On the discrete boundary value problem for anisotropic equation

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1 On the discrete boundary value problem for anisotropic equation Marek Galewski, Szymon G l ab August 4, 0 Abstract In this paper we consider the discrete anisotropic boundary value problem using critical point theory. hirstily we apply the direct method of the calculus of variations and the mountain pass technique in order to reach the existence of at least one non-trivial solution. Secondly we derive some version of a discrete three critical point theorem which we apply in order to get the existence of at least two non-trivial solutions. Introduction In this note we consider an anisotropic difference equation { ( u(k ) p(k ) u(k ) ) = λf(k, u(k)), u(0) = u( + ) = 0, () where λ > 0 is a numerical parameter, f : [0, +] R + R, [a, b] for a < b, a, b Z denotes a discrete interval {a, a+,..., b}, u(k ) = u (k) u(k ) is the forward difference operator; p : [0, + ] R +, p = min k [0, +] p (k) > and p + = max k [0, +] p (k) > ; q : [0, + ] R + stands for the conjugate exponent. We aim at showing that problem () has at least one, at least two nontrivial solutions. Our approach relies on the application of the direct method of the calculus of variations, the mountain pass technique and a modified version of a finite dimensional three critical point theorem which we derive for our special case. Let us mention, far from being exhaustive, the following recent papers on discrete BVPs investigated via critical point theory, [], [3], [8], [], [3], [4], [5], [6]. hese papers employ in the discrete setting the variational techniques already known for continuous problems of course with necessary modifications. he tools employed cover the Morse theory, mountain pass methodology, linking arguments. Continuous version of problems like () are known to be mathematical models of various phenomena arising in the study of elastic mechanics (see [7]), electrorheological fluids (see []) or image restoration (see [4]). Variational continuous anisotropic problems have been started by Fan and Zhang

2 in [5] and later considered by many methods and authors- see [6] for an extensive survey of such boundary value problems. he research concerning the discrete anisotropic problems of type () have only been started, see [7], [0], where known tools from the critical point theory are applied in order to get the existence of solutions. Variational framework for () By a solution to () we mean such a function x : [0, + ] R which satisfies the given equation and the associated boundary conditions. Solutions will be investigated in a space considered with a norm H = {u : [0, + ] R : u(0) = u( + ) = 0} u = ( + ) / u(k ). hen (H, ) becomes a Hilbert space. he functional corresponding to () is J λ (u) = + p(k ) u(k ) p(k ) λ F (k, u(k)), where F (k, u(k)) = u(k) 0 f(k, t)dt. With any fixed λ > 0 functional J λ is differentiable in the sense of Gâteaux and its Gâteaux derivative reads + J λ(u), v = u(k ) p(k ) u(k ) v(k ) λ A critical point to J λ, i.e. such a point u E that J λ(u), v = 0 for all v E f(k, u(k))v (k). is a weak solution to (). Summing by parts we see that any weak solution to () is in fact a strong one. Hence in order to solve () we need to find critical points to J λ and investigate their multiplicity. he following auxiliary result was proved in [0]. Lemma (a) here exist two positive constants C and C such that + u(k ) p(k ) C u p C for every u H with u >.

3 (b) For any m there exists a positive constant c m such that for every u H. + u(k) m c m u(k ) m We note that + m u(k) m u(k ) m () for any m. Since any two norms on a finite-dimensional Banach space are equivalent, then there is a positive constant K m (depending on m) with It can be verified that (K m ) m u m u(k) m. ( + ) /m ( + ) m m u u(k ) m ( + ) m u (3) for any u H and any m. Let us recall some preliminaries from critical point theory. Let E be a reflexive Banach space. Let J C (E, R). For any sequence {u n } E, if {J(u n )} is bounded and J (u n ) 0 as n possesses a convergent subsequence, then we say J satisfies the Palais Smale condition - (PS) condition for short. heorem (Mountain pass lemma)[9] Let J satisfy the (PS) condition. Suppose that. J(0) = 0;. there exist ρ > 0 and α > 0 such that J(u) α for all u E with u = ρ; 3. there exists u in E with u ρ such that J(u ) < α. hen J has a critical value c α. Moreover, c can be characterized as inf max J(u), g Γu g([,0]) where Γ = {g C([, 0], E) : g(0) = 0, g() = u }. heorem 3 [9]If functional J : E R, J is weakly lower semi-continuous and coercive, i.e. lim x J(x) = +, then there exists x 0 such that inf J(x) = J(x 0) x E and x 0 is also a critical point of J, i.e. J (x 0 ) = 0. Moreover, if J is strictly convex, then a critical point is unique. 3

