Existence and multiplicity of solutions for a Neumann-type p(x)-laplacian equation with nonsmooth potential. 1 Introduction

Transcription

1 Electronic Journal of Qualitative Theory of Differential Equations 20, No. 7, -0; Existence and multiplicity of solutions for a Neumann-type p(x)-laplacian equation with nonsmooth potential Bin Ge, Qingmei Zhou 2 Department of Applied Mathematics, Harbin Engineering University, Harbin, 5000, P. R. China 2 Library, Northeast Forestry University, Harbin, 50040, P. R. China Abstract: In this paper we study Neumann-type p(x)-laplacian equation with nonsmooth potential. Firstly, applying a version of the non-smooth three-critical-points theorem we obtain the existence of three solutions of the problem in W,p(x) (). Finally, we obtain the existence of at least two nontrivial solutions, when α > p +. Key words: p(x)-laplacian, Differential inclusion problem, Three critical points theorem, Neumann-type problem. Introduction The study of differential equations and variational problems with variable exponent has been a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [, 2]). It also has wide applications in different research fields, such as image processing model (see e.g. [3, 4]), stationary thermorheological viscous flows (see [5]) and the mathematical description of the processes filtration of an idea barotropic gas through a porous medium (see [6]). The study on variable exponent problems attracts more and more interest in recent years, many results have been obtained on this kind of problems, for example [7-4]. In this paper, we investigate the following Neumann-type differential equation with p(x)- Laplacian and a nonsmooth potential: div( u p(x) 2 u) + u p(x) 2 u λ j(x, u), in, u γ = 0, on, (P) where is a bounded domain of R N with smooth boundary, λ > 0 is a real number, p(x) C() with < p := min p(x) p + := max p(x) < +, j(x, u) is the Clarke subdifferential of x x j(x, ), γ is the outward unit normal to the boundary. Corresponding author: gebin @63.com Supported by the National Science Fund (grant , 00063), Heilongjiang Province Foundation for Distinguished Young Scholars (JC20080), Program of Excellent Team in Harbin Institute of Technology and the Natural Science Foundation of Heilongjiang Province (No. A200803). EJQTDE, 20 No. 7, p.

2 In [7], Dai studied the particular case p(x) C() with N < p. He established the existence of three solutions by using the non-smooth critical -points theorem [5]. In this paper we will study problem (P) in the case when < p(x) < + for any x. We will prove that there also exist three weak solutions for problem (P), and existence of at least two nontrivial solutions, when α > p +. This paper is organized as follows. We will first introduce some basic preliminary results and lemma. In Section 2, including the variable exponent Lebesgue, Sobolev spaces, generalized gradient of locally Lipschitz function and non-smooth three-critical-points theorem. In section 3, we give the main results and their proof. 2 Preliminaries In this part, we introduce some definitions and results which will be used in the next section. Firstly, we introduce some theories of Lebesgue-Sobolev space with variable exponent. The detailed can be found in [8-3]. Assume that p C() and p(x) >, for all x. Set C + () = {h C() : h(x) > for any x }. Define h = min x h(x), h + = max h(x) for any h C + (). x For p(x) C + (), we define the variable exponent Lebesgue space: L p(x) () = {u : u is a measurable real value function u(x) p(x) dx < + }, with the norm u L p(x) () = u p(x) =inf{λ > 0 : u(x) λ p(x) dx }, and define the variable exponent Sobolev space W,p(x) () = {u L p(x) () : u L p(x) ()}, with the norm u = u W,p(x) () = u p(x) + u p(x). We remember that spaces L p(x) () and W,p(x) () are separable and reflexive Banach spaces. Denoting by L q(x) () the conjugate space of L p(x) () with p(x) + q(x) =, then the Hölder type inequality uv dx ( p + q ) u L () v p(x) L (), u L p(x) (), v L q(x) () () q(x) holds. Furthermore, define mapping ρ : W,p(x) R by ρ(u) = ( u p(x) + u p(x) )dx, then the following relations hold Hereafter, let p (x) = u < (=, > ) ρ(u) < (=, > ), (2) u > u p ρ(u) u p+, (3) u < u p+ ρ(u) u p. (4) Np(x), p(x) < N, N p(x) +, p(x) N. Remark 2.. If h C + () and h(x) p (x) for any x, by Theorem 2.3 in [], we deduce that W,p(x) () is continuously embedded in L h(x) (). When h(x) < p (x), the embedding is compact. EJQTDE, 20 No. 7, p. 2

