Existence of Solutions for a Class of p(x)-biharmonic Problems without (A-R) Type Conditions

Transcription

1 International Journal of Mathematical Analysis Vol. 2, 208, no., HIKARI Ltd, Existence of Solutions for a Class of p(x)-biharmonic Problems without (A-R) Type Conditions G.A. Afrouzi Department of Mathematics, Faculty of Mathematical Sciences University of Mazandaran, Babolsar, Iran N.T. Chung Department of Mathematics, Quang Binh University 32 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam M. Mirzapour Department of Mathematics, Faculty of Mathematical Sciences University of Mazandaran, Babolsar, Iran Copyright c 208 G.A. Afrouzi, N.T. Chung and M. Mirzapour. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we study the existence and multiplicity of nontrivial solutions for a class of p(x)-biharmonic problems. The interesting point lines in the fact that we do not need the usual Ambrosetti-Rabinowitz type condition for the nonlinear term f. The proofs are essentially based on the mountain pass theorem and its Z 2 symmetric version. Mathematics Subject Classification: 35P30, 35D05, 35J60 Keywords: p(x)-biharmonic problems, Mountain pass theorem, (A-R) type condition

2 506 G.A. Afrouzi, N.T. Chung, M. Mirzapour Introduction and preliminary results In this paper, we are interested in the existence of nontrivial solutions for a class of p(x)-biharmonic problems { 2 p(x) u = K(x)f(u), x, u = u = 0, x, where is a bounded domain in R N with smooth boundary, N 2, p C() with max{2, N } < 2 p := inf x p(x) p + := sup x p(x), and 2 p(x) u = ( u p(x) 2 u) is the operator of fourth order called the p(x)- biharmonic operator, K L () and f : R R is a continuous function. Recently, Navier problems involving the biharmonic operator have been studied by some authors, see [5, 6, 4] and references therein. In [4], A. Ayoujil and A.R. El Amrouss first studied the spectrum of a fourth order elliptic equation with variable exponent. After that, many authors studied the existence of solutions for problems of this type, see for examples [, 3,, 2, 3, 6]. In [3], A. El Amrouss et al. used the mountain pass theorem to study the existence of nontrivial solutions. For this purpose, the authors need the Ambrosetti- Rabinowitz (A-R) type condition (see [2]) to prove the energy functional satisfies the Palais-Smale (PS) condition. In [2, 6], the authors studied the multiplicity of solutions for a class of Navier boundary value problems involving the p(x)-biharmonic operator. We also refer the readers to recent papers [,, 3], in which the authors study the existence of eigenvalues of the p(x)-biharmonic operator. Motivated by the ideas introduced in [7] and some properties of the p(x)-biharmonic operator in [3, 4, 6], we study the existence and multiplicity of nontrivial solutions for a class of p(x)-biharmonic problems without (A-R) type conditions. Our result here is different from one introduced in the previous paper [5]. For the reader s convenience, we recall some necessary background knowledge and propositions concerning the generalized Lebesgue-Sobolev spaces. We refer the reader to the papers [8, 9, 0]. Let be a bounded domain of R N, denote C + () := {p(x); p(x) C(), p(x) >, x }, p + := max x p(x), p := min x p(x)} and define the space L p(x) () := { u : R; u is a measurable such that () } u(x) p(x) dx < +, with the norm { u L p(x) () = u p(x) := inf µ > 0; u(x) µ p(x) dx }.

