Electronic Journal of Differential Equations, Vol. 204 204), No. 34, pp.. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu STEKLOV PROBLEMS INVOLVING THE px)-laplacian GHASEM A. AFROUZI, ARMIN HADJIAN, SHAPOUR HEIDARKHANI Abstract. Under suitable assumptions on the potential of the nonlinearity, we study the existence and multiplicity of solutions for a Steklov problem involving the px)-laplacian. Our approach is based on variational methods.. Introduction The aim of this article is to study the following Steklov problem involving the px)-laplacian, px) u = ax) u px)2 u in, px)2 u u ν = λfx, u) on,.) where R N is a bounded smooth domain, λ is a positive parameter, p C ), px) u := div u px)2 u) denotes the px)-laplace operator, f : R R is a Carathéodory function, a L ) with ess inf a > 0 and ν is the outer unit normal to. The study of differential equations and variational problems with nonstandard px)-growth conditions is a new and interesting topic. It varies from nonlinear elasticity theory, electro-rheological fluids, and so on see [24, 25]). Many results have been obtained on this kind of problems, for instance we here cite [, 5, 6, 7, 9, 0,, 3, 4, 6, 8, 9]. The inhomogeneous Steklov problems involving the p-laplacian has been the object of study in, for example, [22], in which the authors have studied this class of inhomogeneous Steklov problems in the cases of px) p = 2 and of px) p >, respectively. In this paper, motivated by [], at first, we prove the existence of a non-zero solution of the problem.), without assuming any asymptotic condition neither at zero nor at infinity see Theorem 3.). Next, we obtain the existence of two solutions, possibly both non-zero, assuming only the classical Ambrosetti-Rabinowitz condition; that is, without requiring that the potential F satisfies the usual condition at zero see Theorems 3.2 and 3.3). Finally, we present a three solutions existence result under appropriate condition on the potential F see Theorem 3.4). 2000 Mathematics Subject Classification. 35J60, 35J20. Key words and phrases. px)-laplace operator; variable exponent Sobolev spaces; multiple solutions; variational methods. c 204 Texas State University - San Marcos. Submitted December 26, 203. Published June 0, 204.
2 G. A. AFROUZI, A. HADJIAN, S. HEIDARKHANI EJDE-204/34 Our approach is fully variational method and the main tools are critical point theorems contained in [3] and [8] see Theorems 2. and 2.2 in the next section). A special case of Theorem 3.4 is the following theorem. Theorem.. Let px) = p > N for every x and let f : R R be a nonnegative continuous function. Put F t) := t fξ)dξ for each t R. Assume that 0 F d) > 0 for some d and, moreover, lim inf ξ 0 F ξ) ξ p F ξ) = lim sup ξ + ξ p = 0. Then, there is λ > 0 such that for each λ > λ the problem p u = ax) u p2 u in, p2 u u ν = λfu) admits at least three non-negative weak solutions. 2. Preliminaries on, In this section, we recall definitions and theorems to be used in this paper. Let X, ) be a real Banach space and Φ, Ψ : X R be two continuously Gâteaux differentiable functionals; put I := Φ Ψ and fix r, r 2 [, + ], with r < r 2. We say that the functional I satisfies the Palais-Smale condition cut off lower at r and upper at r 2 [r] PS) [r2] -condition) if any sequence {u n } X such that {Iu n )} is bounded, lim n + I u n ) X = 0, r < Φu n ) < r 2 n N, has a convergent subsequence. If r = and r 2 = +, it coincides with the classical PS)-condition, while if r = and r 2 R it is denoted by PS) [r2] -condition. First we recall a result of local minimum obtained in [3], which is based on [2, Theorem 5.]. Theorem 2. [3, Theorem 2.3]). Let X be a real Banach space and let Φ, Ψ : X R be two continuously Gâteaux differentiable functionals such that inf X Φ = Φ0) = Ψ0) = 0. Assume that there exist r R and ū X, with 0 < Φū) < r, such that sup u Φ ],r[) Ψu) < Ψū) 2.) and, for each λ Λ := ] Φū) Ψū), r r sup u Φ ],r[) Ψu) Φū) [ the functional I λ := Φ λψ satisfies the PS) [r] -condition. Then, for each λ Λ, there is u λ Φ ]0, r[) hence, u λ 0) such that I λ u λ ) I λ u) for all u Φ ]0, r[) and I λ u λ) = 0. Now we point out an other result, which insures the existence of at least three critical points, that has been obtained in [8] and it is a more precise version of [4, Theorem 3.2].
