DSpace at VNU: On a nonlinear and non-homogeneous problem without (A-R) type condition in Orlicz-Sobolev spaces

Applied Mathematics and Computation 219 (2013) 7820–7829 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc On a nonlinear and non-homogeneous problem without (A–R) type condition in Orlicz–Sobolev spaces N.T Chung a,b,⇑, H.Q Toan c a Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam c Department of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam b a r t i c l e i n f o a b s t r a c t Using variational methods, we prove some existence and multiplicity results for a class of nonlinear and non-homogeneous problems without (A–R) type condition in Orlicz–Sobolev spaces Ó 2013 Elsevier Inc All rights reserved Keywords: Nonlinear and non-homogeneous problems Orlicz–Sobolev spaces Existence Multiplicity Variational methods Introduction and preliminaries Let X be a bounded domain in RN ðN P 3Þ with smooth boundary @ X Assume that a : ð0; 1Þ ! R is a function such that the mapping u : R ! R, dened by utị :ẳ & ajtjịt 0; for t – 0; for t ¼ is an odd, increasing homeomorphisms from R onto R In this article, we are concerned with a class of nonlinear and non-homogeneous problems in OrliczSobolev spaces of the form & divajrujịruị ẳ f x; uị in X; uẳ0 on @ X; 1:1ị where f : X  R ! R is a continuous function satisfying some suitable conditions It should be noticed that if ajtjị ẳ jtjp2 ; t R; p > then we obtain the well-known p-Laplace operator Dp u ẳ divjrujp2 ruị and problem (1.1) becomes & Dp u ẳ f x; uị in X; uẳ0 on @ X: ð1:2Þ Since Ambrosetti and Rabinowitz proposed the mountain pass theorem in 1973 (see [2]), critical point theory has become one of the main tools for finding solutions to elliptic problems of variational type Especially, elliptic problems of type (1.2) have been intensively studied for many years One of the very important hypotheses usually imposed on the nonlinearities is the following Ambrosetti–Rabinowitz type condition ((A–R) type condition for short): There exists l > p such that ⇑ Corresponding author at: Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam E-mail addresses: ntchung82@yahoo.com (N.T Chung), hq_toan@yahoo.com (H.Q Toan) 0096-3003/$ - see front matter Ó 2013 Elsevier Inc All rights reserved http://dx.doi.org/10.1016/j.amc.2013.02.011 N.T Chung, H.Q Toan / Applied Mathematics and Computation 219 (2013) 78207829 < lFx; tị :ẳ Z 7821 t f ðx; sÞds f ðx; tÞt ð1:3Þ for all x X and t R n f0g This condition ensures that the energy functional associated to the problem satisfies the Palais– Smale condition ((PS) condition for short) Clearly, if the condition (A–R) is satisfied then there exist two positive constants d1 ; d2 such that Fðx; tÞ P d1 jtjl À d2 ; 8ðx; tÞ X  R: This means that f is p-superlinear at infinity in the sense that lim jtj!