3 Let X be a Banach space and X be its topological dual space and we denote <, > as the duality bracket for pair (X, X). A function ϕ : X R is said to be locally lipschitz, if for every x X, we can find a neighbourhood U of x and a constant k > 0(depending on U), such that ϕ(y) ϕ(z) k y z, y, z U. The generalized directional derivative of ϕ at the point u X in the direction h X is ϕ 0 ϕ(u + λh) ϕ(u ) (u; h) = limsup. u u;λ 0 λ The generalized subdifferential of ϕ at the point u X is defined by ϕ(u) = {u X ; < u, h > ϕ 0 (u; h), h X}, which is a nonempty, convex and w compact set of X. We say that u X is a critical point of ϕ, if 0 ϕ(x). For further details, we refer the reader to [6]. Finally, for proving our results in the next section, we introduce the following lemma: Lemma 2.(see [5]). Let X be a separable and reflexive real Banach space, and let Φ, Ψ : X R be two locally Lipschitz functions. Assume that there exist u 0 X such that Φ(u 0 ) = Ψ(u 0 ) = 0 and Φ(u) 0 for every u X and that there exists u X and r > 0 such that: () r < Φ(u ); (2) sup Ψ(u) < r Ψ(u) Φ(u ), and further, we assume that function Φ λψ is sequentially lower Φ(u)<r semicontinuous, satisfies the (PS)-condition, and (3) lim (Φ(u) λψ(u)) = + u for every λ [0, a], where hr a =, with h >. r Ψ(u ) sup Ψ(u) Φ(u ) Φ(u)<r Then, there exits an open interval Λ [0, a] and a positive real number σ such that, for every λ Λ, the function Φ(u) λψ(u) admits at least three critical points whose norms are less than σ. 3 Existence theorems In this section, we will prove that there also exist three weak solutions for problem (P). Our hypotheses on nonsmooth potential j(x, t) as follows. H(j) : j : R R is a function such that j(x, 0) = 0 a.e. on and satisfies the following facts: (i) for all t R, x j(x, t) is measurable; (ii) for almost all x, t j(x, t) is locally Lipschitz; (iii) there exist α C + () with α + 0 for all x ; (v) there exist q C() such that p + < q q(x) < p (x) and j(x, t) lim = 0 uniformly a.e. x. t 0 t q(x) EJQTDE, 20 No. 7, p. 3

4 Remark 3.. It is easy to give examples satisfying all conditions in H(j). For example, the following nonsmooth locally Lipschitz function j : R R, satisfies hypotheses H(j): j(x, t) = { β(x) t β(x), if t, t, if t >, α(x) t α(x) + α(x) β(x) α(x)β(x) where α, β C + () with α + < p p + < q q + < β β(x) < p (x). In order to use Lemma 2., we define the function Φ, Ψ : W,p(x) () R by Φ(u) = p(x) ( u p(x) + u p(x) )dx, Ψ(u) = j(x, u)dx. and let ϕ(u) = Φ(u) λψ(u), by Fan [4, Theorem 3.], we know that Φ is continuous and convex, hence locally Lipschitz on W,p(x) (). On the other hand, because of hypotheses H(j)(i),(ii),(iii), Ψ is locally Lipschitz (see Clarke [6], p.83)). Therefore ϕ(u) is locally Lipschitz. We state below our main results Theorem 3.. If hypotheses H(j) hold, Then there are an open interval Λ [0. + ) and a number σ such that, for each λ Λ the problem (P) possesses at least three weak solutions in W,p(x) () whose norms are less than σ. Proof: The proof is divided into the following three Steps. Step. We will show that ϕ is coercive in the step. Firstly, for almost all x, by t j(x, t) is differentiable almost everywhere on R and we have d dt j(x, t) j(z, t). Moveover, from H(j)(iii), there exist positive constants c 3, c 4, such that t d j(x, t) = j(x, 0) + 0 dy j(x, y)dy c t + c 2 α(x) t α(x) c 3 + c 4 t α(x) (5) for almost all x and t R. Note that < α(x) α + and u >, u α(x) dx u α+ α(x) u cα+ α+. Hence, from (3) and (5), we have ϕ(u) = p(x) ( u p(x) + u p(x) )dx λ j(x, u)dx as u +. p + ( u p(x) + u p(x) )dx λ j(x, u)dx p + u p λ j(x, u)dx p + u p λc 3 meas() λc 4 c α+ u α+ +, Step 2. We show that (PS)-condition holds. Suppose {u n } n W,p(x) () such that ϕ(u n ) c and m(u n ) 0 as n +. Let u n ϕ(u n) be such that m(u n ) = u n (W,p(x) ()), n, then we know that u n = Φ (u n ) λw n EJQTDE, 20 No. 7, p. 4

