Mariusz Jurkiewicz, Bogdan Przeradzki EXISTENCE OF SOLUTIONS FOR HIGHER ORDER BVP WITH PARAMETERS VIA CRITICAL POINT THEORY

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1 DEMONSTRATIO MATHEMATICA Vol. XLVIII No Mariusz Jurkiewicz, Bogdan Przeradzki EXISTENCE OF SOLUTIONS FOR HIGHER ORDER BVP WITH PARAMETERS VIA CRITICAL POINT THEORY Communicated by E. Zadrzyńska Abstract. This paper is concerned with the existence of at least one solution of the nonlinear 2k-th order BVP. We use the Mountain Pass Lemma to get an existence result for the problem, whose linear part depends on several parameters. 1. Introduction There has recently been an increased interest in studying the existence of solutions for boundary value problems (BVPs) of higher order differential equations (cf. [1, 4 6, 9]). Most of the earlier discussions were devoted to the fourth order BVPs, for example see [3, 7]. In this paper, we consider the depending on real parameters family of 2kth order (k 2) BVP, $ & p 1q k x p2kq ` kř λ j x p2k 2jq p 1q i 1 f `t, x p2i 2q, (1) j 1, % x p2jq pq x p2jq p1q, j,..., k 1, where f : r, 1s Ñ R is a continuous function. Here, i is a fixed integer 1 i k. Notice that the nonlinear term f depends only on the 2i 2-th derivative of unknown function. It is seen that for some unions of lambdas, the differential operator that corresponds to the left-hand side of (1) is invertible and is not for others (see [4]). The second case, commonly called a resonance one, needs additional conditions of Landesman Lazer type and was examined in [3], [4]. Here, we focus on the the nonresonant case. Jurkiewicz [6] established the existence of infinitely many solution for (1), by applying the Rabinowitz s theorem about unbounded sequence of critical points. 21 Mathematics Subject Classification: Primary 34B15; Secondary 35P3. Key words and phrases: multidimensional spectrum, Lidstone BVP, critical point theory, Palais Smale condition. DOI: /dema c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology

2 54 M. Jurkiewicz, B. Przeradzki Theorem 1. [6] Assume that λ P ` psee p. 55q and let f be odd with respect to the second variable, i.e. fpt, uq fpt, uq for all t P r, 1s and u P R. If f satisfies the following conditions (i) lim sup ă π2k ` Λ p1q uñ u π 2i 2 (definition of Λ (2)), (ii) lim inf `8, uñ`8 u (iii) there exist k P r, 1{2q and N ą, such that ş w du kwf pt, wq, for w N, t P r, 1s, then the problem p1q possesses infinitely many solutions. It is well known that in a lot of problems of the form Lu Npuq, where L is a linear operator and N a nonlinear one, the existence of at least m solution is obtained if an asymptotic behaviour of N near and near 8 is similar to µ I and µ 8 I, respectively (I the identity operator), and there are exactly m eigenvalues of L in the interval pµ, µ 8 q. Assumptions (i) and (ii) of Theorem 1 mean that in this interval sit infinitely many eigenvalues, although we generalize the notion of eigenvalue as you will see in Section 2: the eigenvalues are k-dimensional vectors, which forms a sequence ph n q k 1-dimensional hyperplanes and the segment in R k joining points with all coordinates except 2pk ` 1 iq-th, where there are limits µ from condition (i) and µ 8 from (ii), respectively, intersects all these hyperplanes. Thus, it is natural to ask if there is at least m solution to (1), if in assumption (ii), the limit is a number µ 8 such that the above mentioned segment intersect exactly H 1,..., H m. The present paper is an answer to this question for m 1. This enables us to drop the assumption of oddness of f. 2. Preliminaries Assume that k 2 and i 1,... k 1 are fixed positive integer numbers and let kÿ (2) Λ pxq p 1q k j λ `x2 j π 2 k j. j 1 Definition 1. (see [4], [5]) A point λ pλ 1,... λ k q P R k will be called a k-dimensional eigenvalue iff the homogeneous problem $ & p 1q k x p2kq ptq ` λ 1 x p2k 2q ptq ` λ 2 x p2k 4q ptq (3) `... ` λ k 1 x % 2 ptq ` λ k x ptq, x p2jq pq x p2jq p1q, for j... k 1, has a nonzero solution. The set of all such k-tuples will be denoted by σ k.