4 3 Existence of solution by a direct method and a mountain pass lemma In this section we are concerned with the applications of heorems 3 and in order to get the existence results. We introduce firstly some assumptions. H a : [, ] R +, b : [, ] R are such that f(k, t) a(k) t q(k) + b(k) for all t R and all k [, ]. H a : [, ] R +, b : [, ] R are such that f(k, t) a (k) t q(k) + b (k) for all t R and all k [, ]. H3 c : [, ] R + is such that f(k, t) c(k) t q(k) for all t R and all k [, ]. H4 c : [, ] R + is such that f(k, t) c (k) t q(k) for all t R and all k [, ]. heorem 4 Assume that condition H holds. hen (4.) functional J λ is coercive provided p > q + + ; (4.) if p = q + +, then there is λ such that for any λ (0, λ ) functional J λ is coercive. Proof. Note that F (k, u(k)) u(k) f(k, t)dt 0 u(k) 0 f(k, t) dt u(k) (a(k) t q(k) + b(k))dt = a(k) u(k) q(k)+ 0 q(k)+ + b(k)u(k) a + q b + u(k) q+ u(k) a + c q + + q + + u(k) q+ + + b + c u(k), where constants c q + + and c in the last inequality follow from lemma (b). Using the above estimation we obtain J λ (u) + λb + c u(k) p(k ) u(k ) p(k ) λ a+ c q + + q + u(k) q+ + 4

5 Using Lemma (a) we have Hence + u(k ) p(k ) C u p C. J λ (u) C p u p C + λ a+ c q q + + u(k) q+ λb + c u(k). hus J λ (u) as u in case p > q + +. Let us now assume that p = q + +. hen J λ (u) as u in case C p + λa+ c q+ + q + > 0. hus there is λ such that for any λ (0, λ ) the functional J(u) is coercive. In fact we may take λ = C(q +) p + a + c q + +. Now, we immediately obtain the following existence result. heorem 5 Assume that condition H holds and let f (k, 0) 0 for at least one k [, ]. hen problem () has at least one nontrivial solution for all λ > 0 provided p > q + +. When p = q + +, there is λ such that for any λ (0, λ ) problem () has at least one nontrivial solution. Proof. Indeed, functional J λ is continuous differentiable in the sense of Gâteaux. he assertion follows then by heorem 4 and heorem 3. heorem 6 Assume that condition H holds. hen (6.) functional J λ is anti-coercive provided p + < q + ; (6.) if p + = q +, then there is λ such that for any λ > λ the functional J λ is anti-coercive. Proof. As in the proof of heorem 4 we see by () that F (k, u(k)) a (q + +) q + u(k + ) q + b u(k ). Using the above we have for some constant c J λ (u) + a p + u p p λ + (q + + ) q + u q + q + + c u. 5

6 Hence by (3) we see that J λ (u) + + p u p his inequality provides the assertion. he existence result immediately follows. λa ( + ) q + u q + + c u. (4) (q + + ) q + heorem 7 Assume that condition H holds and let f (k, 0) 0 for at least one k [, ]. hen problem () has at least one nontrivial solution for all λ > 0 provided p + < q +. When p + = q +, there is λ such that for any λ > λ problem () has at least one nontrivial solution. When f (k, 0) = 0 for all k [, ] heorem 3 yields the existence of at least one solution which in this case may become trivial. So that we will use some different methodology pertaining to heorem. his however requires some additional assumptions. Corollary 8 (i) Assume that conditions H, H3 hold and let p > q + +. hen for any M < 0 there is positive λ such that for any λ > λ and for all u = we have J λ (u) M. (ii) Assume that conditions H, H4 hold and let q + > p +. hen there is λ, t > 0 and M > 0 such that for any λ (0, λ ) and for all u = t we have J λ (u) M. Proof. We will prove only the first assertion. Let us fix any M > 0. Note that by the proof of heorem 6, see relation (4), we obtain that for all u = when λ λ = Corollary 9 q + J λ (u) + p λc ( + ) (q + + ) q + ( ) M+ + p (q + +) q + c ( +) q +. M (i) Assume that conditions H,H3 hold and let p > q + + and f (k, 0) = 0 for k [, ]. hen problem () has at least one nontrivial solution for all λ > λ = ( ) M+ + p (q + +) q + c ( +) q + (ii) Assume that conditions H, H4 hold and let q + > p +. hen there is λ such that () has at least one nontrivial solution for all λ (0, λ ). Proof. We will show that all assumptions of heorem are satisfied in case of assertion (i). We put E λ = J λ. Now we see that E λ is anti-coercive. Next we note that E λ (0) = 0. hus assumptions. and 3. of heorem are satisfied. Assumption. follows by Lemma 8. Hence the assertion of the theorem follows.. 6