3 p(x)-biharmonic problems without (A-R) type conditions 507 Denote p (x) = { Np(x) N p(x) if p(x) < N, + if p(x) N, p k(x) = { Np(x) N kp(x) if kp(x) < N, + if kp(x) N for any x, k. Proposition. (see [0]). The space (L p(x) (),. p(x) ) is separable, uniformly convex, reflexive and its conjugate space is L q(x) () where q(x) is the conjugate function of p(x), i.e., + =, for all x. For u p(x) q(x) Lp(x) () and v L q(x) (), we have ( uv dx p + q ) u p(x) v q(x) 2 u p(x) v q(x). The Sobolev space with variable exponent W k,p(x) () is defined as where D α u = W k,p(x) () = {u L p(x) () : D α u L p(x) (), α k}, α x α xα xα N N u, with α = (α,..., α N ) is a multi-index and α = N i= α i. The space W k,p(x) () equipped with the norm u k,p(x) := D α u p(x), α k also becomes a separable and reflexive Banach space. For more details, we refer to [0]. Proposition.2 (see [0]). For p, r C + () such that r(x) p k (x) for all x, there is a continuous embedding W k,p(x) () L r(x) (). If we replace with <, the embedding is compact. We denote by W k,p(x) 0 () the closure of C0 () in W k,p(x) (). Note that the weak solutions of problem () are considered in the generalized Sobolev space X = W 2,p(x) () W,p(x) 0 () equipped with the norm { } p(x) u := inf µ > 0 : u(x) µ dx. Proposition.3 (see [6]). If R N is a bounded domain, then the embedding X C() is compact whenever N < 2 p. From Proposition.3, there exists a positive constant c depending on p(x), N and such that u := sup c u, u X. (2) x

4 508 G.A. Afrouzi, N.T. Chung, M. Mirzapour Remark.4. According to [7], the norm. 2,p(x) is equivalent to the norm. p(x) in the space X. Consequently, the norms. 2,p(x),. and. p(x) are equivalent. Proposition.5 (see [3]). If we denote ρ(u) = u p(x) dx, then for u, u n X, we have the following assertions: (i) u < (respectively = ; > ) ρ(u) < (respectively = ; > ); (ii) u u p+ ρ(u) u p ; (iii) u u p ρ(u) u p+ ; (iv) u n 0 (respectively + ) ρ(u n ) 0 (respectively + ). Let us define the functional I(u) = p(x) u p(x) dx. It is well known that J is well defined, even and C in X. operator L = I : X X defined as L(u), v = u p(x) 2 u v dx for all u, v X satisfies the following assertions. Moreover, the Proposition.6 (see [3]). (i) L is continuous, bounded and strictly monotone. (ii) L is a mapping of (S + ) type, namely, u n u and lim sup n + L(u n )(u n u) 0, implies u n u. (iii) L is a homeomorphism. In order to establish the existence of infinitely many solutions for the problem (), we will use the following Z 2 version of the mountain pass theorem in [2]. Proposition.7. Let X be an infinite-dimensional Banach space, and let J C (X, R) be even, satisfy the (P S) condition, and have J(0). Assume that X = V Y, where V is finite dimensional and (i) There are constants ρ, α > 0 such that inf Bρ Y J α; (ii) For each finite-dimensional subspace X X, there is R = R( X) such that J(u) 0 on X\B R( X). Then the functional J possesses an unbounded sequence of critical values. 2 Main result In this section, we state and prove the main result of the paper. We shall use c i to denote general positive constants whose values may be changed from line to line. We first make the definition of weak solutions for ().

