BOUNDED WEAK SOLUTION FOR THE HAMILTONIAN SYSTEM. Q-Heung Choi and Tacksun Jung

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1 Korean J. Math. 2 (23), No., pp BOUNDED WEAK SOLUTION FOR THE HAMILTONIAN SYSTEM Q-Heung Choi and Tacksun Jung Abstract. We investigate the bounded weak solutions for the Hamiltonian system with bounded nonlinearity decaying at the origin and periodic condition. We get a theorem which shows the existence of the bounded weak periodic solution for this system. We obtain this result by using variational method, critical point theory for indefinite functional.. introduction G(t, z(t)) be a C 2 function defined on R R 2n which is 2πperiodic with respect to the first variable t. In this paper we investigate the number of 2π-periodic solutions of the following Hamiltonian system (.) p(t) = G q (t, p(t), q(t)), q(t) = G p (t, p(t), q(t)), where p, q R n, z = (p, q). J be the standard symplectic structure on R 2n, i. e., ( ) In J =, I n Received February 9, 23. Revised March 8, 23. Accepted March 2, Mathematics Subject Classification: 35Q72. Key words and phrases: Hamiltonian system, bounded nonlinearity, variational method, critical point theory for indefinite functional, (P.S.) c condition. This work was supported by Inha University Research Grant. Corresponding author. c The Kangwon-Kyungki Mathematical Society, 23. This is an Open Access article distributed under the terms of the Creative commons Attribution Non-Commercial License ( -nc/3./) which permits unrestricted non-commercial use, distribution and reproduction in any medium, provided the original work is properly cited.

2 82 Q-Heung Choi and Tacksun Jung where I n is the n n identity matrix. Then (.) can be rewritten as (.2) Jż = G z (t, z(t)), where ż = dz dt and G z is the gradient of G. We assume that G C 2 (R R 2n, R ) satisfies the following conditions: (G) G C 2 (R R 2n, R ), (G2) G(t, z(t)) = O( z 2 ) as z, G(t, θ) =, G z (t, θ) = θ, where θ = (,, ), (G3) there exists C > such that G(t, ξ) < C t R, ξ R 3. Several authors ([], [3], [4], [5] etc.) studied the nonlinear hamiltonian system. Jung and Choi ([3], [4]) considered (.) with nonsingular potential nonlinearity or jumping nonlinearity crossing one eigenvalue, or two eigenvalues, or several eigenvalues. Chang ([]) proved that (.) has at least two nontrivial 2πperiodic weak solutions under some asymptotic nonlinearity. Jung and Choi ([3]) proved that (.) has at least m weak solutions, which are geometrically distinct and nonconstant under some jumping nonlinearity. We are looking for the weak solutions of (.) with the conditions (G)-(G3). The 2π-periodic weak solution z = (p, q) E of (.) satisfies i. e., (ż JG z (t, z(t))) Jwdt = for all w E, [(ṗ + G q (t, z(t))) ψ ( q G p (t, z(t))) φ]dt = where E is introduced in section 2. Our main result is as follows: for all ζ = (φ, ψ) E, Theorem.. Assume that G satisfies the conditions (G) (G3). Then system (.) has at least one bounded 2π-periodic solution. For the proof of our main result we approach the variational method and apply the critical point theory to indefinite functional. The outline of the proof of Theorem. is as follows: In Section 2, we introduce the perturbed operator A ɛ such that A ɛ is a compact operator, and the associated functional I(z) corresponding to the operator A ɛ, prove that

3 Bounded weak solution for the Hamiltonian system 83 I(z) satisfies F réchet differentiability, and state the critical point theorem for indefinite functional. In section 3, we show that the associated functional I(z) satisfies the geometrical assumptions of the critical point theorem for indefinite functional, and prove Theorem.. 2. Compact operator and variational approach L 2 ([, 2π], R 2n ) denote the set of 2n-tuples of the square integrable 2πperiodic functions and choose z L 2 ([, 2π], R 2n ). Then it has a Fourier expansion z(t) = k=+ k= a ke ikt, with a k = 2π z(t)e ikt dt 2π C 2n, a k = ā k and k Z a k 2 <. with domain A : z(t) Jż(t) D(A) = {z(t) H ([, 2π], R 2n ) z() = z(2π)} = {z(t) L 2 ([, 2π], R 2n ) k Z(ɛ + k ) 2 a k 2 < + }, where ɛ is a positive small number. Then A is a self-adjoint operator. {M λ } be the spectral resolution of A, and let α be a positive number such that α / σ(a) and [α, α] contains only one element of σ(a). P = α α dm λ, P + = + α dm λ, P = α dm λ. L = P L 2 ([, 2π], R 2n ), L + = P + L 2 ([, 2π], R 2n ), L = P L 2 ([, 2π], R 2n ). For each u L 2 ([, 2π], R 2n ), we have the decomposition u = u + u + + u, where u L, u + L +, u L. According to A, there exists a small number ɛ > such that ɛ / σ(a). us define the space E as follows: E = D( A 2 ) = {z L 2 ([, 2π], R 2n ) k Z(ɛ + k ) a k 2 < }