5 where the nonlinear operator Φ : W,p(x) () (W,p(x) ()) defined as < Φ (u), v >= u p(x) 2 u vdx + u p(x) 2 uvdx, for all v W,p(x) () and w n Ψ(u n ). From Chang [7] we know that w n L α (x) (), where α (x) + α(x) =. Since, ϕ is coercive, {u n } n is bounded in W,p(x) () and there exists u W,p(x) 0 () such that a subsequence of {u n } n, which is still be denoted as {u n } n, satisfies u n u weakly in W,p(x) (). Next we will prove that u n u in W,p(x) (). By W,p(x) () L α(x) (), we have u n u in L α(x) (). Moreover, since u n 0, we get ε n. Note that u n = Φ (u n ) λw n, we obtain < Φ (u n ), u n u > λ w n(u n u)dx ε n, n. Moreover, w n(u n u)dx 0, since u n u in L α(x) () and {w n } n are bounded in L α (x) (), where α(x) + α (x) =. Therefore, lim sup < Φ (u n ), u n u > 0. n But we know Φ is a mapping of type (S + )(see [4, Theorem 3.]). Thus we obtain u n u in W,p(x) (). Step 3. We show that Φ, Ψ satisfy the conditions () and (2) in Lemma 2.. Consider u 0, u W,p(x) (), u 0 (x) = 0 and u (x) = t 0 for any x. A simple computation implies Φ(u 0 ) = Ψ(u 0 ) = 0 and Ψ(u ) > 0. From (3) and (4), we have if u, then p + Φ(u) u p p ; (6) u p+ if u <, then + p + u p Φ(u) p u p. (7) From H(j)(v), there exist η [0, ] and c 5 > 0 such that In view of H(j)(iii), if we put Ψ(u) = j(x, u)dx c 6 Since q > p +, we have j(x, t) c 5 t q(x) c 5 t q, t [ η, η], x. c 6 = max{c 5, sup η t < c 3 + c 4 t α a + a 2 t α+, sup }, t t q t q then we have j(x, t) c 6 t q, t R, x. Fix r such that 0 < r <. And when p max{ u p, u p+ } < r <, by Sobolev Embedding + Theorem (W,p(x) () L q ()), we have (for suitable positive constants c 7, c 8 ) u q dx c 7 u q < c 8 r q p (or c 8 r q p + ). lim r 0 + sup p + max{ u p, u p+ }<r r Ψ(u) = 0. (8) EJQTDE, 20 No. 7, p. 5

6 Fix r 0 such that r 0 r and From (6), we have Hence, sup p + u p <r Ψ(u) r 2 Ψ(u ) r Ψ(u ) p u p+ 2 Φ(u ) < rψ(u ) Φ(u ). {u W,p(x) () : Φ(u) < r} {u W,p(x) () : p + u p+ <r sup Ψ(u) < r Ψ(u ) Φ(u)<r Φ(u ). p + u p < r}. Case 2. When u, fixing r as above, with the role of u p+ above now assumed by u p, we can analogously get sup Ψ(u) r Ψ(u ) 2 r Ψ(u ) p u p 2 Φ(u ) < rψ(u ) Φ(u ). From (7), we have Hence, {u W,p(x) () : Φ(u) < r} {u W,p(x) () : + p + u p < r}. sup Ψ(u) < r Ψ(u ) Φ(u)<r Φ(u ). Thus, Φ and Ψ satisfy all the assumptions of Lemma 2., and the proof is complect. Thus far the results involved potential functions exhibiting p(x)-sublinear. The next theorem concerns problems where the potential function is p(x)-superlinear. The hypotheses on the nonsmooth potential are the following: H(j) : j : R R is a function such that j(x, 0) = 0 a.e. on and satisfies the following facts: (i) for all t R, x j(x, t) is measurable; (ii) for almost all x, t j(x, t) is locally Lipschitz; (iii) there exist α C + () with α > p + and positive constants c, c 2, such that w c + c 2 t α(x) for every t R, almost all x and all w j(x, t); (iv) There exist γ C() with p + < γ(x) lim sup < µ(x), t 0 t γ(x) uniformly for almost all x and all w j(x, t); (v) There exist ξ 0 R, x 0 and r 0 > 0, such that where B r0 (x 0 ) := {x : x x 0 r 0 } ; j(x, ξ 0 ) > δ 0 > 0, a.e. x B r0 (x 0 ) EJQTDE, 20 No. 7, p. 6

