Conference Paper Existence and Multiplicity of Solutions for a Robin Problem Involving the p(x)-laplace Operator

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1 Conference Papers in Mathematics Volume 203, Article ID 23898, 7 pages Conference Paper Existence and Multiplicity of Solutions for a Robin Problem Involving the -Laplace Operator Najib Tsouli, Omar Chakrone, Omar Darhouche, and Mostafa Rahmani Department of Mathematics, University Mohamed I, P.O. Box 77, Oujda 60000, Morocco Correspondence should be addressed to Omar Darhouche; omarda3@hotmail.com Received 6 June 203; Accepted 0 July 203 Academic Editors: G. S. F. Frederico, N. Martins, D. F. M. Torres, and A. J. Zaslavski This Conference Paper is based on a presentation given by Najib Tsouli at The Cape Verde International Days on Mathematics 203 held from 22 April 203 to 25 April 203 in Praia, Cape Verde. Copyright 203 Najib Tsouli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study the following nonlinear Robin boundary-value problem Δ u=λf(x,u)in, u 2 ( u/ v) + (x) u 2 u=0 on, where R N is a bounded domain with smooth boundary, u/ V is the outer unit normal derivative on, λ>0is arealnumber,p is a continuous function on with inf x >, L ( ) with : = inf x (x) > 0,andf: R R is a continuous function. Using the variational method, under appropriate assumptions on f, we obtain results on existence and multiplicity of solutions.. Introduction The purpose of this paper is to study the existence and multiplicity of solutions for the following Robin problem involving the -Laplacian: Δ u=λf(x, u) in, u 2 u ] +(x) u 2 u=0 on, where R N is a bounded smooth domain, u/ ] is the outer unit normal derivative on, λ > 0 is a real number, p is a continuous function on with p := inf x >,and L ( ) with := inf x (x) > 0. The main interest in studying such problems arises from the presence of the -Laplace operator div( u 2 u), whichisanatural extension of the classical p-laplace operator div( u p 2 u) obtained in the case when p is a positive constant. However, such generalizations are not trivial since the -Laplace operator possesses a more complicated structure than p- Laplace operator; for example, it is inhomogeneous. In the recent years increasing attention has been paid to the study of differential and partial differential equations () involving variable exponent conditions. The interest in studying such problems was stimulated by their applications in elastic mechanics, fluid dynamics, and calculus of variations; for information on modelling physical phenomena by equations involving -growth condition we refer to [ 0]. In the past decades a vast amount of literature that deals with the existence for type problems Δ u=f(x,u)with different boundary conditions (Dirichlet, Neumann, Robin, nonlinear, etc.) has appeared. See, for instance, [ 6] and references therein. In [4], by applying the subersolution method and the variational method, under appropriate assumptions on f, the author proves that there exists λ >0such that problem () has at least two positive solutions if λ (0,λ ), has at least one positive solution if λ=λ <+,andhasnopositive solution if λ = λ.recentlyin[], the authors obtain the existence of at least two nontrivial solutions for problem () using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functions. We make the following assumptions on the function f: (H0) f: R R satisfies the Carathéodory condition and there exists a constant C 0such that f (x, s) C(+ s α(x) ) (x, s) R, (2)

