n g (x) u n dx + o n (1), (67) and by Remak 8, we have that thee exists a d>such that <d lim α λ n lim J λ n (u n )=c 1 N SN/, (68) whee d is indeendent of λ n fo all n. It then follows easily that {u n } is unifomly bounded in W 1, (), and since g(x) is continuous on,weobtain λ n g (x) u n dx=o n (1) as. (69) Fom (67) (69), we may assume that thee exist a and b such that u n =a+o n (1), u n =b+o n (1). (7) So (7)and f =1imly that u n f (x) u n dx=a+b+o n (1). (71) By (7), (71), and the Sobolev ineuality, we have a= lim u n Slim u n / Slim ( f (x) u n dx) S(a+b) / Sa /, (72) which imlies a=o a S N/.Ifa=,thenby(72) we have that b=.foma=b=,wecandeducethatc= which is a contadiction. Hence, a S N/. (73) On the othe hand, by J λn (u n )=c+o n (1), c (1/N)S N/, and (69) (71), we get a + b a+b = lim J λ n (u n )=c 1 N SN/. (74) This imlies that a N (a a )+(b b )=c 1 N SN/. (75) Hence, togethe with (73), we get a=s N/ and b=,and then, fom (71)and(72), we also have lim u n = lim f (x) u n dx=a=s N/. (76)

8 8 Abstact and Alied Analysis Set w n =u n / u n ;thenwehave w n =1, lim w n = lim u n u n =S. (77) Using Lemma 14, thee exists a seuence (y n,σ n ) R + such that the seuence V n (x) := σ (N )/ n w n (σ n x+y n ) (78) conveges stongly to V D 1, (R N ), σ n, y n y, and (1/σ n ) dist(y n, ) as. Let n ={x:σ n x+y n }.Sinceσ n, y n y, and (1/σ n ) dist(y n, ) as n,then n R N as n.obsevethatq i (w n )=Q i (u n )= /3. Bythe Lebesgue dominated convegence theoem, we have 3 = lim Q i (w n )= lim φ i (x) w n dx w n dx = lim φ i (x) V n ((x y n )/σ n ) dx V n ((x y n )/σ n ) dx n φ i (σ n x+y n ) V n (x) dx = lim n V =φ i (y), n (x) dx (79) which imlies that y =a i by the definition of φ i (x). Onthe othehand,bythelebesguedominatedconvegencetheoem again and (76), we get 1= lim f (x) w n (x) dx = lim n f(σ n x+y n ) V n (x) dx=f(y), (8) which is imossible, because f(x) is not a constant function by condition (H3). Accoding to Lemma13,we have <α λ α i λ < 1 N SN/ λ >. (81) Accoding to Lemma 15,fo each i {1,2,...,k}, thee exists λ i >such that α i λ > 1 N SN/ λ (, λ i ). (82) Let λ = min 1 i k λi >. Then fo each i {1,2,...,k},by (81)and(82), we obtain that Hence α i λ = α i λ < αi λ λ (, λ ). (83) u N i λ Ni λj λ (u) λ (, λ ). (84) Alying Ekeland s vaiational incile and using the standad comutation, we have the following lemma. Lemma 16. If λ (,λ ),thenfoeachi {1,2,...,k},thee exists a (PS) α i -seuence {ui λ n } Ni λ in W1, () fo J λ. Poof. See Cao and Zhou [17] o Taantello [18]. Poof of Theoem 2. By Lemma 16, foallλ (,λ ),thee exists a (PS) α i -seuence {ui λ n } Ni λ in W1, () fo J λ whee 1 i k.fom(81), we have α i λ (, 1 N SN/ ). (85) Note that J λ satisfies the (PS) c -condition fo c (, (1/N)S N/ ).Hence,weobtainthatJ λ at least k citical oints in M λ fo all λ (,λ ).Setu + = max{u, }.Relace the tems f(x) u dx and g(x) u dx of the functional J λ by f(x)u + dx and g(x)u + dx, esectively. It then follows that (E λ ) has k nonnegative solutions. Alying the maximum incile, (E λ ) admits at least k ositive solutions. Conflict of Inteests The authos declae that thee is no conflict of inteests egading the ublication of this ae. Refeences [1] V. Benci, A. M. Micheletti, and D. Visetti, An Eigenvalue oblem fo a uasilinea ellitic field euation, Jounal of Diffeential Euations,vol.184,no.2, ,22. [2] M.WuandZ.Yang, Aclassof--Lalacian tye euation with otentials eigenvalue oblem in R N,Bound, Bounday Value Poblems, vol. 29, Aticle ID , 29. [3] C. He and G. Li, The existence of a nontivial solution to the --Lalacian oblem with nonlineaity asymtotic to u 1 at inity in R N, Nonlinea Analysis: Theoy, Methods & Alications, vol. 68, no. 5, , 28. [4] L. Gongbao and Z. Guo, Multile solutions fo the -- Lalacian oblem with citical exonent, Acta Mathematica Scientia B,vol.29,no.4, ,29. [5] H.YinandZ.Yang, Aclassof--Lalacian tye euation with concave-convex nonlineaities in bounded domain, Jounal of Mathematical Analysis and Alications,vol.382,no.2, , 211. [6] H. Yin and Z. Yang, Multilicity of ositive solutions to a - -Lalacian euation involving citical nonlineaity, Nonlinea Analysis:Theoy,Methods&Alications,vol.75,no.6, , 212. [7] H. Bézis and L. Nienbeg, Positive solutions of nonlinea ellitic euations involving citical sobolev exonent, Communications on Pue and Alied Mathematics, vol.36,no.4, , [8] C.O.Alves,J.M.doÓ,andO.H.Miyagaki, Onetubations of a class of a eiodic m-lalacian euation with citical gowth, Nonlinea Analysis: Theoy, Methods & Alications, vol.45,no.7, ,21. [9] A. Ambosetti, H. Bézis, and G. Ceami, Combined effects of concave and convex nonlineaities in some ellitic oblems, Jounal of Functional Analysis,vol.122,no.2, ,1994.