3 Existence of solutions for higher order BVP with parameters It can be proved that the set σ k has the form σ k ď npn H n, where H n is a hyperplane of the form H n : λ P R k `n 2 π 2 k ` Λ pnq ( (see [4], [5]). Theorem 2. (see [4]) Let s P N and n 1, n 2,..., n s P N be such that n i n j, i j. (1) If s k, then W Ş s i 1 H n i is an affine subspace in R k and dim W k s. (2) If s ą k, then Ş s i 1 H n i. Shortly, hyperplanes H n are in a general position. Let us put ` : Ş npn tλ P Rk Λpnq u. It is easily seen that `. Indeed, putting $ tλ P R k λ s if s is odd and λ s if s is evenu, & if k is an even number, D` : tλ P R k λ s if s is odd and λ s if s is evenu, % if k is an odd number, we see that D` Ă `. Furthermore, due to the inequality pn 2 π 2 q k `Λpnq ą, n 1, 2,..., we get ` X σ k. Definition 2. (see [8]) A sequence ty n u Ă X is a Palais Smale sequence for ϕ P C 1 px, Rq, if tϕ py n qu is bounded while ϕ 1 py n q Ñ as n Ñ 8. Definition 3. (see [8]) We say ϕ P C 1 px, Rq satisfies (P.-S.) condition, if any Palais Smale sequence has a (strongly) convergent subsequence. Theorem 3. (Mountain Pass Lemma (MPL), [8], Theorem 6.1) Let X be a real Banach space and ϕ P C 1 px, Rq such that ϕ satisfies (P.-S.) condition and ϕpq. Furthermore (1) there exist α ą and r ą such that ϕ pyq α for all y P BB p, rq ; (2) there exists y P X such that ϕ py q ă α with }y } r. Then ϕ possesses a critical value c α. If x is a solution to (1) and y x p2i 2q, then we have (see [4], [5] and [6]) y ptq p 1q i 1 H i pt, sq f ps, y psqq ds,

4 56 M. Jurkiewicz, B. Przeradzki where (4) H i pt, sq Putting 8ÿ n 1 p 1q i 1 pnπq 2i 2 pn 2 π 2 q k sin pnπsq sin pnπtq. ` Λ pnq pt i yq ptq p 1q i 1 H i pt, sq f ps, y psqq ds, we see that T i maps C the space of real continuous functions on r, 1s into itself. Let ph i zqptq p 1q i 1 ş 1 H ipt, sqzpsqds and pfyqpsq f ps, y psqq (both operators act in C), then T i can be described as a composition of H i and f. Consider the unique extension of the operator H i to the Hilbert space L 2 denoted also H i for simplicity. The operator H i is continuous, self-adjoint and σ ph i q σ p ph i q Y σ c ph i q, where σ p ph i q pnπq 2i 2 rpn 2 π 2 q k ` Λpnqs 1 p 1, 2,... ( and σ c ph i q tu (see [6]). Moreover, if λ P ` then σ ph i q Ă r, `8q and H i is a positive operator. This implies that there exists a unique positive and self-adjoint S i such that Si 2 H i (see [2], Theorem 2.2.1). Below, the norm without subscript stands for the norm in the space L 2 ; the supremum norm in the space of continuous functions will be denoted by } } C. Remark. Hereinafter, it will be assumed that λ P `. If we reason in the similar way as in [4], we can deduce that 8ÿ pnπq i 1 ps i zq ptq b sin pnπsq z psq ds sin pnπtq. n 1 pnπq 2k ` Λ pnq (i) The function S i pt, sq 8ÿ n 1 pnπq i 1 b sin pnπsq sin pnπtq pnπq 2k ` Λ pnq is continuous. (ii) Operator S i maps L 2 into C and it has the form ps i zq ptq S i pt, sq z psq ds. It easy to see that the sequence pnπq 2i 2 r`n2 π 2 k`λ pnqs 1 is decreasing, thus by Theorem [2], it is easy to calculate the norms of H i and S i, treating them as operators from L 2 into itself (5) }H i } π 2i 2 After [6], one can notice that π 2k ` Λ p1q 1 and }S i } π i 1 1 π 2k 2 ` Λ p1q.

