ON EXISTENCE OF THREE SOLUTIONS FOR p(x)-kirchhoff TYPE DIFFERENTIAL INCLUSION PROBLEM VIA NONSMOOTH CRITICAL POINT THEORY

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1 TAIWANESE JOURNAL OF MATHEMATICS Vol. 9, No. 2, pp , April 25 DOI:.65/tjm This paper is available online at ON EXISTENCE OF THREE SOLUTIONS FOR px-kirchhoff TYPE DIFFERENTIAL INCLUSION PROBLEM VIA NONSMOOTH CRITICAL POINT THEORY Lian Duan, Lihong Huang* and Zuowei Cai Abstract. In this paper, we study a class of differential inclusion problems driven by the px-kirchhoff with non-standard growth depending on a real parameter. Working within the framework of variable exponent spaces, a new existence result of at least three solutions for the considered problem is established by using the nonsmooth version three critical points theorem.. INTRODUCTION In recent years, various Kirchhoff type problems have been extensively investigated by many authors due to their theoretical and practical importance, such problems are often referred to as being nonlocal because of the presence of the integral over the entire domain. It is well known that this problem is analogous to the stationary problem of a model introduced by Kirchhoff []. More precisely, Kirchhoff proposed a model given by the equation. ρu tt ρ h + E 2L L u 2 xdx u xx =, where ρ, ρ,h,e,l are all positive constants. Equation. is an extension of the classical D Alembert wave equation by considering the changes in the length of the string during the vibrations. For a bounded domain, the problem a + b u 2 dx u = fx, u, x,.2 u =, x, Received November 27, 23, accepted June 26, 24. Communicated by Biagio Ricceri. 2 Mathematics Subject Classification: 49J52, 49J53. Key words and phrases: px-kirchhoff, Differential inclusion, Nonsmooth three critical points theorem, Palais-Smale condition, Locally Lipschitz functional. Research supported by National Natural Science Foundation of China 3727 and Hunan Provincial Innovation Foundation for Postgraduate. *Corresponding author. 397

2 398 Lian Duan, Lihong Huang and Zuowei Cai is related to the stationary analogue of.. Nonlocal elliptic problems like.2 have received a lot of attention due to the fact that they can model several physical and biological systems and some interesting results have been established in, for example, [2,3] and the references therein. Moreover, the study of the Kirchhoff type equation has already been extended to the case involving the p-laplacian operator or px-laplacian operator. For instance, G. Dai and R. Hao in [4] were concerned with the existence and multiplicity of solutions to the following Kirchhoff type equation involving the p-laplacian operator.3 K p u =, u p dx div u p 2 u =fx, u, x, x, they established conditions ensuring the existence and multiplicity of solutions for the addressed problem by means of a direct variational approach and the theory of the variable exponent Sobolev spaces. In [5], by using a direct variational approach, G. Dai and R. Hao established conditions ensuring the existence and multiplicity of solutions for the following problem.4 K px u =, u px dx div u px 2 u =fx, u, x, x. Very recently, G. Dai and R. Ma in [6] studied the existence and multiplicityof solutions to the following px-kirchhoff type problem.5 K px u px + u px dx div u px 2 u u px 2 u = fx, u, x, u n =, x, we refer the reader to, e.g., [7,8] for other interesting results and further research on this subject. However, the nonlinearity in the above mentioned Kirchhoff type problems is continuous or differentiable. As pointed out by K.C. Chang in [9-], many free boundary problems and obstacle problems arising in mathematical physics may be reduced to partial differential equations with discontinuous nonlinearities, among these problems, we have the obstacle problem [9], the seepage surface problem [], and the Elenbaas equation [], and so forth. Associated with this development, the theory of nonsmooth variational has been given extensive attention, for a comprehensive treatment, we refer