ỵ1 Fx; tị ẳ ỵ1: jtjp In recent years, there have been some authors considering p-superlinear problems of type (1.2) without the (A–R) type condition, we refer to some interesting papers on this topic [6,7,10,14,16,17,20,23] and the references cited there Our aim in this paper is to develop the ideas by Miyagaki et al [20] and Li et al [16] to a class of nonlinear and non-homogeneous problems of type (1.1) in Orlicz–Sobolev spaces We assume that f is u0 -superlinear at infinity (see the condition ðf2 Þ in Section 2) but does not satisfy the (A–R) type condition (1.3) as in [8,12] To overcome the difficulties brought, we shall use the mountain pass theorem in [10] and the fountain theorem in [24] with the ðC c Þ condition (see Definition 2.5) Our situation here is different from one’s introduced in the works [15,18], in which the authors consider problem (1.1) in the case when f is u0 -sublinear at infinity In order to study problem (1.1), let us introduce the functional spaces where it will be discussed We will give just a brief review of some basic concepts and facts of the theory of Orlicz and Orlicz–Sobolev spaces, useful for what follows, for more details we refer the readers to the books by Adams [1], Rao and Ren [21], the papers by Clément et al [8,9], Donaldson [11], Gossez et al [13], Miha˘ilescu et al [18,19] and Cammaroto et al [5] For u : R ! R introduced at the start of the paper, we dene Utị ẳ Z t usịds; 8t R: We can see that U is a Young function, that is, U0ị ẳ 0; U is convex, and limt!1 Utị ẳ ỵ1 Furthermore, since Utị ẳ if and only if t ẳ 0; limt!0 Utị ẳ 0, and limt!1 Utị ẳ ỵ1, the function U is then called an N-function The function Uà defined by t t the formula U tị ẳ Z t u1 sịds for all t R is called the complementary function of U and it satises the condition U tị ẳ supfst UðsÞ : s P 0g for all t P 0: We observe that the function Uà is also an N-function in the sense above and the following Young inequality holds st Usị ỵ U tị for all s; t P 0: The Orlicz class defined by the N-function U is the set K U Xị :ẳ & ' Z u : X ! R measurable : UðjuðxÞjÞdx < X and the Orlicz space LU ðXÞ is then defined as the linear hull of the set K U ðXÞ The space LU ðXÞ is a Banach space under the following Luxemburg norm &  ' Z  uðxÞ kukU :¼ inf k > : U dx k X or the equivalent Orlicz norm  &Z ' Z   kukLU :ẳ sup  uxịv xịdx : v K Uà ðXÞ; Uà ðjv ðxÞjÞdx : X X For Orlicz spaces, the Hölder inequality reads as follows (see [21]): Z X uv dx 2kukLU ðXÞ kukLà ðXÞ U for all u LU ðXÞ and v LU ðXÞ: à The Orlicz–Sobolev space W LU ðXÞ building upon LU ðXÞ is the space dened by W LU Xị :ẳ & ' @u u LU Xị : LU Xị; i ẳ 1; 2; ; N @xi 7822 N.T Chung, H.Q Toan / Applied Mathematics and Computation 219 (2013) 7820–7829 and it is a Banach space with respect to the norm kuk1;U :ẳ kukU ỵ kjrujkU : Now, we introduce the Orlicz–Sobolev space W 10 LU ðXÞ as the closure of C ðXÞ in W LU ðXÞ It turns out that the space W 10 LU ðXÞ can be renormed by using as an equivalent norm kuk :¼ kjrujkU : For an easier manipulation of the spaces defined above, we define the numbers t uðtÞ UðtÞ u0 :¼ inf t>0 and u0 :¼ sup t>0 t uðtÞ : UðtÞ Throughout this paper, we assume that < u0 t uðtÞ u0 < 1; UðtÞ t P 0; ð1:4Þ which assures that U satisfies the D2 -condition, i.e., Uð2tÞ K