5 p(x)-biharmonic problems without (A-R) type conditions 509 Definition 2.. We say that u X is a weak solution of problem () if for all v X, it holds that u p(x) 2 u v dx K(x)f(u)v dx = 0 The main result of the paper can be formulated as follows. Theorem 2.2. Assume that K, f satisfy the following conditions: (H) K L () and there exists k 0 > 0 such that K(x) k 0 for all x ; (H2) f C(R, R) and there exist a constant s 0 0 and a decreasing function θ(s) C(R\( s 0, s 0 ), R) such that 0 < (p + + θ(s))f (s) f(s)s, s s 0, s where θ(s) > 0 and lim s + θ(s) s = +, lim s + s 0 F (s) = s f(t) dt; 0 (H3) lim s 0 f(s) s p+ = 0. θ(t) t dt = +, Then problem () has a nontrivial weak solution. If further, f is odd, then () has infinitely many pairs of weak solutions. If inf s s0 θ(s) > 0, then it follows from condition (H2) that 0 < (p + + inf s s 0 θ(s))f (s) f(s)s, s s 0, and thus we have the well-known Ambrosetti-Rabinowitz type condition as in [3]. In this paper, we are interested in the case inf s s0 θ(s) = 0. For this reason, we may assume throughout this work that s 0 and there is a constant N 0 > 0 such that θ(s) N 0 for all s R\( s 0, s 0 ). Note that the result here is different from one introduced in [5]. Our idea is to prove Theorem 2.2 by using the mountain pass theorem and its Z 2 symmetric version stated in the celebrated paper [2]. For this purpose, we introduce the following energy functional J : X R defined by J(u) = p(x) u p(x) dx K(x)F (u) dx, where F (s) = s f(t) dt. Then J is of 0 C (X, R) and its derivative is given by J (u)(v) = u p(x) 2 u v dx K(x)f(u)v dx for all u, v X. Hence, weak solutions of problem () are exactly the critical points of the functional J.

6 50 G.A. Afrouzi, N.T. Chung, M. Mirzapour Lemma 2.3. There exist positive constants ρ and α such that J(u) α for all u X with u = ρ. Proof. From (2) we have that u 0 if u 0. By the hypothesis (H3), for any ɛ > 0, there exists δ > 0 such that Hence, f(s) ɛ s p+, s < δ. F (s) ɛ p + s p+, s < δ. (3) Combining (H) with Proposition.5, we deduce for u X with u < min{, δ } (c is given by (2) that c J(u) = p(x) u p(x) dx K(x)F (u) dx p + u p+ K(x) F (u) dx p + u p+ ɛ K u p+ dx p + p + u p+ ɛ K µ() (4) u p+ p + p + u p+ ɛ K µ()c p+ u p+ p ( + ) = p ɛ K µ()c p+ u p+. + p + From (4), there exist positive constants ρ and α such that J(u) α for all u X with u = ρ. Let S = {w X : w = }. We note that, for all w S and a.e. x we have w(x) L for some L > 0. There is a number s λ {s R : s λl } such that θ(s λ ) = min s0 s λl θ(s). Then λ s λ and s L λ + when λ +. When s s 0 we have 0 < (p + + θ(s))f (s) f(s)s. Hence, ( ) s θ(t) F (s) C s p+ exp dt = C s p+ G( s ), (5) s 0 t ( ) s θ(t) where G( s ) = exp s 0 dt. Then by (H2), it follows that G( s ) increases t when s increases and lim s + G( s ) = +.

7 p(x)-biharmonic problems without (A-R) type conditions 5 Lemma 2.4. For any w S there exist δ w > 0 and λ w > 0 such that, for all v S B(w, δ w ) and for all λ λ w, we have J(λv) < 0, where B(w, δ w ) = {v X : v w < δ w }. Proof. Fix w S. By w =, we know that µ({x : w(x) 0}) > 0 and that there exists a λ w > s 0 such that µ({x : λ w w(x) s 0 }) > 0, where µ is the Lebesgue measure. Let w := { x : λ w w(x) < s 0 }, 2 w := { x : λ w w(x) s 0 }. Then µ( w) > 0. When x w we have w(x) s 0 λ w. Let δ w = s 0. Then, for any v S B(w, δ w ), v w L v w < s 0. Hence, when x w, we observe that v(x) s 0 and v(x) p+ ( ) p + s0 = C 2. (6) When λ, one has λv s 0 in 2 w. By the condition (H) and (5), (6) we know that λ p+ K(x)F (λv) dx C K(x) v p+ G( λv ) dx 2 w 2 w C C 2 K(x)G( λv ) dx 2 w C C 2 µ( 2 w)k 0 G ( ) s0 λ, (7) since G( s ) increases when s increases and λv(x) s 0 λ. There exists a C 3 > 0 such that F (s) C 3 when s s 0. However, F (s) > 0 if s s 0, so K(x)F (λv) dx K(x)F (λv) dx C 3 K. w w {x : λv(x) s 0 } Hence, by Proposition.5, for any v S B(w, δ w ) and λ >, we have J(λv) = p(x) (λv) p(x) dx K(x)F (λv) dx p λ p+ v p+ K(x)F (λv) dx K(x)F (λv) dx w 2 w ( ) = λ p+ p λ p+ K(x)F (λv) dx K(x)F (λv) dx 2 w w