EJDE-204/34 STEKLOV PROBLEMS 3 Theorem 2.2 [8, Theorem 3.6]). Let X be a reflexive real Banach space, Φ : X R be a continuously Gâteaux differentiable, coercive and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X, Ψ : X R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact, moreover Φ0) = Ψ0) = 0. Assume that there exist r R and ū X, with 0 < r < Φū), such that i) sup u Φ ],r]) Ψu) r < Ψū) Φū) ii) for each λ Λ := ] Φū) Ψū), r sup u Φ ],r]) Ψu) [ the functional I λ = Φ λψ is coercive. Then, for each λ Λ, the functional I λ has at least three distinct critical points in X. Here and in the sequel, we suppose that p C ) satisfies the following condition: N < p := inf px) px) x p+ := sup px) < +. 2.2) Define the variable exponent Lebesgue space L px) ) by { } L px) ) := u : R : u is measurable and ux) px) dx < +. We define a norm, the so-called Luxemburg norm, on this space by the formula u L px) ) = u px) := inf { λ > 0 : ux) px) dx }. λ x Define the variable exponent Sobolev space W,px) ) by W,px) ) := { u L px) ) : u L px) ) } equipped with the norm u W,px) ) := u px) + u px). It is well known [7] that, in view of 2.2), both L px) ) and W,px) ), with the respective norms, are separable, reflexive and uniformly convex Banach spaces. When a L ) with ess inf a > 0, for any u W,px) ), define u a := inf { λ > 0 : ux) px) + ax) ux) λ λ px)) dx Then, it is easy to see that u a is a norm on W,px) ) equivalent to u W,px) ). In the following, we will use u a instead of u W,px) ) on X = W,px) ). As pointed out in [20] and [7], X is continuously embedded in W,p ) and, since p > N, W,p ) is compactly embedded in C 0 ). Thus, X is compactly embedded in C 0 ). So, in particular, there exists a positive constant m > 0 such that u C 0 ) m u a 2.3) for each u X. When is convex, an explicit upper bound for the constant m is m 2 p { ) d p p max p, a p p N ) p a } p ) + a N }.
4 G. A. AFROUZI, A. HADJIAN, S. HEIDARKHANI EJDE-204/34 where d := diam) and is the Lebesgue measure of for details, see [0]), a := ax)dx and a := sup x ax). Lemma 2.3 [7]). Let Iu) = u px) + ax) u px) )dx. For u X we have i) u a < = ; > ) Iu) < = ; > ); ii) If u a < u p+ a Iu) u p a ; iii) If u a > u p a Iu) u p+ a. We refer the reader to [5, 7] for the basic properties of the variable exponent Lebesgue and Sobolev spaces. Throughout this article, we assume the following condition on the Carathéodory function f : R R: F0) fx, s) αx) + b s βx) for all x, s) R, where α L βx) βx) ), b 0 is a constant and β C) such that < β := inf βx) βx) β + := sup βx) < p. 2.4) x x We recall that f : R R is a Carathéodory function if x fx, ξ) is measurable for all ξ R and ξ fx, ξ) is continuous for a.e. x. Put for all x, t) R. F x, t) := t 0 fx, ξ)dξ, Theorem 2.4 [, Theorem 2.9]). Let f : R R be a Carathéodory function satisfying F0). For each u X set Ψu) = F x, ux))dσ. Then Ψ C X, R) and Ψ u)v) = fx, ux))vx)dσ for every v X. Moreover, the operator Ψ : X X is compact. We say that a function u X is a weak solution of problem.) if u px)2 u v dx + ax) u px)2 uv dx = λ fx, u)v dσ for all v X. We cite the very recent monograph by Kristály et al. [2] as a general reference for the basic notions used in the paper. 3. Main results In this section we present our main results. First, we establish the existence of one non-trivial solution for the problem.). Theorem 3.. Let f : R R be a Carathéodory function satisfying F0). Assume that there exist d and c m with d p+ a < p p c + m, such that )p max t c F x, t)dσ c ) p < p F x, d)dσ. 3.) p + d p+ a m Then, for each ] d p+ a c p λ Λ := p, m) F x, d)dσ p [ + max, 3.2) t c F x, t)dσ