7 4 General multiplicity result In this section we are concerned with the existence of multiple solutions. We will use the version of the three critical point theorem. In order to do so we shall somehow generalize the main result, namely heorem 3 from [] which is proved for some special functional containing term p. We note however, that this term can be replaced by any nonnegative, coercive function being 0 at 0. Our proof in fact follows the ideas employed in []. Let us start from the following abstract result which is used in the proof heorem 3 from [] and which we also use. heorem 0 Let (X, τ) be a Hausdorff space and Φ, J : X R be functionals; moreover, let M be the (possibly empty) set of all the global minimizers of J and define α = inf Φ(x), x X { infx M Φ(x), if M, β = sup x X Φ(x), if M =. Assume that the following conditions are satisfied: (0.) for every σ > 0 and every ρ R the set {x X : Φ(x) + σj(x) ρ} is sequentially compact (if not empty); (0.) α < β. hen at least one of the following conditions holds: (0.3) there is a continuous mapping h : (α, β) X with the following property: for every t (α, β), one has Φ(h(t)) = t and for every x Φ (t) with x h(t), J(x) > J(h(t)); (0.4) there is σ > 0 such that the functional Φ + σ J admits at least two global minimizers in X. heorem Let H be a finite dimensional Banach space. Assume that J, µ C (H), µ (x) 0 for all x H, µ(0) = 0 and µ is coercive. Let 0 < r < s be constants. Assume that (.) lim sup µ(u) J(u) µ(u) 0; (.) inf u H J(u) < inf µ(u) s J(u); (.3) J(0) inf r µ(u) s J(u). 7

8 hen there is λ > 0 such that the functional u µ(u)+λ J(u) has at least three critical points in H. Proof. Define a continuous functional Φ : H R in the following way µ(u), if µ(u) < r, Φ(u) = r, if r µ(u) s, µ(u) s + r, if µ(u) > s. Let M, α and β be defined as in heorem 0. At first we show that β > r. We consider two cases: Case. Assume that M. he set M of minimizers of continuous functional J is closed, and therefore there is u M with Φ(u) = β. Hence by (.) we obtain that µ(u) > s, and thus β = Φ(u) = µ(u) s + r > r. Case. Assume that M =. hen β = > r. Now we will show that the assumptions of heorem 0 are fulfilled. Note that for any σ > 0 we have lim inf µ(u) (Φ(u) + σj(u)) = lim inf µ(u) (µ(u) s + r + σj(u)) = ( ) lim inf µ(u) µ(u) s r µ(u) + σ J(u) µ(u) ) lim inf µ(u) µ(u) lim inf µ(u) ( s r µ(u) + σ J(u) µ(u) lim inf µ(u) µ(u) =. Hence by the above and by the continuity of Φ and J we obtain that the set {u X : Φ(u) + σj(u) ρ} is closed and bounded, and consequently compact. By the definition of Φ we have α = inf u H Φ(u) = 0. hus since β > r > 0 we obtain that β > α. By heorem 0 one of the two conditions (0.3) and (0.4) holds. We will show that (0.3) cannot hold. Suppose to the contrary that (0.3) holds true, i.e. there is a mapping h : (0, β) H with Φ(h(t)) = t for every t (0, β). Since r (0, β), then herefore t < r Φ(h(t)) < r µ(h(t)) < r, t = r Φ(h(t)) = r r µ(h(t)) s, t > r Φ(h(t)) > r µ(h(t)) > s. lim sup µ(h(t)) r = µ(h(r)), t r lim inf µ(h(t)) s > µ(h(r)). t r + But this contradicts the continuity of h. Hence the condition (0.4) holds, that is there is σ > 0 such that the functional Φ + σ J has at least two minimizers, say u and u (u u ). 8