5 Existence of solutions for higher order BVP with parameters (1) has a solution if T i H i f has a fixed point; fixed points of the operator S i f S i in L 2 are also fixed points of T i H i f in C; if ψ : C Ñ R is defined by the formula ψy ż yptq dudt, then functionals ψ and ψ S i are Fréchet differentiable on C and L 2, respectively and pψ 1 pyqq h xfy, hy L 2, pψ S i q 1 y ps i f S i q y. Let ϕ i : L 2 Ñ R, be the functional of the form (6) ϕ i pyq 1 2 xy, yy L 2 pψ S iq y. We have (7) ϕ 1 i pyq y ps i f S i q y, for y P L 2. Due to the above arguments, solutions to (1) are critical points of ϕ i. 3. Main result Lemma 1. [6] Assume that there are k P r, 1{2q and N ą, such that for w N, we have du kwf pt, wq then the functional ϕ i satisfies the (P.S.) condition. Remark. If f is odd then functions R Q w ÞÑ ş w du and R Q w ÞÑ w f pt, wq are even for any t P r, 1s. Therefore, to verify assumption of Lemma 1, it is sufficient to check the inequality for w N ą. Now we can prove the main theorem. Theorem 4. Assume that λ P ` and let f : r, 1sˆR Ñ R be a continuous function that satisfies the following conditions (i) lim sup uñ u ă π2k ` Λ p1q π 2i 2, (ii ) lim inf ą π2k ` Λ p1q uñ`8 u π 2i 2, (iii) there exist k P r, 1{2q and N ą, such that for w N, we have du kwf pt, wq, for each t P r, 1s. Then problem p1q possesses at least one nonzero solution.

6 58 M. Jurkiewicz, B. Przeradzki Proof. We shall use the Mountain Pass Lemma. The considered functional has the form ϕ i pyq ptq 1 2 }y}2 ż psi yqptq dudt. It is obvious that ϕ i pq, furthermore we explained on the page 56, that π 2k ` Λ p1q π 2i 2 }H i } 1. Due to assumption, there exist δ ą and ε 1 P p, }H i } 1 q such that This implies that and u }H i } 1 ε 1, for u δ. p}h i } 1 ε 1 qu, for u P r, δs, p}h i } 1 ε 1 qu, for u P r δ, q. Thus we have, for w P r, δs, and, for w P r δ, q, fpt, uqdu du p}h i } 1 ε 1 q ż w fpt, uqdu p}h i } 1 ε 1 q The above calculations imply that (8) for w δ. udu p}h i } 1 ε 1 q 1 2 w2, ż du p}h i } 1 ε 1 q 1 2 w2, w udu p}h i } 1 ε 1 q 1 2 w2. Let y P L 2, the kernel S i of S i is a positive continuous function. Therefore, let B i ą be its maximal value. This and the Hölder s inequality imply that ˆ }S i y} C B i y psq ds B i y psq 2 ds Therefore, if }y} r, where r B 1 δ, we get }S i y} C δ. 1 2 Bi }y}. Now, the last estimate together with (8), give us the following condition

7 Existence of solutions for higher order BVP with parameters ϕ i pyq 1 2 }y}2 ż psi yqptq 1 2 }y}2 p}h i } 1 ε 1 q 1 2 dudt pps i yq ptqq 2 dt 1 2 }y}2 p}h i } 1 ε 1 q 1 2 xs iy, S i yy L }y}2 p}h i } 1 ε 1 q 1 2 xh iy, yy L }y}2 p}h i } 1 ε 1 q 1 2 }H i} }y} }H i} }y} 2 ε }H i} r 2 ε 1. Thus ϕ i pyq α, α 1 2 }H i} r 2 ε 1, for y P BB p, rq, r B 1 i δ, and assumption (1) of MPL holds. Assumption (ii) implies the existence of such N ą and ε 2 ą, that }H } 1 i p1 ` ε 2 q, for u ą N and t P r, 1s. u Then (9) }H i } 1 p1 ` ε 2 q u, for u ą N and t P r, 1s. Because the function }H i } 1 p1 ` ε 2 q u is continuous on r, 1s ˆ r, Ns, there exists M ą such that (1) }H i } 1 p1 ` ε 2 q u M, for u P r, Ns and t P r, 1s. The inequalities (9) and (1) led us to the following conclusion }H i } 1 p1 ` ε 2 q u M, for u, t P r, 1s. Integrating the above inequality, we get (11) du 1 2 }H i} 1 p1 ` ε 2 q w 2 Mw, for w and uniformly for t P r, 1s. We have explained that 8ÿ H i u ξn 2 xe n, uy L2 e n and S i u where n 1 pnπq 2i 2 8ÿ ξ n xe n, uy L2 e n, n 1 ξn 2 pnπq 2k and e n ptq? 2 sin pnπtq. ` Λ pnq Furthermore ξ1 2 π 2i 2 π 2k ` Λ p1q }H i}.