3 On Existence of Three Solutions for px-kirchhoff Type Differential Inclusion Problem 399 to the monographs of [2,3], as well as for updated list of references [4-2] and the references therein. Therefore, it is natural from both a physical and biological standpoint as well as a theoretical view to give considerable attention to a synthesis involving px-kirchhoff type differential inclusion problem. Motivated by the above discussions, the main purpose of this paper is to establish a new result for the existence of at least three solutions to the following differential inclusion problem Kt Δ px u u px 2 u F x, u+λ F 2 x, u, x, P λ u n =, x, where Δ px u = div u px 2 u is said to be px-laplacian operator, Kt is a continuous function with t := px u px + u px dx, λ> is a parameter, R N N>2 is a nonempty bounded domain with a boundary of class C, px >,px C with <p =inf x px,p+ =suppx, F x, t and F 2 x, t x are locally Lipschitz functions in the t-variable integrand, F x, t and F 2 x, t are the subdifferential with respect to the t-variable in the sense of Clarke [22], and n is the outward unit normal on. To study problem P λ, we should overcome the difficulties as follows: Firstly, the px-laplacian possesses more complicated nonlinearities than p-laplacian, in general, the property of the first eigenvalue of px-laplacian is not the same as the p-laplacian, namely the first eigenvalue is not isolated see [23], therefore, the first difficulty is that we cannot use the eigenvalue property of px-laplacian. Secondly, the lack of differentiability of nonlinearity causes several technical obstructions, that is to say, we can not use variational methods for C functional, because in our case, the energy functional is only locally Lipschitz continuous, and so, our approach, which is variational, is based on the nonsmooth critical point theory as was developed by K. C. Chang [9] and S.A. Marano et al. [5]. On the other hand, to the best of our knowledge, there are no papers concerning the problem P λ by the nonsmooth three critical points theorem. So even in the case of a constant exponent, our results are also new. This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Lebesgue and Sobolev spaces and generalized gradient of locally Lipschitz function. In Section 3, we give the main result and its proof of this paper. 2. PRELIMINARIES 2.. Variable exponent spaces and px-kirchhoff-laplacian operator In this subsection, we recall some preliminary results about Lebesgue and Sobolev variable exponent spaces and the properties of px-kirchhoff-laplace operator, which

4 4 Lian Duan, Lihong Huang and Zuowei Cai are useful for discussing problem P λ. Set C + = { h h C : hx > for all x }, for any h C +, we will denote h =minhx, x h + =maxhx. x Let p C +, the variable exponent Lebesgue space is defined by { } L px = u : R is measurable, ux px dx < + furnished with the Luxemburg norm { u L px = u px =inf λ>: u } px dx, λ and the variable exponent Sobolev space is defined by { } W,px = u L px : u L px equipped with the norm u W,px = u px + u px. Proposition 2.. See [24]. The spaces L px and W,px are separable and reflexive Banach spaces. Proposition 2.2. See [24]. If qx C + and qx p x, for all x, then the embedding from W,px to L qx is continuous and if qx < p x, for all x, the embedding from W,px to L qx is compact, where Npx p x = N px, px <N, +, px N. 2 If p x,p 2 x C +, and p x p 2 x, x, then L p 2x L p x, and the embedding is continuous. In the following, we will discuss the px-kirchhoff-laplace operator div u J p u = K px u px + u px dx px 2 u u px 2 u.

5 On Existence of Three Solutions for px-kirchhoff Type Differential Inclusion Problem 4 Denote t Ju := K px u px + u px dx where Kt = Kτdτ, Kt satisfies the following condition: K Kt :[, + k,k is a continuous and increasing function with k > k >. For simplicity, we write X = W,px, denoteby X the norm of X, u n u and u n u the weak convergence and strong convergence of sequence {u n } in X, respectively. It is obvious that the functional J is a continuously Gâteaux differentiable whose Gâteaux derivative at the point u X is the functional J u X,givenby J u,v =K px u px + u px dx u px 2 u v+ u px 2 uv dx, where, is the duality pairing between X and X. Therefore, the px-kirchhoff- Laplace operator is up to the minus sign the derivative operator of J in the weak sense. We have the following properties about the derivative operator of J. Lemma 2.. See [6]. If K holds, then i J : X X is a continuous, bounded and strictly monotone operator; ii J is a mapping of type S +, i.e., if u n uin X and lim sup J u n n + J u,u n u, thenu n u in X; iii J : X X is a homeomorphism; iv J is weakly lower semicontinuous Generalized gradient Let X, be a real Banach space and X be its topological dual. A function ϕ : X R is called locally Lipschitz if each point u X possesses a neighborhood U and a constant L u > such that ϕu ϕu 2 L u u u 2, for all u,u 2 U. We know from convex analysis that a proper, convex and lower semicontinuous function g : X R = R {+ } is locally Lipschitz in the interior of its effective domain dom g = {u X : gu < + }. We define the generalized directional derivative of a locally Lipschitz function φ at x X in the direction h X by φ x; h = lim sup x x λ + φx + λh φx. λ

6 42 Lian Duan, Lihong Huang and Zuowei Cai It is easy to see that the function h φ x; h is sublinear and continuous on X, soby Hahn-Banach Theorem, it is the support function of a nonempty, convex, w -compact set φx ={x X : x,h φ x; h for all h X}. The multifunction x φx is known as the generalizedor Clarke subdifferential of φ. Now, we list some fundamental properties of the generalized grandient and directional derivative which will be used throughout this paper. Proposition 2.3. See [22]. i φ u; z=φ u; z for all u, z X; ii φ u; z=max{ x,z : x φu} for all u, z X; iii Let κ be a locally Lipschitz function, φ + κ u; z φ u; z+κ u; z, for all u, z X; iv Let j : X R be a continuously differentiable function. Then ju = {j u},j u; z = j u,z, and φ + j u; z =φ u; z + j u,z for all u, z X; v Lebourg s mean value theorem Let u and v be two points in X. Then there exists a point w in the open segment between u and v, and x w φw such that φu φv= x w,u v ; vi The function u, h φ u; h is upper semicontinuous. Let I be a function on X satisfying the following structure hypothesis: H I =Φ+Ψ,whereΦ:X R is locally Lipschitz while Ψ:X R {+ } is convex, proper, and lower semicontinuous. We say that u X is a critical point of I if it satisfies the following inequality Φ u; v u+ψv Ψu, for all v X. Set and K := {u X u is a critical point of I} K c = K I c. A number c R such that K c is called a critical value of I. Definition 2.. See [5]. I =Φ+Ψis said to satisfy the Palais-Smale condition at level c R PS c for short if every sequence {u n } in X satisfying Iu n c and Φ u n ; v u n +Ψv Ψu n ɛ n v u n, for all v X,

7 On Existence of Three Solutions for px-kirchhoff Type Differential Inclusion Problem 43 for a sequence {ɛ n } in [, with ɛ n +, contains a convergent subsequence. If PS c is verified for all c R, I is said to satisfy the Palais-Smale condition shortly, PS. Finally, we recall a non-smooth version three critical points theorem due to S. A. Marano and D. Motreanu, which represents the main tool to investigate problem P λ. Theorem 2.. See [5]. Let X be a separable and reflexive Banach space, let I := Φ +Ψ and I 2 := Φ 2 be like in H, let Λ be a real interval. Suppose that b Φ is weakly sequentially lower semicontinuous while Φ 2 is weakly sequentially continuous. b 2 For every λ Λ the function I + λi 2 fulfills PS c,c R, together with I u+λi 2 u =+. lim u + b 3 There exists a continuous concave function h :Λ R satisfying sup inf I u+λi 2 u+hλ < inf sup I u+λi 2 u+hλ. λ Λ u X u X λ Λ Then there is an open interval Λ Λ such that for each λ Λ the function I + λi 2 has at least three critical points in X. Moreover, if Ψ, then there exist an open interval Λ Λ and a number σ> such that for each λ Λ the function I + λi 2 has at least three critical points in X having norms less than σ>. Theorem 2.2. See [25]. Let X be a non-empty set and Φ, Ψ two real functionals on X. Assume that there are r>,u,u X, such that Φu =Ψu =, Φu >r, and sup Ψu <r Ψu u Φ,r] Φu. Then, for each ρ satisfying one has sup inf λ u X sup Ψu <ρ<r Ψu u Φ,r] Φu, Φu+λρ Ψu < inf sup Φu+λρ Ψu. u X λ