8 52 G.A. Afrouzi, N.T. Chung, M. Mirzapour ( ( )) λ p+ p C C 2 µ( 2 s0 w)k 0 G λ + C 3 K. (8) From (8), J(λv) uniformly for v S B(w, δ w ) as λ +. Therefore, there exists a λ w > such that J(λv) < 0 for any v S B(w, δ w ) and λ λ w. Lemma 2.5. The functional J satisfies the (PS) condition. Proof. Let {u m } be a (PS) sequence of the functional J, that is, J(u m ) c and J (u m )(v) ɛ m v (9) for all v X with ɛ m 0 as m. We shall prove that {u m } is bounded in X. Indeed, if {u m } is not bounded, we may assume that u m + as m. Let {λ m } R such that u m = λ m w m, w m S. Then λ m + as m. Let us define the sets m = {x : λ m w m (x) L} and 2 m = {x : λ m w m (x) < L}. Then we have ɛ m λ m = ɛ m u m J (u m )(u m ) = u m p(x) dx K(x)f(u m )u m dx = (λ m w m ) p(x) dx K(x)f(λ m w m )λ m w m dx m K(x)f(λ m w m )λ m w m dx, (0) 2 m which implies that K(x)f(λw m )λ m w m dx (λ m w m ) p(x) dx + ɛ m λ m m K(x)f(λ m w m )λ m w m dx 2 m p + p(x) (λ mw m ) p(x) dx + ɛ m λ m K(x)f(λ m w m )λ m w m dx. () We know that 0 < (p + + θ(s λm ))F (λ m w m ) f(λ m w m )λ m w m in m. 2 m

9 p(x)-biharmonic problems without (A-R) type conditions 53 Combining this with () we then have J(u m ) = J(λ m w m ) = p(x) (λ mw m ) p(x) dx K(x)F (λ m w m ) dx K(x)F (λ m w m ) dx m 2 m p(x) (λ mw m ) p(x) dx K(x)f(λ p + m w m )λ m w m dx + θ(s λm ) m K(x)F (λ m w m ) dx 2 m p(x) (λ mw m ) p(x) p + dx p + + θ(s λm ) p(x) (λ mw m ) p(x) dx ɛ m λ m p + + θ(s λm ) + K(x)f(λ p + m w m )λ m w m dx + θ(s λm ) 2 m K(x)F (λ m w m ) dx 2 m = θ(s λ m ) p + + θ(s λm ) p(x) (λ mw m ) p(x) dx ɛ m λ m p + + θ(s λm ) + ψ(λ mw m ) θ(s λm ) (p + + θ(s λm ))p λ m p+ ɛ m λ m + p + + θ(s λm ) + ψ(λ mw m ) [ θ(s λm ) λ m (p + + N 0 )p λ m p+ ɛ ] m + ψ(λ + p + m w m ), (2) where ψ(λ m w m ) = 2 m ( ) p + + θ(s λm ) K(x)f(λ mw m )λ m w m K(x)F (λ m w m ) dx. By the condition (H2), the sequence {ψ(λ m w m )} is bounded from below. On the other hand, we know that λ m +, and so s λm + as m +. By (H2), lim λ m p+ s λm θ(s λm ) θ(s λm ) lim = +. m + m L Hence, J(u m ) +, and we obtain the contradiction. Now, {u m } is bounded in X. Since X is compactly embedded into C() there exist a function u X and a subsequence still denoted by {u m } of {u m } such that it converges strongly towards u in C(). From this and the continuity of f, we then have K(x)f(u m )(u m u) dx K max f(s) u m u 0 (3) s u + when m. Combining (9) and (3) imply that u m p(x) 2 u m ( u m u) dx 0 as m. (4)