8 6 M. Jurkiewicz, B. Przeradzki It easily seen that (12) S i e 1 ξ 1 e 1 a }H i }e 1. For any real τ, we have from (11) and (12) ϕ i pτe 1 q 1 2 xτe 1, τe 1 y L 2 ż psi pτe 1 qqptq 1 2 τ }Hi} 1 p1 ` ε 2 q ` M ps i pτe 1 qq ptq dt dudt ps i pτe 1 qq 2 ptq dt 1 2 τ τ 2 }H } 1 i p1 ` ε 2 q xs i e 1, S i e 1 y L 2 ` Mτ ps i pe 1 qq ptq dt 1 2 τ τ 2 }H } 1 i p1 ` ε 2 q }H i } ` Mτ a }H i } 1 2 ε 2τ 2 ` M a }H i } 2? 2 π τ. e 1 ptq dt Let τ ą. The above estimate implies that ϕ i pτe 1 q Ñ 8 and }τe 1 } τ Ñ `8 as τ Ñ `8. Therefore, there exists τ ą r such that ϕ i py q ϕ i pτ e 1 q ă ă α and assumption (2) of MPL is satisfied. Due to MPL, we get the assertion. It has to be emphasized that we do not assume oddness of a nonlinear part. Example. Let us consider the following problem # u p8q π 2 u p6q ` π 4 u p4q π 6 u 2 ` π 8 u f `t, u p4q, where u p2jq pq u p2jq p1q, for j... 3, $ & w 3 ` 5π 4 ` 1, for w ă 1, f pt, wq w % 2 ` 5π 4 1, for w P r 1, 1s, w 2 arctan pwq ` 5π 4 1 4π, for w ą 1. It is seen that }H 3 } 1 5π 4, furthermore it is easy to verify that f is continuous, satisfies conditions (i) (iii) of Theorem 4 and does not satisfy assumptions of Theorem 1.

9 Existence of solutions for higher order BVP with parameters References [1] R. P. Agarwal, D. O Regan, S. Stanĕk, Singular Lidstone boundary value problem with given maximal values for solutions, Nonlinear Anal. 55 (23), [2] O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, Springer-Verlag, Berlin, Heidelberg, [3] M. Jurkiewicz, On solutions of fourth-order Lidstone boundary value problem at resonance, Ann. Polon. Math. 95 (29), [4] M. Jurkiewicz, Existence result for the Lidstone boundary value problem at resonance, J. Math. Anal. Appl. 394 (212), [5] M. Jurkiewicz, Some remarks on exact solution of Lidstone boundary value problem, Biuletyn WAT 61 (212), [6] M. Jurkiewicz, Existence of infinitely many solutions to Lidstone BVP, (submitted). [7] Y. Li, Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl. 281 (23), [8] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th Ed., Springer-Verlag, Berlin, Heidelberg, 28. [9] Y.-M. Wang, On 2nth-order Lidstone boundary value problems, J. Math. Anal. Appl. 312 (25), M. Jurkiewicz INSTITUTE OF MATHEMATICS AND CRYPTOLOGY MILITARY UNIVERSITY OF TECHNOLOGY Gen. Sylwestra Kaliskiego 2-98 WARSZAWA 49, POLAND mjurkiewicz@wat.edu.pl B. Przeradzki INSTITUTE OF MATHEMATICS LODZ UNIVERSITY OF TECHNOLOGY Wólczańska ŁÓDŹ, POLAND bogdan.przeradzki@p.lodz.pl Received December 16, 213; revised version June 4, 214.