8 44 Lian Duan, Lihong Huang and Zuowei Cai 3. MAIN RESULT In this section, we shall prove the existence of at least three solutions to problem P λ. Let us first introduce some notations. It is clear that P λ is the Euler-Lagrange equation of the functional ϕ : X R defined by ϕu = K 3. where Kt = t Ju = K Kτdτ. Set px u px + u px dx F 2 x, udx, F x, udx λ px u px + u px dx, F u = F x, udx, 3.2 F 2 u = F 2 x, udx, and 3.3 h = J + F, h 2 = F 2, then, under these notations, ϕ = h + λh 2.LetV px = {u W,px : udx = }, thenv px is a closed linear subspace of W,px with codimension, andwe have W,px = R V px see [23]. If we define the norm by { u u px =inf λ>: λ + u λ px } dx, by Proposition 2.6 in [23], it is easy to see that u is an equivalent norm in V px. Thereafter, we will also use to denote the equivalent norm u in V px and one can easily see that when u V px,wehave i if u <, then k p + u p+ Ju k p u p+ ; ii if u >, then k p + u p Ju k p u p+. Moreover, let q + <px and denote λ := inf u V px \{}, u > px u px + u px dx, u qx dx

9 On Existence of Three Solutions for px-kirchhoff Type Differential Inclusion Problem 45 by continuous embedding of W,px in L p and, by Proposition 2.22, the above inequalities of i, ii, one has λ >. Now we are in a position to present our main result. In order to reduce our statements, we need the following assumptions: A : F : R R is a function such that F x, t satisfies F x, = and also A for all t R, x F x, t R is measurable; A 2 for almost all shortly, a.a. x, R t F x, t R is locally Lipschitz; A 3 for a.a. x, allt R and all w F x, t, there holds w ax t p, with ax L + ; A 4 there exist qx,sx C + satisfying q + <px <s,forallx, such that F x, t lim sup t t qx < 2λ, and F x, t âx t p lim sup <, with âx L +, t + t s uniformly for a.a. x. A 2 : F 2 : R R is a function such that F 2 x, t satisfies F 2 x, = and also A 2 for all t R, x F 2 x, t R is measurable; A 2 2 for a.a. x, R t F 2 x, t R is locally Lipschitz; A 2 3 for a.a. x, allt R and all v F 2 x, t, there holds v bx t px, with bx L + ; A 2 4 there exists R >, forall < t < R, such that pxf 2 x, t >, and pxf 2 x, t=o t p+ as t, and uniformly for a.a. x. F 2 x, t lim sup <, t + t p+ Remark 3.. A simple example of nonsmooth locally Lipschitz function satisfying hypothesis A and A 2 is for simplicity we drop the x-dependence: F t = { 2λ t qx, if t, âx t p ln t s, if t >,

10 46 Lian Duan, Lihong Huang and Zuowei Cai and { 2 t px, if t, F 2 t = ln t p+, if t >. One can easily check that all conditions in A and A 2 are satisfied. The main result can be formulated as follows: Theorem 3.. Suppose that the assumptions K, A and A 2 hold, then there exists an open interval Λ [, +, such that for each λ Λ, problem P λ has at least three nontrivial solutions in V px. Before proving Theorem 3., we give some preliminary lemmas which are useful to prove the main result. Lemma 3.. Since F i are locally Lipschitz functions which satisfy A i 3,thenF i in 3.2 are well defined and they are locally Lipschitz. Moreover, let E be a closed subspace of X and F i E the restriction of F i to E, wherei =or 2. Then F i E u; v F i x, ux; vxdx, for all u, v E. The proof of the above lemma is similar to that of [26, Lemma 4.2], we omit the details here. Lemma 3.2. For any ɛ>, and qx as mentioned in A 4, there exists a u ɛ V px with u ɛ >, such that k p + u ɛ p + λ u ɛ qx dx + k λ + k ɛ p +. λ + ɛ Proof. Since λ := inf px u px + u px dx, then by the u V px \{}, u > u qx dx definition of infimum, for any ɛ>, there exists a u ɛ V px with u ɛ >, such that px u ɛ px + u ɛ px dx <λ + ɛ, u ɛ qx dx which produces λ λ + ɛ < λ u ɛ qx dx px u ɛ px + u ɛ px dx λ u ɛ qx dx. p u + ɛ p