EJDE-204/34 STEKLOV PROBLEMS 5 problem.) admits at least one non-trivial weak solution ū X such that max ū x) < c. x Proof. Our aim is to apply Theorem 2. to.). To this end, for each u X, let the functionals Φ, Ψ : X R be defined by Φu) := u px) + ax) u px)) dx, px) Ψu) := F x, ux))dσ, and put I λ u) := Φu) λψu), u X. Note that the weak solutions of.) are exactly the critical points of I λ. The functionals Φ and Ψ satisfy the regularity assumptions of Theorem 2.. Indeed, by standard arguments, we have that Φ is Gâteaux differentiable and its Gâteaux derivative at the point u X is the functional Φ u) X, given by ) Φ u)v) = u px)2 u v + ax) u px)2 uv dx for every v X. Moreover, Φ is sequentially weakly lower semicontinuous and its inverse derivative is continuous since it is a continuous convex functional) and, thanks to Lemma 2.3, the functional Φ turns out to be coercive. On the other hand, by Theorem 2.4, the functional Ψ is well defined, continuously Gâteaux differentiable and with compact derivative, whose Gâteaux derivative at the point u X is given by Ψ u)v) = fx, ux))vx)dσ for every v X. So, owing to [2, Proposition 2.], the functional I λ satisfies the PS) [r] -condition for all r R. We will verify condition 2.) of Theorem 2.. Let w be the function defined by wx) := d for all x and put r := c ) p p +. m Clearly, w X and from our assumption one has 0 < Φw) = px) ax)dpx) dx p a d p + < r. For all u X with Φu) < r, owing to Lemma 2.3, definitively one has min { } u p+ a, u p a < rp +. Then and so, by 2.3), Therefore, u a < max { p + r) p +, p + r) p } = c m, max ux) m u a < c. x sup u Φ ],r[) Ψu) max t c F x, t)dσ r ) p p + c m
6 G. A. AFROUZI, A. HADJIAN, S. HEIDARKHANI EJDE-204/34 On the other hand, taking into account that Φw) + p dp a, we have Ψw) Φw) F x, d)dσ. p d p+ a Therefore, by the assumption 3.), condition 2.) of Theorem 2. is verified. Therefore, all the assumptions of Theorem 2. are satisfied. So, for each ] Φw) λ Λ Ψw), r [, sup u Φ ],r[) Ψu) the functional I λ has at least one non-zero critical point ū X such that max x ū x) < c that is the weak solution of the problem.). The following result, in which the global Ambrosetti-Rabinowitz condition is also used, ensures the existence at least two weak solutions. Theorem 3.2. Assume that all the assumptions of Theorem 3. hold. Furthermore, suppose that f, 0) 0 in, and AR) there exist two constants µ > p + and R > 0 such that for all x and s R, 0 < µf x, s) sfx, s). Then, for each λ Λ, where Λ is given by 3.2), the problem.) has at least two non-trivial weak solutions ū, ū 2 X such that max ū x) < c. x Proof. Fix λ as in the conclusion. So, Theorem 3. ensures that the problem.) admits at least one non-trivial weak solution ū which is a local minimum of the functional I λ. Now, we prove the existence of the second local minimum distinct from the first one. To this end, we must show that the functional I λ satisfies the hypotheses of the mountain pass theorem. Clearly, the functional I λ is of class C and I λ 0) = 0. We can assume that ū is a strict local minimum for I λ in X. Therefore, there is ρ > 0 such that inf uū =ρ I λ u) > I λ ū ), so condition [23, I ), Theorem 2.2] is verified. From AR), by standard computations, there is a positive constant C such that F x, s) C s µ 3.3) for all x and s > R. In fact, setting γx) = min ξ =R F x, ξ) and by AR), for every x and s > R one has Therefore, ϕ s t) = F x, ts) t > 0, 3.4) 0 < µϕ s t) = µf x, ts) tsfx, ts) = tϕ st) t > 0. R/ s ϕ st) ϕ s t) dt µ R/ s t dt.
EJDE-204/34 STEKLOV PROBLEMS 7 Then ϕ s ) ϕ s R s ) s µ. Taking into account 3.4), we obtain F x, s) F x, R ) s s s µ γx) s µ C s µ, and 3.3) is proved. Now, by choosing any u X \ {0} and t >, one has I λ tu) = Φ λψ)tu) t px) = px) t p+ px) u px) + ax) u px)) dx λ u px) + ax) u px)) dx Ct µ λ F x, tux))dσ ux) µ dσ. Since µ > p +, the functional I λ is unbounded from below. So, condition [23, I 2 ), Theorem 2.2] is verified. Therefore, I λ satisfies the geometry of mountain pass. Now, to verify the PS)-condition it is sufficient to prove that any PS)-sequence is bounded. To this end, suppose that {u n } X is a PS)-sequence; i.e., there is M > 0 such that sup I λ u n ) M, I λu n ) 0 as n +. Let us show that {u n } is bounded in X. Using hypothesis AR), since I λ u n ) is bounded, we have for n large enough: M + I λ u n ) µ I λu n ), u n + µ I λu n ), u n = u n px) + ax) u n px)) dx λ px) [ u n px) + ax) u n px)) dx λ µ + µ I λu n ), u n p + µ p + µ ) un p a µ I λu n ) X u n a c ) un p a c 2 µ u n a c, F x, u n x))dσ ] fx, u n x))u n x)dσ where c and c 2 are two positive constants. Since µ > p +, from the above inequality we know that {u n } is bounded in X. Hence, the classical theorem of Ambrosetti and Rabinowitz ensures a critical point ū 2 of I λ such that I λ ū 2 ) > I λ ū ). So, ū and ū 2 are two distinct weak solutions of.) and the proof is complete. Here we give the following result as a direct consequence of Theorem 3.2 in the autonomous case. Theorem 3.3. Let f : R R be a continuous function satisfying f0) 0 and fs) α + b s β for all s R, where α > 0, b 0 and < β < p are three constants. Put F t) := t fξ)dξ for all t R. Under the following conditions 0
8 G. A. AFROUZI, A. HADJIAN, S. HEIDARKHANI EJDE-204/34 i) there exist d and c m with d p+ a < p p + c m) p, such that max t c F t) c ) p < p F d) ; p + d p+ a m ii) there exist two constants µ > p + and R > 0 such that for all s R, and for each the problem λ 0 < µf s) sfs), ] d p + a p F d), c m) p p + max t c F t) px) u = ax) u px)2 u in, px)2 u u ν = λfu) on, admits at least two non-trivial weak solutions ū, ū 2 X such that max ū x) < c. x Now, we point out the following result of three weak solutions. Theorem 3.4. Let f : R R be a Carathéodory function satisfying F0). Assume that there exist d and c m with d p a > c m) p, such that the assumption 3.) in Theorem 3. holds. Then, for each λ Λ, where Λ is given by 3.2), the problem.) has at least three weak solutions. Proof. Our goal is to apply Theorem 2.2. The functionals Φ and Ψ defined in the proof of Theorem 3. satisfy all regularity assumptions requested in Theorem 2.2. So, our aim is to verify i) and ii). Arguing as in the proof of Theorem 3., put r := c p p m) and wx) := d for all x, bearing in mind that d p a + > c m )p, we have Φw) = px) ax)dpx) dx p + dp a > r > 0. Therefore, the assumption i) of Theorem 2.2 is satisfied. We prove that the functional I λ is coercive for all λ > 0. If u X, then by condition 2.4) and the embedding theorem see [2, Theorem 2.]) we have u L βx) ). Then there is some constant C > 0 such that u L βx) ) C u a, u X. Now, by using Hölder inequality see [7]) and condition F0), for all u X such that u a, we have ux) ) Ψu) = F x, ux))dσ = fx, t)dt dσ 0 αx) ux) + b βx) ux) βx)) dσ 2 α βx) u L βx) ) + b β ux) βx) dσ L βx) ) [,