2 2 Conference Papers in Mathematics where α(x) C() and < α(x) < p (x) for all x. (H) There exist R>0, μ>p + := x such that, for all s R and x, 0<μF(x, s) f(x, s) s. (3) (H2) One has f(x, s) = o( s p+ ) as s 0and uniformly for x. (H3) One hasf(x, s) = f(x, s), x, s R. (H4) One has t R t f(x, s) ds > 0,forallx. 0 (H5) There exists q C() such that p + < q := inf x q(x) q(x) < q (x) and lim t 0 t 0 x f (x, s) ds t q(x) <+. (4) Themainresultsofthispaperareasfollows. Theorem. If (H0), (H), and (H2) hold and α >p +,then, for any λ (0,+ ), () has at least a nontrivial weak solution. Theorem 2. If (H0), (H), and (H3) hold and α >p +,then, for any λ (0,+ ), () has infinite many pairs of weak solutions. Theorem 3. If (H0), (H4), and (H5) hold and p >α +,then there exist an open interval Λ (0, ) and a positive real number ρ such that, for each λ Λ, () has at least three solutions whose norms are less than ρ. To prove our results, we will use a variational method and the theory of variable exponent Sobolev spaces. For the proof of Theorem, wewillusethemountainpasstheorem(see [7, 8]). For the proof of Theorem 2,wewillusetheFountain Theorem (see [8, 9]). For the proof of Theorem 3,wewilluse Ricceri three critical points Theorem (see [20, 2]). This paper is organized as follows. First, we will introduce some basic preliminary results and lemmas in Section 2. In Section 3,wewillgivetheproofsofourmainresults. 2. Preliminaries For completeness, we first recall some facts on the variable exponent spaces L () and W, (). Supposethat is a bounded open domain of R N with smooth boundary and p C + (),where C + () = {p C (), inf p (x) >}. (5) x Denote p := inf x and p + := x. Definethe variable exponent Lebesgue space L () by L () ={u: R is a measurable, u dx<+ }, (6) with the norm u L () = u = inf {τ >0; u dx }. (7) τ Define the variable exponent Sobolev space W, () by W, () = {u L () : u L ()}, (8) with the norm u u u = inf {τ > 0; ( + )dx }, τ τ u = u + u. We refer the reader to [4, 22, 23] for the basic properties of the variable exponent Lebesgue and Sobolev spaces. Lemma 4 (see [23]). Both (L (), ) and (W, (), )are separable, reflexive, and uniformly convex Banach spaces. Lemma 5 (see [23]). Hölderinequalityholds,namely, uv dx 2 u V p (x) u L (), V L p (x) (), (0) where / + /p (x) =. Now,weintroduceanorm,whichwillbeusedlater.For u W, (), define u (x) u = inf {λ > 0 : dx λ u (x) + (x) dσ λ x }. (9) () Then, by Theorem 2. in [4], u is also a norm on W, () which is equivalent to u. Hereafter, let p (x) = { Np (x) { N, { +, if p (x) <N, if p (x) N, p (x) = { (N ) p (x), if p (x) <N, { N { +, if p (x) N. (2) Lemma 6 (see [2, 6, 23]). () If q C + () and q(x) < p (x) for any x, then the imbedding from W, () to L q(x) () is compact and continuous. (2) If q C + () and q(x) < p (x) for any x,then the trace imbedding from W, () to L q(x) ( ) is compact and continuous.

3 Conference Papers in Mathematics 3 An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the mapping defined by the following. Lemma 7 (see [4]). Denoting I (u) = u dx+ (x) u dσ x, u W, (), with >0,then () u u p (2) u u p+ I (u) u p+, I (u) u p, (3) (3) u 0ifandonlyifI (u) 0 (as k ), (4) u(x) if and only if I (u) (as k ). Lemma 8 (see [20, 2, 24]). Let X be a separable and reflexive real Banach space; φ : X R is a continuous Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X ; ψ : X R is a continuous Gâteaux differentiable functional whose Gâteaux derivative is compact. Assume that (i) lim u X (φ(u) + λψ(u)) = for all λ>0, (ii) there exist r Rand u 0,u Xsuch that φ(u 0 )<r< φ(u ), (iii) inf ψ (u) u φ (,r] > (φ (u ) r)ψ(u 0 ) + (r φ (u 0 )) ψ (u ). φ(u ) φ(u 0 ) (4) Then there exist an open interval Λ (0, ) and a positive constant ρ>0such that for any λ Λthe equation φ (u) + λψ (u) = 0 has at least three solutions in X whose norms are less than ρ. Theorem 9. Let X=W, () and f: R R be a continuous function with primitive F(x, u) = u f(x, t)dt. If 0 the following condition hold: f (x, s) C(+ s α(x) ), (x, s) R, (5) where C 0is a constant and α(x) C + () such that, for all x, α(x) < p (x), thenψ(u) = F(x, u(x))dx C (X, R) and Dψ(u, φ) = ψ (u), φ = f(x, u(x))φ dx; moreover, the operator ψ :X X is compact. Proof. It is easily adapted from Theorem 2. in [3]. Let X=W, () and (x) φ (u) = p (x) u dx+ p (x) u dσ x, ψ (u) = F (x, u) dx, J (u) =φ(u) +λψ(u), where F(x, t) = t f(x, s) ds. 0 Obviously φ C (X, R) and (φ (u), V) = u 2 u V dx + (x) u 2 uv dσ x, (ψ (u), V) = f (x, u) V dx. Moreover, we have the following. (6) (7) Proposition 0 (see []). () φ :X X is a continuous, bounded, and strictly monotone operator. (2) φ : X X is a mapping of type (S) + ;thatis,if u n uin X and lim n (φ (u n ) φ (u), u n u) 0, then u n uin X. (3) φ :X X is a homeomorphism. Definition. One says that u W, () is a weak solution of problem ()ifforallv W, () u 2 u Vdx+ (x) u 2 uv dσ x =λ f (x, u) V dx, where dσ x is the measure on the boundary. 3. Proof of Main Results (8) To prove Theorem, wehavetocheckthatthefunctionalj satisfies the following compactness condition (PS). Definition 2. One says that the C -functional H:X R satisfies the Palais-Smale condition ((PS) condition for short) if any sequence (u n ) n N Xfor which (H(u n )) n N R is bounded and H (u n ) 0 as n has a convergent subsequence. Lemma 3. If (H0), (H) hold, then for any λ (0,+ )the functional J satisfies the Palais-Smale condition (PS). Proof. Suppose that (u n ) Xis a (PS) sequence; that is, J(u n) M, J (u n ) 0 as n. (9)

4 4 Conference Papers in Mathematics We claim that (u n ) is bounded in X. Using hypothesis (H), since J(u n ) is bounded, we have for n large enough M+ J(u n ) μ J (u n ),u n + μ J (u n ),u n = p (x) u n dx+ λ F(x,u n )dx (x) p (x) u n dσ x μ (I (u n ) λ f(x,u n )u n dx) + μ J (u n ),u n p + I u (u n ) λ n μ f(x,u n)dx μ (I (u n ) λ f(x,u n )u n dx) + μ J (u n ),u n ( p + μ ) u n p μ J (u n ) X u n C ( p + μ )C 3 u n p C 2 μ u n C, (20) where C, C 2,andC 3 are three positive constants. Hence (u n ) is bounded in X since μ>p +. Without loss of generality, we assume that u n u,thenψ (u n ) ψ (u) since ψ :X X is completely continuous. The hypothesis J (u n ) 0, that is, φ (u n )+λψ (u n ) 0 and the fact that ψ (u n ) ψ (u) imply that φ (u n ) λψ (u). FromProposition 0, φ is a homeomorphism, then u n u,andsoj satisfies the (PS) condition. The proof is complete. Lemma 4. There exists r,c >0such that J(u) C for all u Xsuch that u =r. Proof. Conditions (H0) and (H2) assure that F (x, s) ε s p+ +C(ε) s α(x) (x, s) R. (2) For u small enough, we have J (u) p + u p+ λ F (x, u) dx p + u p+ λ (ε u p+ +C(ε) u α(x) )dx. (22) Note that <p + <α α(x) < p (x), forallx ;then, by Lemma 6,wehaveX L p+ () and X L α(x) () with a continuous and compact embedding. Furthermore, there exists C 4, C 5 >0such that u L p + () C 4 u, (23) u L α(x) () C 5 u, u X. Since u is small enough, we deduce u α(x) dx max { u α α(x), u α+ α(x) } C 6 u α. (24) Replacing in (22), it results that J (u) p + u p+ λεcp+ 4 u p+ λc(ε) C 6 u α. (25) Choosing ε>0small enough such that 0 < λεc p+ 4 </2p +, then we obtain J (u) + 2p + u p λc(ε) C 6 u α u p+ ( 2p + λc(ε) C 6 u α p + ). (26) Since p + <α,thefunctiont (/2p + λc(ε)c 6 t α p + ) is strictly positive in a neighborhood of zero. It follows that there exist r >0and C >0such that J (u) C u X : u =r. (27) The proof is complete. Proof of Theorem. To apply the Mountain Pass Theorem [7, 8], we need to prove that J(tu) as t +,fora certain u X. From condition (H), we obtain F (x, s) c s μ (x, s) R. (28) Letting u Xand t>,wehave J (tu) = t p (x) u dx+ t (x) u dσ p (x) x λ F (x, tu) dx (x) t p+ ( p (x) u dx+ p (x) u dσ x ) t μ λc u μ dx. (29) The fact μ>p + implies for any λ (0,+ ), J(tu) as t +. It follows that there exists e Xsuch that e >r and J(e) < 0. According to the Mountain Pass Theorem, J admits acriticalvalueτ C which is characterized by τ=inf J (h (t)), h Γ (30) t [0,] where Γ={h C([0, ],X) :h(0) =0,h() =e}. (3) This completes the proof.