11 On Existence of Three Solutions for px-kirchhoff Type Differential Inclusion Problem 47 Consequently, we have k p + u ɛ p + λ u ɛ qx dx p + k + λ u ɛ p. λ + ɛ Lemma 3.3. There exists a ū V px with ū and r > such that h ū > r, where h = J + F is as mentioned in 3.3. Proof. In view of A 4, there exists a δ > small enough without loss of generality we assume δ <, such that for a.a. x, wehave 3.4 F x, t 2λ t qx, for all t δ, we also know from A 4 that there exists τ> such that F x, t âx t p lim sup < 2τ, t + t s uniformly for a.a. x. Then we can find a M>large enough such that for a.a. x, all t M such that which leads to F x, t âx t p lim sup t + t s < τ, 3.5 F x, t âx t p τ t s, for all t >M. On the other hand, since F x, = and A 3 holds, from Lebourg s mean value theorem, for a.a. x, it follows that 3.6 F x, t ax t p, for all δ < t M. Note that ax L +, p >q + and, δ < imply 2λ x, we obtain from 3.4 and 3.6 that 3.7 F x, t 2λ t qx + < 2λ, for a.a. ax+ 2λ t p, for all t M. When t >M >δ,sinceδ < and p >q +,then t p > t qx,whichshows δ that 3.8 2λ t qx + ax+ 2λ t p >.

12 48 Lian Duan, Lihong Huang and Zuowei Cai Thus, through 3.5, for a.a. x, and 3.8, we have 3.9 F x, t ax+âx+ 2λ t p τ t s 2λ t qx, for all t >M. Due to q + <p <s, it follows from 3.7 that F x, t 2λ t qx + ax+ 2λ t p ax t p M p τ t s + ax t qx + τ t M s M M 2λ + ax+τ t qx M qx + ax ax + 2λ M p qx τ M s t s t p, for all t M. Therefore, note that M>, from 3.9 and 3., for a.a. x and all t R, one can derive F x, t 2λ + ax+τ t qx M qx + ax+âx+ 2λ we take M >Msufficiently large such that ax+τ 3.2 F x, t λ t qx + M qx t p τ M s t s, <λ, without loss of generality we suppose that ax > âx >, for all x, it follows from 3. that 2ax+ 2λ t p τ t s. M s We now distinguish two cases to complete the proof. Case. We claim that there exists a u V px with u > such that h u > r. In fact, for any u V px with u >, we obtain from 3.2 that 3.3 h u =Ju+F u = K k p + u p + λ u px + u px dx px 2ax+ 2λ u qx dx + u p dx. τ M s u s dx F x, udx

13 On Existence of Three Solutions for px-kirchhoff Type Differential Inclusion Problem 49 Since the imbedding X L p is compact, then there exists a constant C >, such that u p C u, forallu X. By virtue of Lemma 3.2, taking a ɛ small enough, we can reach λ 3.4 p + λ + ɛ > 2 a + 2λ C. Recall that ax L +,where denotes the norm of L, there exists a u ɛ V px with u ɛ >, combining 3.3 with 3.4 yields h u ɛ =Ju ɛ +F u ɛ k + p + uɛ px + u ɛ px dx + λ px τ M s k + λ u ɛ s dx λ + ɛ 2 a + 2λ k p + u ɛ p. u ɛ p + C u ɛ p 2ax+ 2λ τ M s u ɛ p dx u ɛ s dx u ɛ qx dx Taking u = u ɛ and a constant <r < k,wehaveh p + u >r. Case 2. We also claim that there exists a u 2 V px with u 2 such that h u 2 >r 2. In fact, for any u V px with u, we know from 3.3 that 3.5 Set h u k p + u p+ + τ M s 2 a + 2λ { K = t, t < min u s dx + λ u p dx. λ 2 a +2λ then we can choose a u 2 K V px with u 2 <, satisfying λ u 2 qx 2 a + 2λ u 2 p >, u qx dx p q },