7 (vi) For almost all x, all t R and all w j(x, t), we have j(x, t) ν(x) with ν L β(x) (), β(x) < p. Remark 3.2. It is easy to give examples satisfying all conditions in H(j). For example, the following nonsmooth locally Lipschitz function j : R R, satisfies hypotheses H(j) : sin( π 2 t γ(x) ), t, j(x, t) = 2 t 3, t >, 2 Theorem 3.2. If hypotheses H(j) hold, then there exists a λ 0 > 0 such that for each λ > λ 0, the problem (P) has at least two nontrivial solutions. Proof: The proof is divided into the following five Steps. Step. We will show that ϕ is coercive in the step. By H(j) (vi), for all u W,p(x) (), u >, we have ϕ(u) = p(x) ( u p(x) + u p(x) )dx λ j(x, u)dx p + λ ν(x)dx, as u. u p Step 2. We will show that the ϕ is weakly lower semi-continuous. Let u n u weakly in W,p(x) (), by Remark 2., we obtain the following results: W,p(x) () L p(x) (); u n u in L p(x) (); u n u for a.e. x ; j(x, u n (x)) j(x, u(x)) for a.e. x. By Fatou s Lemma, lim sup j(x, u n (x))dx j(x, u(x))dx. n Thus, lim inf n) = liminf n n p(x) ( u n p(x) + u n p(x) )dx limsup λ j(x, u n )dx n p(x) ( u p(x) + u p(x) )dx λ j(x, u)dx = ϕ(u). Hence, by The Weierstrass Theorem, we deduce that there exists a global minimizer u 0 W,p(x) () such that ϕ(u 0 ) = min ϕ(u). u W,p(x) () Step 3. We will show that there exists λ 0 > 0 such that for each λ > λ 0, ϕ(u 0 ) < 0. By the condition H(j) (v), there exists ξ 0 R such that j(x, ξ 0 ) > δ 0 > 0, a.e. x B r0 (x 0 ). It is clear that 0 < M := max {c t + c 2 t α +, c t + c 2 t α } < +. t ξ 0 Now we denote M ξ t 0 = ( ) 0 ξ 0 N, K(t) := max{( δ 0 + M r 0 ( t) )p, ( r 0 ( t) )p+ } EJQTDE, 20 No. 7, p. 7

8 and λ 0 = max t [t,t 2] K(t)( t N ) + max{ξ p 0, ξp+ [δ 0 t N M ( t N )] where t 0 < t < t 2 < and δ 0 is given in the condition H(j) (v). A simple calculation shows that the function t δ 0 t N M ( t N ) is positive whenever t > t 0 and δ 0 t N 0 M ( t N 0 ) = 0. Thus λ 0 is well defined and λ 0 > 0. We will show that for each λ > λ 0, the problem (P) has two nontrivial solutions. In order to do this, for t [t, t 2 ], let us define 0, if x \B r0 (x 0 ), η t (x) = ξ 0, if x B tr0 (x 0 ), ξ 0 r 0 ( t) (r 0 x x 0 ), if x B r0 (x 0 )\B tr0 (x 0 ). By conditions H(j) (iii) and (v) we have j(x, η t (x))dx = j(x, η t (x))dx + j(x, η t (x))dx B tr0 (x 0) w N r N 0 tn δ 0 M ( t N )w N r N 0 =w N r N 0 (δ 0 t N M ( t N )). 0 } B r0 (x 0)\B tr0 (x 0) Hence, for t [t, t 2 ], ϕ(η t ) = p(x) η t p(x) dx + p(x) η t p(x) dx λ j(x, η t (x))dx p ( η t p(x) + η t p(x) )dx λw N r0 N (δ 0 t N M ( t N )) ξ 0 max{[, [ }w r 0 ( t) ]p r 0 ( t) ]p+ N r0 N ( tn ) ξ 0 + max{ξ p 0, ξp+ 0 }w Nr N 0 λw N r N 0 (δ 0 t N M ( t N )) =w N r N 0 [K(t)( tn ) + max{ξ p 0, ξp+ 0 } λ(δ 0t N M ( t N ))], so that ϕ(η t ) < 0 whenever λ > λ 0. Step 4. We will check the C-condition in the following. Suppose {u n } n W,p(x) 0 () such that ϕ(u n ) c and ( + u n )m(u n ) 0. Since, ϕ is coercive, {u n } n is bounded in W,p(x) () and passed to a subsequence, still denote {u n } n, we may assume that there exists u W,p(x) (), such that u n u weakly in W,p(x) (). Next we will prove that u n u in W,p(x) (). By W,p(x) () L p(x) (), we have u n u in L p(x) (). Moreover, since u n 0, we get ε n. Note that u n = Φ (u n ) λw n, we have < Φ (u n ), u n u > λ w n(u n u)dx ε n, n. Moreover, w n(u n u)dx 0, since u n u in L p(x) () and {w n } n in L p (x) () are bounded, where p(x) + p (x) =. Therefore, lim sup < Φ (u n ), u n u > 0. n From [4, Theorem 3.], we have u n u as n. Thus ϕ satisfies the nonsmooth C-, EJQTDE, 20 No. 7, p. 8