5 Conference Papers in Mathematics 5 Since X is a separable and reflexive Banach space [22, 25], there exist {e n } n= Xand {f n} n= X such that For k=,2,...denote f n (e m )=δ n,m ={, if n=m, 0, if n =m. X=span {e n :n=,2,...}, X = span W {f n :n=,2,...}. X n = span {e n }, Y n = Z n = X j. j=n n j= X j, (32) (33) Lemma 5. For α(x) C + (), α(x) < p (x),andx,let Then lim k γ k =0. γ k = { u L α(x) () : u =,u Z k }. (34) Proof. It is clear that 0<γ k+ <γ k,soγ k converges to γ 0. Let u k Z k such that u k =, 0 γ k u k L α(x) () < k. (35) Then, there exists a subsequence, noted also by (u k ),suchthat u k uin X and f i,u = lim f i,u k = 0, i =, 2,.... (36) k + Thus u=0and so u k 0in X. According to Lemma 6,there is a compact embedding of X into L α(x) (), which assure that u k 0in L α(x) ().Hencewegetγ k 0as k +. Proof of Theorem 2. We will use the Fountain theorem [8, 9]. Obviously, J is an even functional and satisfies the (PS) conditionaccordingto(h)and(h2).wewillprovethatifk is large enough, then there exist ρ k >r k >0such that (A) b k := inf{j(u) : u Z k, u =r k } + as k +, (A2) a k := max{j(u) : u Y k, u =ρ k } 0as k +. (A) For u Z k such that u =r k >, by condition (H0), we have (x) J (u) = p (x) u dx+ p (x) u dσ x λ F (x, u) dx p + u p λ C(+ u α(x) )dx p + u p λc u α(x) dx C 7. (37) If u α(x) then u α(x) dx u α α(x).andif u α(x) > then u α(x) dx u α+ α(x) (γ k u ) α+.soweobtainthat { p J (u) + u p C 8 (λ) C 7 if u α(x) { { p + u p C 8 (λ) (γ k u ) α+ C 7 if u α(x) > p + u p C 8 (λ) (γ k u ) α+ C 9 =r p k ( p + C 8 (λ) γ α+ p k rα+ k ) C 9. (38) If we take C 8 (λ)γ α+ p k rα+ k = /α +,thatis,r k = (C 8 (λ) α + γ α+ α +) k )/(p,weobtain J (u) r p k ( p + α + ) C 9. (39) Since γ k 0and p p + <α α +,wehaver k + as k +.Consequently, J (u) + as u +, u Z k. (40) So (A) holds. (A2) Condition (H) implies that there exist positive constants C 0, C such that F (x, s) C 0 s μ C, (x, s) R. (4) Let u Y k be such that u =ρ k >r k >.Then J (u) p u p+ p u p+ λ (C 0 u μ C )dx λc 0 u μ dx+c 2. (42) Note that the space Y k has finite dimension; then all norms are equivalents and we obtain Finally, J (u) p u p+ λc 3 u μ +C 2. (43) J (u) as u +, u Y k (44) since μ > p +. The assertion (A2) is then satisfied and the proof of Theorem 2 is complete. Proof of Theorem 3. We will use Ricceri three critical points Theorem (see [20, 2]). Notice that φ is a continuous convex functional, so it is weakly lower semicontinuous and its inverse derivative is continuous. From Theorem 9 the precondition of Lemma 8 is satisfied. Now we only need to verify that conditions (i), (ii), and (iii) in Lemma 8 are fulfilled.