9 Abstact and Alied Analysis 9 [1] V. Benci and G. Ceami, The effect of the domain toology on the numbe of ositive solutions of nonlinea ellitic oblems, Achive fo Rational Mechanics and Analysis, vol.114,no.1, , [11] O. Rey, A multilicity esult fo a vaiational oblem with lack of comactness, Nonlinea Analysis: Theoy, Methods & Alications,vol.13,no.1, ,1989. [12] C. O. Alves and Y. H. Ding, Multilicity of ositive solutions to a -Lalacian euation involving citical nonlineaity, Jounal of Mathematical Analysis and Alications, vol.279,no.2, , 23. [13] T.-S. Hsu, Multilicity esults fo -Lalacian with citical nonlineaity of concave-convex tye and sign-changing weight functions, Abstact and Alied Analysis, vol.29,aticleid 65219,24ages,29. [14] P. Han, Multile solutions to singula citical ellitic euations, Isael Jounal of Mathematics,vol.156,no.1, , 26. [15] G. Talenti, Best constant in Sobolev ineuality, Annali di Matematica Pua ed Alicata,vol.11,no.1, ,1976. [16] P. H. Rabinowitz, Minimax Methods in Citical Point Theoy with Alications to Diffeential Euations, vol.65ofregional Confeence Seies in Mathematics, Ameican Mathematical Society, Povidence, RI, USA, [17] D.-M. Cao and H.-S. Zhou, Multile ositive solutions of nonhomogeneous semilinea ellitic euations in R N, Poceedings of the Royal Society of Edinbugh A: Mathematics, vol. 126, no. 2, ,1996. [18] G. Taantello, On nonhomogeneous ellitic euations involving citical sobolev exonent, Annales de l Institut Heni Poincaé: Analyse Non Linéaie Oen Achive, vol.9,no.3, , [19] H. Bézis and E. Lieb, A elation between ointwise convegence of functions and convegence of functionals, Poceedings of the Ameican Mathematical Society, vol. 88, no. 3, , [2] P. Dábek and Y. X. Huang, Multilicity of ositive solutions fo some uasilinea ellitic euation in R N with citical Sobolev exonent, Jounal of Diffeential Euations, vol. 14, no. 1, , [21] M. Stuwe, Vaiational Methods, Singe, Heidelbeg, Gemany, 2nd edition, [22] M. Willem, Minimax Theoems,Bikhäuse, 1996.

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