UðtÞ; 8t P 0; ð1:5Þ where K is a positive constant, see [19, Proposition 2.3] In this paper, we also need the following condition pffiffi the function t # Uð t Þ is convex for all t ẵ0; 1ị: 1:6ị We notice that Orlicz–Sobolev spaces, unlike the Sobolev spaces they generalize, are in general neither separable nor reflexive A key tool to guarantee these properties is represented by the D2 -condition (1.5) Actually, condition (1.5) assures that both LU ðXÞ and W 10 LU ðXÞ are separable, see [1] Conditions (1.5) and (1.6) assure that LU ðXÞ is a uniformly convex space and thus, a reflexive Banach space (see [19]); consequently, the Orlicz–Sobolev space W 10 LU ðXÞ is also a reflexive Banach space We also find that with the help of condition (1.4), the Orlicz–Sobolev space W 10 LU ðXÞ is continuously embedded in 1;u the classical Sobolev space W 0 ðXÞ, as a result, W 10 LU ðXÞ is continuously and compactly embedded in the classical Lebesgue q à space L ðXÞ for all q < u0 , where ( uÃ0 :¼ N u0 NÀu0 if u0 < N; ỵ1 if u0 P N: The following lemma plays an essential role in our arguments Proposition 1.1 (see [5,18,19]) Let u W 10 LU ðXÞ Then we have R (i) kuku X UðjruðxÞjÞdx kuku0 if kuk < 1, R u0 (ii) kuk X UðjruðxÞjÞdx kuku if kuk > Main results In this section, we state and prove the main result of this paper Let us introduce the following hypotheses: ðf0 Þ f : X  R ! R is a continuous function and satisfies the subcritical growth condition jf ðx; tịj C1 ỵ jtjq1 ị; 8x; tị X  R; where u < q < u and C is a positive constant; ðf1 Þ f ðx; tị ẳ ojtju ị; t ! 0, uniformly a.e x X; ẳ ỵ1 uniformly a.e x X, i.e., f is u0 -superlinear at infinity; ðf2 ị limjtj!ỵ1 Fx;tị u0 jtj f3 ị There exists a constant l1 > such that Gðx; tÞ Gðx; sị ỵ l1 for any x X; < t < s or s < t < 0, where Gx; tị :ẳ tf x; tị u0 Fx; tị and Fx; tị :ẳ f4 ị f x; tị ẳ Àf ðx; tÞ for all ðx; tÞ X  R Rt f ðx; sÞds; It should be noticed that the condition ðf3 Þ is a consequence of the following condition, which was firstly introduced by Miyagaki et al [20] for problem (1.2) in the case p ¼ and developed by Li et al [16] in the case when p > is arbitrary: ðf30 Þ There exists t > such that f ðx;tÞ jtju À2 t is nondecreasing in t P t and nonincreasing in t Àt for any x X N.T Chung, H.Q Toan / Applied Mathematics and Computation 219 (2013) 7820–7829 7823 The readers may consult the proof and comments on this assertion in the papers by Li et al [16] or Miyagaki et al [20] and the references cited there In order to prove the energy functional verifying the ðC c Þ condition, we assume that the functions u and U satisfy the following condition: ðHÞ There exists a positive constant l2 such that HðtsÞ HðtÞ þ l2 for all t P and s ẵ0; 1, where Htị ẳ u0 Utị utịt Before stating and proving the main results of this paper, we give some examples of functions u : R ! R which are odd, increasing homeomorphism from R onto R and satisfy conditions (1.4) and (1.6) and ðHÞ, the readers can nd them in [5,18] Example 2.1 (1) Let utị ẳ pjtjpÀ2 t; t R; p > A simple computation shows that u0 ¼ u0 ¼ p In this case, the corresponding Orlicz space LU ðXÞ is the classical Lebesgue space Lp ðXÞ while the Orlicz–Sobolev space W 10 LU ðXÞ is the classical Sobolev space W 1;p Xị We have Htị ẳ for all t R and then the condition ðHÞ holds (2) Let uðtÞ ẳ log1 ỵ t2 ịt; t R Then we can deduce that u0 ¼ and u0 ¼ Some simple computations show that the function U is given by Utị ẳ ỵ t ị log1 ỵ t ị t2 : ln 10 Then Htị ẳ u0 Utị utịt ẳ þ t Þ logð1 þ t Þ À 2t P for all t P 0: ln 10 For each fixed t > 0, the function s ẵ0; # Htsị is continuous with respect to s So, there exists s0 ½0; 1Š such that Hts0 ị ẳ maxs2ẵ0;1 Htsị It is clear that s0 If s0 ẳ then Htsị Htị for all s ½0; 1Š and t P If s0 0; 1ị, since 0ị limt!ỵ1 Hts ẳ s20 < there exists t > large enough such that HðtÞ Hðts0 Þ HðtÞ < for all t > t or Hðts0 Þ < HðtÞ for all t > t Now, set l2 :ẳ ỵ maxt;sị2ẵ0;tẵ0;1 Htsị we have Htsị < Htị ỵ l2 for all t P and s ½0; 1Š and then the condition ðHÞ holds Definition 2.2 A function u W 10 LU ðXÞ is said to be a weak solution of problem (1.1) if it holds that Z aðjrujÞru Á rv dx À X Z f ðx; uÞv dx ¼ X for all v W 10 LU ðXÞ Our main results in this paper are given by the following two theorems Theorem 2.3 Assume that the conditions (1.4) and (1.6) and ðHÞ; ðf0 Þ—ðf3 Þ are satisfied Then problem (1.1) has a non-trivial weak solution Theorem 2.4 Assume that the conditions (1.4) and (1.6) and ðHÞ; ðf0 Þ; ðf2 Þ—ðf4 Þ are satisfied Then problem (1.1) has infinitely many weak solutions fuk g satisfying Z Uðjruk jÞdx X Z Fx; uk ịdx ! ỵ1; k ! 1: X Our Theorem 2.3 is exactly an extension from the results Miyagaki et al [20] and Li et al [16] to problem (1.1) considered in Orlicz–Sobolev spaces (note that in this paper, we not use the parameter k as in [16,20]), while our Theorem 2.4 seems to be new even in the special case utị ẳ pjtjp2 t, i.e the well-known problem with p-Laplace operator Dp u We emphasize that the extension from the p-Laplace operator to the differential operators involved in (1.1) is not trivial, since the new operators have a more complicated structure than the p-Laplace operator, for example they are non-homogeneous By the presence of the hypothesis ðf2 Þ, it is clear that our results in this paper are also different from the earlier ones in the paper by Cammaroto et al [5] since the authors required in [5] that f ðx; tÞ satisfies the following condition (see the condition ða1 Þ in [5, Theorem 3.1]): ( max lim sup jtj!0 supx2X Fðx; tÞ jtju ; lim sup jtj!ỵ1 supx2X Fx; tị jtju ) 0: 7824 N.T Chung, H.Q Toan / Applied Mathematics and Computation 219 (2013) 7820–7829 Moreover, the method for study of problem (1.1) in the paper [5] is essentially based on the three critical points theorem by B Ricceri [22] Regarding the problem (1.1) with Neumann boundary conditions, we refer the readers to [3,4], in which the authors studied the multiplicity of weak solutions under the condition u0 > N In order to prove the main theorems, we recall some useful concepts and results Definition 2.5 Let ðX; k Á kÞ be a real Banach space, J C ðX; RÞ We say that J satisfies the ðC c Þ condition if any sequence fum g & X such that Jðum Þ ! c and kJ ðum ịk ỵ kum kị ! as m ! has a convergent subsequence Proposition 2.6 (see [10]) Let ðX; k Á kÞ be a real Banach space, J C ðX; RÞ satisfies the ðC c Þ condition for any c > 0; J0ị ẳ and the following conditions hold: (i) There exists a function / X such that k/k > q and Jð/Þ < 0; (ii) There exist two positive constants q and R such that JðuÞ P R for any u X with kuk ¼ q Then the functional J has a critical value c P R, i.e there exists u X such that J uị ẳ and Juị ¼ c In order to prove Theorem 2.4 we will use the following fountain theorem, see [24] for details Let ðX; k Á kÞ be a real reflexive Banach space presenting by X ¼ Èj2N X j with dimðX j ị < ỵ1 for any j N For each k N, we set Y k ¼ Èkj¼0 X j and Z k ¼ È1 j¼k X j Proposition 2.7 (see [24]) Let ðX; k Á kÞ be a real reflexive Banach space, J C ðX; RÞ satisfies the ðC c Þ condition for any c > and J is even If for each sufficiently large k N, there exist qk > rk > such that the following conditions hold: (i) ak :¼ inf fu2Zk :kukẳrk g Juị ! ỵ1 as k ! 1; (ii) bk :ẳ maxfu2Y k :kukẳqk g Juị Then the functional J has an unbounded sequence of critical values, i.e there exists a sequence fuk g & X such that J uk ị ẳ and Juk ị ! ỵ1 as k ! ỵ1 In the rest of this paper we will use the letter X to denote the Orlicz–Sobolev space W 10 LU ðXÞ Let us define the energy functional J : X ! R by the formula Juị ẳ Z Ujrujị X Z Fðx; uÞdx: ð2:1Þ X By Proposition 1.1 and the continuous embeddings obtained from the hypothesis ðf0 Þ, some standard arguments assure that the functional J is well-defined on X and J C ðXÞ with the derivative given by J uịv ị ẳ Z ajrujịru rv dx À X Z f ðx; uÞv dx X for all u; v X, see for example [Lemma 4.2] [19] Thus, weak solutions of problem (1.1) are exactly the critical points of the functional J Lemma 2.8 Assume that the conditions ðf0 Þ–ðf2 Þ are satisfied Then we have the following assertions: (i) There exists / X; / > such that Jt/ị ! as t ! ỵ1; (ii) There exist q > and R > such that JðuÞ P R for any u X with kuk ẳ q Proof (i) From f2 ị, it follows that for any M > there exists a constant C M ẳ CMị > depending on M, such that Fðx; tÞ P Mjtju À C M ; 8x X; 8t R: ð2:2Þ Take / X with / > 0, from (2.2) and Proposition 1.1 we get Jt/ị ẳ Z Ujrt/jịdx Z X X Fðx; t/Þdx kt/ku À M Z jt/ju dx ỵ C M jXj t u X   Z 0 k/ku À M j/ju dx ỵ C M jXj; 2:3ị X where t > is large enough to ensure that kt/k > 1, and jXj denotes the Lebesgue measure of X From (2.3), if M is large enough such that k/ku À M Z X then we have j/ju dx < 0; 7825 N.T Chung, H.Q Toan / Applied Mathematics and Computation 219 (2013) 78207829 lim Jt/ị ẳ 1; t!