9 We will show that µ(u i ) < r or µ(u i ) > s for i =,. contrary, that r µ(u i ) s for some i =,. hen Suppose to the Φ(u i ) + σ J(u i ) = r + σ J(u i ) > σ J(u i ) = Φ(0) + σ J(0), which contradicts with the fact that u i is a minimizer of Φ + σ J. Put E(u) = µ(u) + σ J(u). Note that u and u are local minimizers of E. Moreover E C (H, R). We need only to show that E has at least one critical point u 3 H \ {u, u }. If both u and u are strict local minimizers of E, then using Mountain Pass technique (see [, heorem ]) we will find a critical point u 3 / {u, u }. If one of u and u is not strict local minimizer, then E admits infinitely many local minimizers at the same level. 5 Applications of multiplicity result for anisotropic problems In order to apply heorem for () we introduce the following notation. Let µ(u) = + p(k ) u(k ) p(k ). Clearly µ C (H, R), µ(0) = 0, µ 0 and µ is coercive. For numbers r, s > 0 we put r = inf{ u : µ(u) r} and s = sup{ u : µ(u) s}. Hence µ(u) r = u r, µ(u) s = u s. Let F (k, t) = t 0 f(k, τ)dτ, J(u) = F (k, u(k)), and E λ(u) = µ(u) + λj(u). heorem Let 0 < r < s. Assume that (.) lim sup t F (k,t) t p 0 for any k [, ]; (.) sup t s F (k, t) < sup t R F (k, t); (.3) sup r t s F (k, t) h k sup t s F (h, t) for any k [, ]. hen there exists λ such that the functional E λ points, two of which must be nontrivial. has at least three critical Proof. By Lemma (a), µ(u) c u p c for some positive constants c, c and u >. Since there is a positive c 3 with u p c 3 u(k) p, 9

10 then µ(u) c c 3 u(k) p c c c 3 provided µ(u) is sufficiently large. Let ε > 0. Using (.) we will find K > 0 with F (k, t) t p < c c 3 ε u(k) p for any k [, ] and t > K. Let M = max{f (k, t) : k [, ], t K}. hen for µ(u) > M/ε with u we obtain J(u) µ(u) = F (k, u(k)) F (k, u(k)) µ(u) µ(u) u(k) K F (k, u(k)) u(k) >K F (k, u(k)) µ(u) u(k) p u(k) K M M/ε u(k) K ε c c 3 u(k) >K u(k) >K F (k, u(k)) c c 3 u(k) p ε = ε. Hence we get condition (.). o show condition (.) we consider two cases. Case. Suppose that inf u H J(u) >. We will show that for any σ > 0 the following equality holds Note that for any u σ we have inf J(u) = sup F (k, t). (5) u σ t σ J(u) = F (k, u(k)) sup F (k, t) t σ On the other hand for any ε > 0 and k [, ] there is t k σ with F (k, t k ) > sup F (k, t) ε t σ. Define ũ H by ũ(k) = t k for k [, ]. hen ũ σ and J(ũ) = F (k, ũ(k)) < sup F (k, t) ε. 0 t σ

11 Hence we obtain (5). Similarly one can show that Now, using (.), (5) and (6) we obtain t R inf J(u) = sup F (k, t). (6) u H t R inf J(u) = sup F (k, t) < sup F (k, t) = u H t s inf J(u) u s inf J(u). µ(u) s Case. Suppose that inf u H J(u) =. hen (.) holds since continuous functional J attains its minimum on compact set {u : u s }. Now we will show (.3). Note that J(0) = 0. For any u H with r µ(u) s we have r u s. ake k [, ] with u = u(k). By (.3) we obtain J(u) = h= F (h, u(h)) = F (k, u(k)) h k sup F (k, t) sup F (h, t) 0. r t s h k t s F (h, u(h)) Finally, by heorem there is λ > 0 such that the functional E λ has at least three critical points in H. Corollary 3 Let 0 < r < s. Suppose that there is a positive constant c such that r < ( ) cp p + and s > + (cp + ) p + and conditions (.) (.3) are fulfilled. hen there is λ > 0 such that E λ has at least three critical points, two of which must be nontrivial. Proof. Let r > 0. hen µ(u) r is equivalent to + herefore there is k [, + ] with p(k ) u(k ) p(k ) r. p(k ) u(k ) p(k ) r +. Hence ( ) rp(k ) p(k ) u(k ). +