10 54 G.A. Afrouzi, N.T. Chung, M. Mirzapour By Proposition.6, the sequence {u m } converges strongly to u in X and the functional J satisfies the (P S) condition. Proof of Theorem 2.2. By Lemmas 2.3, 2.4 and 2.5, the functional J satisfies the conditions of the classical mountain pass theorem due to Ambrosettiand Rabinowitz [2]. Thus, we obtain a nontrivial weak solution of problem (). If, further, f is odd, then J is even. By Lemma 2.3, the functional J satisfies Proposition.7(i) and the (P S) condition. For any finite-dimensional subspace X X, S X = {w X : w = } is compact. From Lemma 2.5 and the finite covering theorem, it is easy to verify that J satisfies condition (ii) of Proposition.7. Therefore, J has a sequence of critical points {u m }. That is, problem () has infinitely many pairs of solutions. Acknowledgements. This research is supported by Quang Binh University Foundation for Science and Technology Development (Grant. N.CS ). References [] Z. El Allali, S. Taarabti, K.B. Haddouch, Eigenvalues of the p(x)- biharmonic operator with indefinite weight under Neumann boundary conditions, Bol. Soc. Paran. Mat., 36 (208), no., [2] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 4 (973), [3] A. El Amrouss, F. Moradi, M. Moussaoui, Existence of solutions for fourth-order PDEs with variable exponentsns, Electron. J. Differ. Equa., 53 (2009), -3. [4] A. Ayoujil, A.R. El Amrouss, On the spectrum of a fourth order elliptic equation with variable exponent, Nonlinear Analysis: Theory, Methods & Applications, 7 (2009), [5] G.M. Bisci, D. Repov s, Multiple solutions of p-biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equa., 59 (204), no. 2, [6] P. Candito, L. Li, R. Livrea, Infinitely many solutions for a perturbed nonlinear Navier boundary value problem involving the p-biharmonic, Nonlinear Analysis: Theory, Methods & Applications, 75 (202),

11 p(x)-biharmonic problems without (A-R) type conditions 55 [7] Z. Chen, W. Zou, A note on the Ambrosetti-Rabinowitz condition for an elliptic system, Appl. Math. Letters, 25 (202), [8] D.E. Edmunds, J. Rakosnik, Density of smooth functions in W k,p(x) (), Proc. R. Soc. A, 437 (992), [9] X.L. Fan, J.S. Shen, D. Zhao, Sobolev embedding theorems for spaces W k,p(x) (), J. Math. Anal. Appl., 262 (200), [0] X.L. Fan, D. Zhao, On the spaces L p(x) and W m,p(x), J. Math. Anal. Appl., 263 (200), [] B. Ge, Q.M. Zhou, Y.H. Wu, Eigenvalues of the p(x)-biharmonic operator with indefinite weight, Z. Angew. Math. Phys., 66 (205), no. 3, [2] S. Heidarkhani, M. Ferrara, A. Salari, G. Caristi, Multiplicity results for p(x)-biharmonic equations with Navier boundary conditions, Complex Var. Elliptic Equa., 6 (206), no., [3] L. Kong, Eigenvalues for a fourth order elliptic problem, Proc. Amer. Math. Soc., 43 (205), [4] C. Li, C.L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Analysis: Theory, Methods & Applications, 72 (200), [5] L. Li, C.L. Tang, Existence and multiplicity of solutions for a class of p(x)-biharmonic equations, Acta Math. Sci., 23 (203), no. 33, [6] H. Yin, Y. Liu, Existence of three solutions for a Navier boundary value problem involving the p(x)-biharmonic, Bull. Korean Math. Soc., 50 (203), no. 6, [7] A. Zang, Y. Fu, Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces, Nonlinear Analysis: Theory, Methods & Applications, 69 (2008), Received: September 2, 208; Published: October 22, 208