14 4 Lian Duan, Lihong Huang and Zuowei Cai which, together with 3.5, leads to 3.6 h u 2 k p + u p+ + τ M s u s dx k p + u p+ >. We can easily see from 3.6 that there exists a r 2 > satisfying h u 2 > r 2. Combining the above two cases, there exist a ū V px with ū and a r > such that h ū > r. The proof of Lemma 3.3 is completed. Remark 3.2. From the proof of Lemma 3.3, it is easy to see that h is coercive, i.e., h u + as u +. In fact, since p <s, by Young inequality, we have 3.7 2ax+ 2λ u p p ɛ where c =2 a + 2λ that h u k p + u p + λ Let ɛ< s τ p M s s u p s p p ɛ s u s + c 2, + s p s ɛ p s s p s c p >,c 2 = c 2 c,ɛ >. Therefore, we know from 3.3 u qx dx + τ M s, one can easily get the coercivity of h. Lemma 3.4. There exists a r> with r< r such that sup u h,r] V px p ɛ u s dx s u s dx c 2. h2 u <r h 2ū h ū, where ū, r and h 2 are as mentioned in Lemma 3.3 and 3.3, respectively. Proof. Firstly, from the assumptions of A 2 4,foranyɛ>, there exists a δ >, for any < t δ = min{δ, }, for a.a. x, wehave 3.8 F 2 t, x ɛ px t p+. By considering again A 2 4, for the above ɛ>, there exists a N> large enough, for a.a. x, such that 3.9 F 2 t, x ɛ t p+, for all t >N.

15 On Existence of Three Solutions for px-kirchhoff Type Differential Inclusion Problem 4 Secondly, since F 2 x, = and taking into account A 2 3, for a.a. x, bythe Lebourg mean value theorem, it follows that 3.2 F 2 x, t cx t αx, for all δ < t N, where cx L + and p + <αx <p x, forallx. Combining 3.9 and 3.2, for a.a. x, wehave 3.2 F 2 x, t ɛ t p+ + cx t αx, for all t >δ, we get from 3.8 and 3.2 that 3.22 F 2 x, t ɛ px t p+ + ɛ t p+ + cx t αx, for all t δ, and 3.23 F 2 x, t ɛ px t p+ + ɛ t p+ + cx t αx, for all t >δ, and hence, for a.a. x and all t R, from 3.22 and 3.23, we have 3.24 F 2 x, t + ɛ t p+ + cx t αx. px Define the function g :[, + R by { } gt =sup h 2 u :u V px with u p+ ηt, where η is an arbitrary constant satisfying η>. Note that αx <p x, forall x, due to the embedding X L αx is compact and the embedding X L p+ is continuous, then there exist constants C 2 > and C 3 > such that 3.25 u αx C 2 u, u p + C 3 u, for all u X. Since h 2 u =F 2 u, then from 3.24 and 3.25, we have gt p + p ɛ u p+ + c p + 3 max { u α+ αx, } u α αx ɛc p+ 3 ηt + c 3 max { C α+ 2 η α + p + t α+ p +,C α 2 η α p + t α } p + where c 3 = c. On the other hand, by virtue of A 2 4, gt > for t >. Furthermore, due to α >p + and the arbitrariness of ɛ>, we deduce gt 3.27 lim =. t + t