ỵ1 which ends the proof of (i) (ii) Since the embeddings X,!Lu ðXÞ and X,!Lq ðXÞ are continuous, there exist constants C ; C > such that kukLu0 ðXÞ C kuk; Let <  < u0 2C kukLq ðXÞ C kuk: ð2:4Þ , where C is given by (2.4) From ðf0 Þ and ðf1 Þ, we have Fðx; tÞ jtju ỵ Cịjtjq ; 8x; tị X R: ð2:5Þ From (2.5), for all u X with kuk < 1, we have Z Z Fðx; uÞdx P kuku À  X X   0 À CðÞC q2 kukqÀu kuku ; P J k uị ẳ Ujrujịdx Z juju dx CðÞ X Z X 0 u jujq dx P kuku À C u À CðÞC q2 kukq kuk ð2:6Þ where C > is given by (2.4) From (2.6) and the fact that q > u0 , we can choose R > and q > such that JðuÞ P R > for all u X with kuk ¼ q The proof of Lemma 2.8 is complete h Lemma 2.9 Assume that the conditions (1.4) and (1.6), ðHÞ; ðf0 Þ, ðf2 Þ–ðf3 Þ are satisfied Then the functional J satisfies the ðC c Þ condition for any c > Proof Let fum g & X be a ðC c Þ sequence of the functional J, that is, kJ ðum Þkà ð1 þ kum kÞ ! as m ! 1; Jðum Þ ! c; which shows that c ¼ Jðum Þ ỵ o1ị; J um ịum ị ẳ o1ị; 2:7ị where oð1Þ ! as m ! We shall prove that the sequence fum g is bounded in X Indeed, if fum g is unbounded in X, we may assume that kum k ! as m ! We define the sequence fwm g by wm ¼ kuumm k ; m ¼ 1; 2; It is clear that fwm g & X and kwm k ¼ for any m Therefore, up to a subsequence, still denoted by fwm g, we have fwm g converges weakly to w X and wm ðxÞ ! wðxÞ; a:e: in X; m ! 1; wm ! w strongly in Lq ðXÞ; m ! 1; wm ! w strongly in Lu ðXÞ; m ! 1: ð2:8Þ ð2:9Þ ð2:10Þ Let X :ẳ fx X : wxị 0g If x X– then it follows from (2.8) that jum xịj ẳ jwm xịjkum k ! ỵ1 as m ! Moreover, from ðf2 Þ, we have lim Fðx; um xịị m!1 jum xịj u0 jwm xịju ẳ þ1; x X– : ð2:11Þ Using the condition ðf2 Þ, there exists t > such that Fðx; tÞ jtju >1 ð2:12Þ for all x X and jtj > t > Since Fðx; tÞ is continuous on X  ½Àt0 ; t0 Š, there exists a positive constant C such that jFðx; tÞj C ð2:13Þ for all ðx; tÞ X  ½Àt ; t Š From (2.12) and (2.13) there exists C R such that Fðx; tÞ P C ð2:14Þ for all ðx; tÞ X  R From (2.14), for all x X and m, we have Fðx; um ðxÞÞ À C kum ku or P0 7826 N.T Chung, H.Q Toan / Applied Mathematics and Computation 219 (2013) 7820–7829 Fðx; um ðxÞÞ jum ðxÞj u0 C4 jwm ðxÞju À kum ku 8x X; 8m: P 0; ð2:15Þ By (2.7), Proposition 1.1 we have c ẳ Jum ị ỵ o1ị ẳ Z Ujrum jịdx X Z Fx; um ịdx ỵ o1ị P kum ku0 Z X Fx; um ịdx ỵ o1ị X or Z Fx; um ịdx P kum ku0 c ỵ o1ị ! ỵ1 as m ! 1: 2:16ị X We also have c ẳ Jum ị ỵ o1ị ẳ Z Ujrum jịdx X Z Fx; um ịdx ỵ o1ị kum ku X Z Fx; um ịdx ỵ o1ị X or kum ku P Z Fx; um ịdx ỵ c À oð1Þ > for m large enough: ð2:17Þ X We claim that jX– j ¼ In fact, if jX– j – 0, then by (2.11), (2.15) and (2.17) and the Fatou lemma, we have ỵ ẳ ỵ1ịjX j Z Z Fx; um xịị C4 u0 ẳ lim inf jw ðxÞj dx À lim sup dx m 0 u m!1 m!1 X– X– kum ku jum ðxÞj ! Z Fx; um xịị C4 dx ẳ lim inf jwm ðxÞju À u0 m!1 X– kum ku jum ðxÞj ! Z Fðx; um ðxÞÞ C4 lim inf jwm ðxÞju À dx u0 m!1 X– kum ku jum ðxÞj ! Z Fðx; um ðxÞÞ C4 dx jwm ðxÞju À lim inf u0 m!1 X kum ku jum ðxÞj Z Z Fðx; um ðxÞÞ C4 ¼ lim inf dx À lim sup dx u u0 m!1 m!1 X X kum k kum k Z Fx; um xịị dx ẳ lim inf m!1 X ku ku R m X Fðx; um ðxÞÞdx : lim inf R m!1 X Fx; um ịdx ỵ c À oð1Þ ð2:18Þ From (2.16) and (2.18), we obtain þ1 1; which is a contradiction This shows that jX j ẳ and thus wxị ẳ a.e in X Since Jðtum Þ is continuous in t ½0; 1Š, for each m there exists t m ½0; 1Š, m ¼ 1; 2; , such that Jt m um ị :ẳ maxJtum ị: 2:19ị t2ẵ0;1 It is clear that t m > and Jðtm um ị P c > ẳ J0ị ẳ J0:um Þ If t m < then tm ¼ 1, then J um ịum ị ẳ o1ị So we always have d Jtum ịjtẳtm dt J tm um Þðtm um Þ ¼ oð1Þ: ¼ which gives J t m um ịtm um ị ẳ If ð2:20Þ Let fRk g be a positive sequence of real numbers such that Rk > for any k and limk!1 Rk ẳ ỵ1 Then kRk wm k ẳ Rk > for any k and m Fix k, since wm ! strongly in the spaces Lq ðXÞ and Lu ðXÞ as m ! 