6 6 Conference Papers in Mathematics Firstly for u Xsuch that u,wehave ψ (u) = F (x, u) dx= C ( u (x) + α (x) u α(x) )dx u(x) ( f (x, t) dt) dx 0 C u (x) dx+ C α u (x) α(x) dx. (45) Using the Hölder inequality and the Sobolev embedding theorem, we have for some positive constants C 4 and C 5 C u (x) dx 2C α (x) u α(x) C 4 u, C α u α(x) dx C max { u α+ α α(x), u α α(x) } C 5 u α+, (46) where /α(x)+/α (x) =.Combiningalltogetherweobtain ψ (u) C 4 u +C 5 u α+. (47) On the other hand, φ (u) = (x) p (x) u dx+ p (x) u dσ x p + u p. (48) Then for any λ>0, φ (u) +λψ(u) p + u p From p >α + we obtain λc 4 u λc 5 u α+. (49) lim u (φ (u) +λψ(u)) =, (50) then (i) of Lemma 8 is verified. Secondly, letting u 0 =0,thenobviouslywehaveφ(u 0 )= ψ(u 0 )=0. We claim that there exist r > 0 and u X such that φ(u 0 )>rand inf ψ (u) >r ψ(u ) u φ ((,r]) φ(u ). (5) From (H5), there exits η [0,]and C 6 >0such that F (x, t) C 6 t q(x) C 6 t q, t [ η, η] uniformly for x. In view of (H0), if we put C 7 := max { { C 6, η t < { A(+ t α ) t q (52) A(+ t α+ )}, }, t t q } (53) where A is a positive constant, then we have F (x, t) C 7 t q, t R uniformly for x. (54) Fix r such that 0<r<.If(/p + ) u p+ r<,then,bythe Sobolev embedding theorem (X L q () is continuous), we have F (x, u) dx C 7 u q dx C 8 u q C 9r q /p +, (55) where C 8 and C 9 are two positive constants. Since q >p +,thenwehave lim r 0 + r (/p + ) u p+ F (x, u) dx=0. r (56) By (H4), we can choose a constant b X \ {0} such that F(x, b)dx > 0. Fix r 0 such that r 0 <(/p + ) min{ b p+, b p,}. If b >,thenwehave p + b p φ(b) p b p+. (57) From (56) and(57), we deduce that, when 0<r<r 0,then φ(b) > r and Thus (/p + ) u p+ (/p + ) u p+ r F (x, u) dx r 2 r Since r<r 0,weget Then F (x, u) dx. (58) (/p ) b p+ F (x, u) dx r F (x, u) dx. (59) (/p ) b p+ φ ((, r]) {u X: p + u p with u =bwhich implies that + r}. (60) ψ(u) < r ψ(u ) φ(u) r φ(u ), (6) inf ψ (u) >r ψ(u ) u φ ((,r]) φ(u ). (62) So we can find r>0, u =b,andφ(b) > r satisfying (ii) and (iii) of Lemma 8. If b,wehave p + b p + φ(b) p b p. (63)

7 Conference Papers in Mathematics 7 From (56) and(63), we deduce that, when 0<r<r 0,then φ(b) > r and Thus (/p + ) u p+ (/p + ) u p+ r F (x, u) dx r 2 r Since r<r 0,weget Then with u =b. Therefore Then we obtain inf u φ ((,r]) F (x, u) dx. (64) (/p ) b p F (x, u) dx r F (x, u) dx. (65) (/p ) b p φ ((, r]) {u X: p + u p + r}. (66) ψ(u) < r ψ(u ) φ(u) r φ(u ), (67) inf ψ (u) >r ψ(u ) u φ ((,r]) φ(u ). (68) ψ (u) > (φ (u ) r)ψ(u 0 ) + (r φ (u 0 )) ψ (u ). φ(u ) φ(u 0 ) (69) This means that condition (iii) in Lemma 8 is verified. Now since all the assumptions of Lemma 8 are verified, there exist an open interval Λ (0, ) and a positive constant ρ>0 such that for any λ Λthe equation φ (u) + λψ (u) = 0 has at least three solutions in X whosenormsarelessthanρ. References [] E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Archive for Rational Mechanics and Analysis,vol.56,no.2,pp.2 40,200. [2] C. Pfeiffer, C. Mavroidis, Y. Bar-Cohen, and B. Dolgin, Electrorheological fluid based force feedback device, in Proceedings of the Telemanipulator and Telepresence Technologies VI Conference, vol. 3840ofProceeding of SPIE, pp , Boston, Mass, USA, July 999. [3] L. Diening, Theorical and numerical results for electrorheological fluids [Ph.D. thesis], University of Freiburg, Breisgau, Germany, [4] M. R. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer, Berlin, Germany, [5] T. C. Halsey, Electrorheological fluids, Science, vol.258,no. 5083, pp , 992. [6] T. G. Myers, Thin films with high surface tension, SIAM Review, vol. 40, no. 3, pp , 998. [7] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Mathematics of the USSR- Izvestiya,vol.29,pp.33 66,987. [8] V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals,translatedfrom Russian by G. A. Yosifian, Springer, Berlin, Germany, 994. [9] Y. M. Chen, S. Levine, and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM Journal on Applied Mathematics,vol.66,no.4,pp ,2006. [0] W. M. Winslow, Induced fibration of suspensions, Applied Physics,vol.20,no.2,pp.37 40,949. [] B. Ge and Q.-M. Zhou, Multiple solutions for a Robin-type differential inclusion problem involving the -Laplacian, Mathematical Methods in the Applied Sciences,203. [2] J. H. Yao, Solutions for Neumann boundary value problems involving -Laplace operators, Nonlinear Analysis, Theory, Methods and Applications,vol.68,no.5,pp ,2008. [3] L.-L. Wang, Y.-H. Fan, and W.-G. Ge, Existence and multiplicity of solutions for a Neumann problem involving the -Laplace operator, Nonlinear Analysis, Theory, Methods and Applications,vol.7,no.9,pp ,2009. [4] S.-G. Deng, Positive solutions for Robin problem involving the -Laplacian, Mathematical Analysis and Applications,vol.360,no.2,pp ,2009. [5] X.-L. Fan and Q.-H. Zhang, Existence of solutions for - Laplacian Dirichlet problem, Nonlinear Analysis, Theory, Methods and Applications,vol.52,no.8,pp ,2003. [6] X. L. Fan and D. Zhao, On the generalized Orlicz-sobolev spaces W, (), Gansu Education College, vol.2, no.,article6,998. [7] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, Functional Analysis,vol.4,no.4,pp ,973. [8] M. Willem, Minimax Theorems, Birkhäuser, Basel, Switzerland, 996. [9] T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Analysis,vol.20,no.0,pp ,993. [20] B. Ricceri, On a three critical points theorem, Archiv der Mathematik,vol.75,no.3,pp ,2000. [2] C. Ji, Remarks on the existence of three solutions for the - Laplacian equations, Nonlinear Analysis, Theory, Methods and Applications,vol.74,no.9,pp ,20. [22] X. L. Fan, J. S. Shen, and D. Zhao, Sobolev embedding theorems for spaces W k, (), Mathematical Analysis and Applications,vol.262,no.2,pp ,200. [23] X. L. Fan and D. Zhao, On the spaces L and W m;, Journal of Mathematical Analysis and Applications, vol.263,pp , 200. [24] G. Bonanno and P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-laplacian, Archiv der Mathematik,vol.80,no.4,pp ,2003. [25] J. Chabrowski and Y. Fu, Existence of solutions for - Laplacian problems on a bounded domain, Mathematical Analysis and Applications,vol.306,pp ,2005.

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