EJDE-204/34 STEKLOV PROBLEMS 9 2C α βx) L βx) ) u a + b β ux) βx) dσ. On the other hand, there is a constant C > 0 such that } ux) βx) dσ max { u β+ L βx) ), u β C u β+ L βx) ) Then, Since Φu) = Ψu) 2C α for every λ > 0 we have βx) L βx) ) u a + b β C u β + a. u px) + ax) u px)) dx px) p + u p a, I λ u) p + u p a 2λC α βx) L βx) ) u a λbc β + u β a. Since p > β +, the functional I λ is coercive. Then also condition ii) holds. So, for each λ Λ, the functional I λ admits at least three distinct critical points that are weak solutions of problem.). Remark 3.5. If we assume that f : R R is a non-negative Carathéodory function satisfying F0), then the previous theorems guarantee the existence of non-negative weak solutions. In fact, let ū be a weak solution of the problem.). We claim that it is non-negative. Arguing by contradiction and setting A := {x : ūx) < 0}, one has A. Put v := min{ū, 0}, one has v X. So, taking into account that ū is a weak solution and by choosing v = v, one has ū px) dx + ax) ū px) dx = λ fx, ūx))ūx)dσ 0, A A that is, ū W,px) A) = 0 which is absurd. Hence, our claim is proved. Also, when f is a non-negative function, condition 3.) becomes F x, c)dσ ) p < p F x, d)dσ. p + d p+ a c m In this case, the previous theorems ensure the existence of non-negative solutions to the problem.) for each ] d p+ a c p λ p, m) F x, d)dσ p [ + F x, c)dσ. Remark 3.6. Theorems 3. and 3.4 ensure more precise conclusions rather than [, Theorems. and.3]. In fact, Theorem. of [] proves that for any λ ]0, + [, the problem.), when a, has at least a non-trivial weak solution. Also, Theorem 3. of [] establishes that there exists an open interval Λ ]0, + [ such that, for every λ Λ, the problem.), when a, admits at least three solutions. Hence, a location of the interval Λ in ]0, + [ is not established. Proof of Theorem.. Fix λ > λ := dp a p F d) Since lim inf ξ 0 F ξ) ξ p = 0, a. for some d such that F d) > 0.
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EJDE-204/34 STEKLOV PROBLEMS [20] O. Ková cik, J. Rákosník; On spaces L px) and W k,px), Czechoslovak Math. J., 4 99), 592 68. [2] A. Kristály, V. Rădulescu, Cs. Varga; Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 36, Cambridge University Press, Cambridge, 200. [22] N. Mavinga, M. N. Nkashama; Steklov spectrum and nonresonance for elliptic equations with nonlinear boundary conditions, Electron. J. Differential Equations, Conf., Vol. 200 200), No. 9, pp. 97 205. [23] P. H. Rabinowitz; Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. Math. Soc., Providence, RI, 986. [24] M. Ru zi cka; Electro-rheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., Vol. 784, Springer-Verlag, Berlin, 2000. [25] V. V. Zhikov; Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR. Izv., 29 987), 33-66. Ghasem A. Afrouzi Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran E-mail address: afrouzi@umz.ac.ir Armin Hadjian Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, P.O. Box 339, Bojnord 9453, Iran E-mail address: hadjian83@gmail.com Shapour Heidarkhani Department of Mathematics, Faculty of Sciences, Razi University, 6749 Kermanshah, Iran E-mail address: s.heidarkhani@razi.ac.ir