1, using (2.5), we deduce that there exists a constant C > such that Z Fðx; Rk wm Þdx C X Z Rk jwm ju dx ỵ C X Z X Rqk jwm jq dx ! as m ! 1; which yield lim m!1 Z X Fðx; Rk wm ịdx ẳ 0: 2:21ị 7827 N.T Chung, H.Q Toan / Applied Mathematics and Computation 219 (2013) 7820–7829 Since kum k ! as m ! 1, we also have kum k > Rk or < kuRmk k < for m large enough Hence, using (2.21), Proposition 1.1, it follows that   Z Z Z Rk UðjrRk wm jÞdx À Fðx; Rk wm Þdx P kRk wm ku0 Fx; Rk wm ịdx um ẳ JRk wm ị ẳ kum k X X X Z u0 u0 ẳ Rk Fx; Rk wm ịdx P Rk X Jðt m um Þ P J ð2:22Þ for any m large enough From (2.22), letting m; k ! we have lim Jt m um ị ẳ þ1: ð2:23Þ m!1 On the other hand, using the conditions ðf3 Þ; ðHÞ and relations (1.4) and (2.7), for all m large enough, we have Jðt m um Þ ¼ Jðt m um Þ À J ðt m um ịt m um ị ỵ o1ị u Z Z Z Z 1 ẳ Ujrt m um jịdx À Fðx; t m um Þdx À aðjrtm um jịjrt m um j2 dx ỵ f x; tm um ịt m um dx ỵ o1ị u X u X X X Z Z 1 Hðt jru jịdx ỵ Gx; tm um ịdx ỵ o1ị u X m m u X Z Z À Á 1 Hjrum jị ỵ l2 dx ỵ Gx; um ị ỵ l1 dx ỵ o1ị u X u X Z  Z Z Z l ỵl ẳ Ujrum jịdx Fx; um ịdx À aðjrum jÞjrum j2 dx À f ðx; um ịum dx ỵ jXj ỵ o1ị u X X l ỵ l l ỵ l ẳ Jum ị J um ịum ị ỵ jXj ỵ o1ị ! c ỵ jXj as m ! 1: u u u X X u ð2:24Þ From (2.23) and (2.24) we obtain a contradiction This shows that the sequence fum g is bounded in X Now, by conditions (1.4) and (1.6), the Banach space X is reflexive Thus, there exists u X such that passing to a subsequence, still denoted by fum g, it converges weakly to u in X and converges strongly to u in the space Lq ðXÞ Using the condition ðf0 Þ and the Hưlder inequality, we have Z  Z Z      f ðx; um Þðum À uÞdx kum À ukLq ðXÞ jf x; um ịjjum ujdx C ỵ jum jq1 ịjum ujdx C ỵ kum kqÀ1   Lq ðXÞ X X X ! as m ! 1; which yield lim m!1 Z f x; um ịum uịdx ẳ 0: 2:25ị X From (2.7) and (2.25) we get lim m!1 Z aðjrum jÞrum rum uịdx ẳ 0: 2:26ị X By (2.26) and the fact that fum g converges weakly to u in X, we can apply the result of Miha˘ilescu [18, Lemma 5] in order to deduce that the sequence fum g converges strongly to u in X Therefore, the functional J satisfies the ðC c Þ condition for any c > The proof of Lemma 2.9 is complete Proof of Theorem 2.3 By Lemmas 2.8 and 2.9, the functional J satisfies all the assumptions of the mountain pass theorem Therefore, the functional J has a critical value c P R > Hence, problem (1.1) has at least one non-trivial weak solution in X Next, because X is a reflexive and separable Banach space, there exist fej g & X and feÃj g & X à such that X ¼ spanfej : j ¼ 1; 2; ; g; X à ¼ spanfeÃj : j ¼ 1; 2; ; g and D E & 1; if i ¼ j; ei ; eÃj ¼ 0; if i – j: For convenience, we write X j ¼ spanfej g and define for each k N the subspaces Y k ¼ Èkj¼1 X j and Z k ¼ È1 X The following j¼k j result is useful for our arguments 7828 N.T Chung, H.Q Toan / Applied Mathematics and Computation 219 (2013) 7820–7829 Lemma 2.10 If u0 < q < uÃ0 then we have n o ak :ẳ sup kukLq Xị : kuk ẳ 1; u Z k ! as k ! 