16 42 Lian Duan, Lihong Huang and Zuowei Cai By Lemma 3.3, we know h ū >, it is obvious that ū. Thus, by A 2 4,it shows that h 2 ū >. In view of 3.27, for h 2ū h >, there exists a t ū > such that gt < h 2ū t h ū, for all t<t, that is, 3.28 { sup } h 2 u <t h 2ū h Vpx ū. u u p+ ηt Now, we choose a constant r with <r<min{t, r}where r is the one in Lemma 3.3. If h u r, then by the coercivity of h, there exists a constant c 4 > such that u p+ c 4 r, therefore, we have h { },r] Vpx u V px, u p+ c 4 r, from which it follows that 3.29 u h,r] sup h 2 u sup h 2 u. Vpx u { u p+ c 4 r} V px In view of the fact that r < t and c 4 >, one can deduce from 3.28 and the arbitrariness of η> that 3.3 sup u { u p+ c 4 r} V px h 2 u <r h 2ū h ū. Therefore, by 3.29 and 3.3, the desired inequality of Lemma 3.4 is obtained. This completes the proof of Lemma 3.4. By a standard argument, one can easily show that h and h 2 are locally Lipschitz under the assumptions of A and A 2. Here, we consider the indicator function of closed subspace V px, i.e., ψ : X, + ], {, if u Vpx, ψ u = +, otherwise. Obviously, ψ is convex, proper, and lower semicontinuous. Denote I u :=h u+ψ u, and I 2 u :=h 2 u, u X. It is clear that I and I 2 satisfy H in Section 2. function I + λi 2 complies with H as well. Therefore, for every λ> the

17 On Existence of Three Solutions for px-kirchhoff Type Differential Inclusion Problem 43 Lemma 3.5. If the assumptions of A and A 2 hold, then for every λ> the function I + λi 2 satisfies PS c in V p x,c R, and I + λi 2 Vpxu =+, for all u V px. lim u + Proof. Firstly, we shall prove that I + λi 2 is coercive in V p x, for every λ>. Indeed, for every u V px and without loss of generality we assume u >, because of the definition of ψ and 3.3, we obtain 3.3 I + λi 2 Vpxu = K u px + u px dx px F x, udx λ k p + u p + λ 2ax+ 2λ F 2 x, udx u qx dx + u p dx λ τ M s u s dx F 2 x, udx. On the one hand, by assumption of A 2 4, there exists a M >, for a.a. x and all t >M, such that F 2 x, t <. On the other hand, from the Lebourg mean value theorem, for a.a. x and all t R, one has 3.32 F2 x, t F 2 x, v t, for some v F 2 x, θt, <θ<. On account of A 2 3 and F 2 x, =, thenfor a.a. x and all t such that t M, it follows from 3.32 that F 2 x, t c 5, where c 5 = c 5 M, b >. Then it follows that 3.33 F 2 x, udx = F 2 x, udx + M s { u >M } { u M } F 2 x, udx c 5. By applying 3.7 and 3.33 to 3.3, we have I + λi 2 Vpxu k p + u p + λ u qx dx τ + u s dx p ɛ s u s dx c 2 + λc 5.

18 44 Lian Duan, Lihong Huang and Zuowei Cai s τ Let ɛ<, then it is easy from 3.33 to see that I p + λi 2 is coercive in V px, M s for every λ>. Next, we prove that I + λi 2 Vpxu satisfies PS c,c R. Let{u n } V px be a sequence such that 3.34 I u n +λi 2 u n c and for every v V px,wehave 3.35 h + λh 2 u n ; v u n +ψ v ψ u n ɛ n v u n, for a sequence {ɛ n } in [, + with ɛ n +. By the coerciveness of the function I +λi 2 in V p x, 3.34 implies that the sequence {u n } is bounded in V px. Therefore, there exists an element u V px such that u n u in V px. Since J is a continuous differentiable function in 3.2, then by Proposition 2.3iii and iv we have 3.36 h + λh 2 u n ; v u n h u n; v u n +λh 2 u n; v u n = J u n,v u n X + F u n ; v u n +λf 2u n ; v u n. Choose in particular v = u in 3.35 and the definition of ψ, 3.36 becomes J u n,u n u X ɛ n u u n + F u n ; u u n +λf 2u n ; u u n. Since the embedding X L p and X L px are compact, passing to a subsequence if necessary, we may assume that {u n } converges strongly to u in L p and L px. By Lemma 3., Proposition 2.3, A 3 and Hölder inequality, one can deduce that F u n ; u u n F x, u n ; u u n dx = = F x, u n ; u n udx max { ζ n x,u n u X : ζ n x F x, u n x } dx ax u n p u n u dx a u n p p u n u p.