9 condition. Step 5. We will show that there exists another nontrivial weak solution of problem (P). From Lebourg Mean Value Theorem, we obtain j(x, t) j(x, 0) = w, t for some w j(x, ϑt) and 0 < ϑ <. Thus, from H(j) (iv), there exists β (0, ) such that j(x, t) w, t µ(x) t γ(x), t < β and a.e. x. (0) On the other hand, by the condition H(j) (iii), we have j(x, t) c t + c 2 t α(x) c t β α(x) t + c 2 t α(x) =c β α+ t α(x) + c 2 t α(x) () =c 5 t α(x) for a.e. x, all t β with c 5 > 0. Combining (0) and (), it follows that for a.e. x and all t R. j(x, t) µ(x) t γ(x) + c 5 t α(x) Thus, For all λ > λ 0, u <, u γ(x) < and u α(x) <, we have ϕ(u) = p(x) u p(x) dx λ j(x, u(x))dx p + λ µ(x) u γ(x) dx λc u p+ 5 u α(x) dx p + u p+ λc 6 u γ λc 7 u α. So, for ρ > 0 small enough, there exists a ν > 0 such that ϕ(u) > ν, for u = ρ and u 0 > ρ. So by the Nonsmooth Mountain Pass Theorem, we can get u W,p(x) () satisfies ϕ(u ) = c > 0 and m(u ) = 0. Therefore, u is second nontrivial critical point of ϕ. References [] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, [2] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory. Math. USSR. Izv, 987, 29(): [3] P. Harjulehto, P. Hästö, V. Latvala, Minimizers of the variable exponent, non-uniformly convex Dirichlet energy. J. Math. Pures Appl, 2008, 89: [4] Y. Chen, S. Levine, M. Rao, Variable exponent linear growth functionals in image restoration. SIAM J. Appl. Math, 2006, 66(4): EJQTDE, 20 No. 7, p. 9

10 [5] S. N. Antontsev, J. F. Rodrigues, On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII, Sci. Mat, 2006, 52: [6] S. N. Antontsev, S. I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: Existence uniqueness and localization properties of solutions. Nonlinear Anal. TMA, 2005, 60: [7] G. Dai, Three solutions for a Neumann-type differential inclusion problem involving the p(x)-laplacian. Nonlinear Anal, 2009, 70(0): [8] D. E. Edmunds, J. Lang, A. Nekvinda, On L p(x) () norms. Proc. R. Soc. Ser. A, 999, 455: [9] D. E. Edmunds, J. Rákosník, Sobolev embedding with variable exponent. Studia Math, 2000, 43: [0] X. L. Fan, J. Shen, D. Zhao, Sobolev embedding theorems for spaces W,p(x). J. Math. Anal. Appl, 200, 262(2): [] X. L. Fan, D. Zhao, On the space L p(x) () and W k,p(x) (). J. Math. Anal. Appl, 200, 263(2): [2] X. L. Fan, D. Zhao, On the generalized Orlicz-Sobolev space W,p(x) (). J. Gansu Educ. College, 998, 2():-6. [3] O. Kováčik, J. Rákosník, On spaces L p(x) W,p(x). Czechoslovak Math. J, 99, 4: [4] X. L. Fan, Q. H Zhang, Existence of solutions for p(x)-laplacian Dirichlet problem. Nonlinear Anal, 2003, 52(8): [5] G. Bonanno, P. Candito, On a class of nonlinear variational-hemivariational inequalities. Appl. Anal, 2004, 83: [6] F. H. Clarke, Optimization and Nonsmooth Analysis. Wiley, New York, 983. [7] K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl, 98, 80(): (Received December 4, 200) EJQTDE, 20 No. 7, p. 0