1: Proof Obviously, for any k N; < akỵ1 ak , so ak ! a P as k ! Let uk Z k ; k ¼ 1; 2; satisfy kuk k ¼ and ak À kuk kLq ðXÞ < : k ð2:27Þ Then there exists a subsequence of fuk g, still denoted by fuk g such that fuk g converges weakly to u in X and D E D E eÃj ; u ¼ lim eÃj ; uk ; j ¼ 1; 2; k!1 Since Z k is a closed subspace of X, by Mazur’s theorem, we have u Z k for any k Consequently, we get u \1 k¼1 Z k ¼ f0g, and so fuk g converges weakly to in X as k ! Since u0 < q < uÃ0 , the embedding X,!Lq ðXÞ is compact, then fuk g converges strongly to in Lq ðXÞ Hence, by (2.27), we have limk!1 ak ¼ h Lemma 2.11 Assume that the conditions ðf0 Þ and ðf2 Þ are satisfied Then there exist qk > r k > such that (i) ak :ẳ inf fu2Zk :kukẳrk g Juị ! ỵ1 as k ! 1; (ii) bk :ẳ maxfu2Y k :kukẳqk g Juị Proof (i) By f0 Þ, there exists C > such that jFðx; tịj C jtj ỵ jtjq ị 2:28ị for all ðx; tÞ X  R; u0 < q < uÃ0 From (2.28), for any u Z k with kuk > we have Juị ẳ Z UðjrujÞdx À X Z Fðx; uÞdx P kuku0 À C X Z jujq dx À C X Z X jujdx P kuku0 À C aqk kukq À C kuk; 2:29ị where n o ak :ẳ sup kukLq Xị : kuk ẳ 1; u Z k : Now, for any u Z k with kuk ẳ rk ẳ 2C aqk ịu0 q we have À Á u0 À Á q À Á JðuÞ P kuku0 À C aqk kukq À C kuk ¼ 2C aqk u0 Àq À C aqk 2C aqk u0 Àq À C 2C aqk u0 Àq Á u0 À Á 1À u ¼ 2C aqk u0 Àq À C 2C aqk u0 Àq ¼ r k À C r k : 2 ð2:30Þ By Lemma 2.10, ak ! as k ! and uÃ0 > q > u0 > u0 > we have that r k ! ỵ1 as k ! Therefore, by (2.30) it follows that ak ! ỵ1 as k ! ðiiÞ By (2.2), for any w Y k with kwk ¼ and t > 1, we have Jtwị ẳ Z X  kwku ¼t u0 Z Fðx; twÞdx ktwku À M X  Z M jwju dx ỵ C M jXj: Ujrtwjịdx Z jtwju dx ỵ C M jXj X X It is clear that we can choose M > large enough such that kwku À M Z jwju dx < 0: X For this choice, it follows from (2.31) that lim Jtwị ẳ 1: t!ỵ1 Hence, there exists t > r k > large enough such that JðtwÞ and thus, if we set qk ¼ t we conclude that bk :ẳ max Juị 0: fu2Y k :kukẳqk g The proof of Lemma 2.11 is complete h ð2:31Þ N.T Chung, H.Q Toan / Applied Mathematics and Computation 219 (2013) 7820–7829 7829 Proof Proof of Theorem 2.4 By Lemma 2.11, the functional J satisfies all the assumptions of the fountain theorem By Lemma 2.9, the functional J satisfies the ðC c Þ condition for every c > Moreover, from the condition ðf4 Þ; J is even Hence, we can apply the fountain theorem in order to obtain a sequence of critical points fuk g & X of J such that Juk ị ! ỵ1 as k ! 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Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti– Rabinowitz condition, Nonlinear Anal 72 (2010) 4602–4613... (1.1) has a non-trivial weak solution Theorem 2.4 Assume that the conditions (1.4) and (1.6) and ðHÞ; ðf0 Þ; ðf2 Þ—ðf4 Þ are satisfied Then problem (1.1) has in nitely many weak solutions fuk g satisfying