19 On Existence of Three Solutions for px-kirchhoff Type Differential Inclusion Problem 45 Because {u n } is bounded in X, it follows that {u n } is bounded in L p, and recall that {u n } converges strongly to u in L p. Thus, and so u n u p, as n F u n ; u u n, as n +. By a similar argument as above, it follows from Lemma 3., Proposition 2.3, A 2 3 and Hölder inequality that F 2u n ; u u n F 2 x, u n ; u u n dx = = F 2 x, u n ; u n udx max { ξ n x,u n u X : ξ n x F 2 x, u n x } dx bx u n px u n u dx b u n px px u n u px, also by the compact embedding of X L px,weget 3.38 F 2 u n; u u n, as n +. Because of the sequence ɛ n as n +, together with 3.37 and 3.38, we conclude that 3.39 lim sup J u n,u n u X. n + From u n u, one can easily see that which, together with 3.39, implies that lim n + J u,u n u X =, lim sup J u n J u,u n u X. n + We know from Lemma 2. that u n u as n +, which implies that I + λi 2 satisfies PS c in V px,c R.

20 46 Lian Duan, Lihong Huang and Zuowei Cai We are now in a position to prove the main result of this section. Proof of Theorem 3.. Firstly, from Lemma 2. iv and a standard argument, the function h is locally Lipschitz and weakly sequentially lower semicontinuous. Since A 2 3 holds and X is compactly embedded in L px, the assertion remains true regarding h 2 as well, thus b of Theorem 2. is satisfied. Secondly, it is clear from Lemma 3.5 that b 2 in Theorem 2. holds. Finally, by Lemmas 3.3 and 3.4, and the definition of ψ, we know that there exist a r>and ū V px such that I ū >r. Moreover, keep in mind that F x, = F 2 x, =, thenwehave I = I 2 =. Choose ρ satisfying sup u I,r] V px I2 u <ρ<r I 2ū I ū, by Theorem 2.2, one has 3.4 sup inf I u+λρ + I 2 u inf sup I u+λρ + I 2 u, λ u V px u V px λ it is easy to see from Lemma 3.4 that ρ>. If we define h :[, + R by hλ=ρλ and Λ=[, +, thenh and 3.4 satisfy the condition b 3 intheorem 2.. Therefore, all conditions in Theorem 2. are satisfied, then there is an open interval Λ Λ such that for each λ Λ the function I + λi 2 has at least three critical points in V px, then the energy function ϕ = h +λh 2 corresponding to problem P λ possesses at least three critical points in V px, which implies that problem P λ has at least three nontrivial solutions in V px. The proof of Theorem 3. is completed. ACKNOWLEDGMENTS The authors would like to thank the Editor-in-Chief, the associate editor and the anonymous reviewer for their valuable comments and constructive suggestions, which helped to enrich the content and greatly improve the presentation of this paper. REFERENCES. G. Kirchhoff, Mechanik, Teubner, Leipzig, C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 25, X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in R 3, J. Differential Equations, , D. Liu and P. Zhao, Multiple nontrivial solutions to a p-kirchhoff equation, Nonlinear Anal., 75 22,

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22 48 Lian Duan, Lihong Huang and Zuowei Cai 22. F. H. Clarke, Nonsmooth Analysis and Optimization, Wiley, New York, X. Fan, Eigenvalues of the px-laplacian Neumann problems, Nonlinear Anal., 67 27, X. Fan and S. Deng, Remarks on Ricceri s variational principle and applications to the px-laplacian equations, Nonlinear Anal., 67 27, B. Ricceri, Existence of three solutions for a class of elliptic eigenvalue problems, in: Advanced Topics in Nonlinear Operator Theory, Math. Comput. Modelling, , A. Kristály, Multiplicity results for an eigenvalue problem for hemi-variational inequalities in strip-like domains, Set-Valued Anal., 3 25, Lian Duan a, Lihong Huang a,b and Zuowei Cai b a College of Mathematics and Econometrics Hunan University Changsha, Hunan 482 P. R. China b Department of Information Technology Hunan Women s University Changsha, Hunan 44 P. R. China lianduan96@63.com huanglihong234@26.com caizuowei@26.com