12 hus there is k [, ] with Consequently Hence u(k) ( rp(k ) + inf{ u : µ(u) r} ) ( ) p(k ) rp p +. + ( rp Now, let s > 0. hen µ(u) s is equivalent to + + p(k ) u(k ) p(k ) s. p(k ) u(k ) p(k ) s for every k [, + ]. hus for every k [, + ] we obtain u(k ) (sp(k )) p(k ). ) p +. (7) Note that if u(k ) a for some a and every k [, + ], then u(k) a. Using this observation we obtain that + hus u(k) + (sp(k )) u + (sp + ) p. herefore sup{ u : µ(u) s} + Suppose that c > 0 is such that r < ( cp + ) p(k ). (sp + ) p. p + and s > + (cp + ) p Since the following functions t inf{ u : µ(u) t} and t sup{ u : µ(u) t} are continuous and their ranges are equal to [0, ), then there are r < c < s with sup{ u : µ(u) s} = s and inf{ u : µ(u) r} = r. Example. Consider the following equation ( u(0) u(0) ) ( ) = λ u 3 () 0 u(), ( u() 3 u() ) = 4λ ( u() 0) 3, u(0) = u(3) = 0.

13 ( ) hen =, p(0) = 4, p() = 5, p = 4, p + t = 5, f(, t) = 3 0 t and f(, t) = 4 ( t 0) 3. hus aking c = we obtain F (, t) = t4 0 t and F (, t) = ( 0 4 t ) 4. 0 ( ) cp p + = ( ) and (cp + ) p = ake r = 0. and s = 3. Note that sup 0. t 3 F (, t) < 0, sup t 3 F (, t) = 0 and sup t R F (, t) =. Moreover sup 0. t 3 F (, t) = 0 and sup t 3 F (, t) = sup t R F (, t) = 0. Hence by Corollary 3 the equation has at least three 4 solutions for some λ > 0. References [] R.P. Agarwal, K. Perera, D. O Regan, Multiple positive solutions of singular discrete p-laplacian problems via variational methods, Adv. Difference Equ. 005 () (005) [] A. Cabada, A. Iannizzotto, S. ersian, Multiple solutions for discrete boundary value problems. J. Math. Anal. Appl. 356 (009), no., [3] X. Cai, J. Yu, Existence theorems of periodic solutions for second-order nonlinear difference equations, Adv. Difference Equ. 008 (008) Article ID [4] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image processing, SIAM J. Appl. Math. 66 (006), No. 4, [5] X.L. Fan, H. Zhang, Existence of Solutions for p(x) Lapacian Dirichlet Problem, Nonlinear Anal., heory Methods Appl. 5, No. 8, A, (003). [6] P. Harjulehto, P. Hästö, U. V. Le and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 7 (00), [7] B. Kone. S. Ouaro, Weak solutions for anisotropic discrete boundary value problems, to appear J. Difference Equ. Appl. [8] J.Q. Liu, J.B. Su, Remarks on multiple nontrivial solutions for quasi-linear resonant problemes, J. Math. Anal. Appl. 58 (00) 09. [9] J. Mawhin, Problèmes de Dirichlet variationnels non linéaires, Les Presses de l Université de Montréal,

14 [0] M. Mihǎilescu, V. Rǎdulescu, S. ersian, Eigenvalue problems for anisotropic discrete boundary value problems. J. Difference Equ. Appl. 5 (009), no. 6, [] M. Růžička, Electrorheological fluids: Modelling and Mathematical heory, in: Lecture Notes in Mathematics, vol. 748, Springer-Verlag, Berlin, 000. [] P. Stehlík, On variational methods for periodic discrete problems, J. Difference Equ. Appl. 4 (3) (008) [3] Y. ian, Z. Du, W. Ge, Existence results for discrete Sturm-Liouville problem via variational methods, J. Difference Equ. Appl. 3 (6) (007) [4] Y. Yang and J. Zhang, Existence of solution for some discrete value problems with a parameter, Appl. Math. Comput. (009), [5] G. Zhang, S.S. Cheng, Existence of solutions for a nonlinear system with a parameter, J. Math. Anal. Appl. 34 () (006) [6] G. Zhang, Existence of non-zero solutions for a nonlinear system with a parameter, Nonlinear Anal. 66 (6) (007) [7] V.V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv. 9 (987), Marek Galewski Institute of Mathematics, echnical University of Lodz, Wolczanska 5, Lodz, Poland, marek.galewski@p.lodz.pl Szymon G l ab Institute of Mathematics, echnical University of Lodz, Wolczanska 5, Lodz, Poland, szymon.glab@p.lodz.pl 4