4 84 Q-Heung Choi and Tacksun Jung with the scalar product and the norm (z, w) E = ɛ(z, w) L 2 + ( A 2 z, A 2 w)l 2 z = (z, z) 2 E = ( k Z(ɛ + k ) a k 2 ) 2. The space E endowed with this norm is a real Hilbert space continuously embedded in L 2 ([, 2π], R 2n ). The scalar product in L 2 naturally extends as the duality pairing between E and E = W 2,2 ([, 2π], R 2n ). We note that the operator (ɛ + A ) is a compact linear operator from L 2 ([, 2π], R 2n ) to E such that ((ɛ + A ) w, z) E = A ɛ = ɛi + A. (w(t), z(t))dt. E = A ɛ 2 L, E + = A ɛ 2 L+, E = A ɛ 2 L. Then E = E E + E and for z E, z has the decomposition z = z + z + + z E, where Thus we have z = A ɛ 2 u, z + = A ɛ 2 u+, z = A ɛ 2 u. z E = u L, z + E+ = u + L+, z E = u L and that E, E +, E are isomorphic to L, L +, L, respectively. The associated functional of (.2) on E is as follows: (2.) I(z) = 2 ( A 2 ɛ z + 2 L + A 2 2 ɛ M + z 2 (A ɛ ) 2 z 2 L 2 (A ɛ ) 2 M z 2 ) ψ ɛ (z), = 2 ( u M + u 2 M u 2 u 2 ) ψ ɛ (z), where ψ ɛ (z) = ψ(z) + ɛ 2 z 2 L 2, ψ(z) = G(t, z(t))dt. F (z) = G z (t, z(t)).

5 Bounded weak solution for the Hamiltonian system 85 By G C 2, ψ(z) = G(t, z(t))dt C 2 (S D, R ). Then (.2) can be rewritten as F ɛ (z) = ɛi + F (z) = ɛi + G z (t, z(t)). (2.2) A ɛ (z) = F ɛ (z). The Euler equation of the functional I(z) is the system (2.3) u + = A ɛ 2 P+ F ɛ (z), (2.4) u = A ɛ 2 P F ɛ (z), (2.5) M + u = A ɛ 2 M+ P F ɛ (z) M u = A ɛ 2 M P F ɛ (z). Thus z = z +z + +z is a solution of (2.2) if and only if u = u +u + +u is a critical point of I. The system (2.3)-(2.5) is reduced to (2.6) A ɛ z + = P + F ɛ (z +z + +z ) or z + = (A ɛ ) P + F ɛ (z +z + +z ), (2.7) A ɛ z = P F ɛ (z +z + +z ) or z = (A ɛ ) P F ɛ (z +z + +z ), (2.8) A ɛ M + z = M + P F ɛ (z + z + + z ), A ɛ M z = M P F ɛ (z + z + + z ). By the following Lemma 2., the weak solutions of (.2) coincide with the critical points of the functional I(z). Lemma 2.. Assume that G satisfies the conditions (G) (G3). Then I(z) is continuous and F réchet differentiable in E with F réchet derivative (2.9) DI(z)w = Moreover DI C. That is, I C. (A ɛ z F ɛ (z)) wdt for all w E,

6 86 Q-Heung Choi and Tacksun Jung Proof. First we prove that I(z) = [ 2 A ɛz G(t, z(t)) ɛ 2 z2 ]dt is continuous and F réchet differentiable in E. For z, w E, I(z + w) I(z) = = We have (2.) 2 A ɛ(z + w) (z + w)dt 2 A ɛ(z) zdt + [G(t, z + w) + ɛ 2 (z + w)2 ]dt 2 [A ɛ(z) w + A ɛ (w) z + A ɛ (w) w]dt [G(t, z + w) G(t, z)]dt [G(t, z) + ɛ 2 z2 ]dt [G(t, z + w) G(t, z) + ɛ 2 (2z w + w2 ))]dt. [G z (t, z(t)) w + O( w R 2n)]dt = O( w R 2n). Thus we have I(z + w) I(z) = O( w R 2n). Next we shall prove that I(z) is F réchet differentiable in E. For z, w E, I(z + w) I(z) DI(z)w = 2 A ɛ(z + w) (z + w)dt 2 A ɛ(z) zdt + A ɛ (z) wdt + = 2 [A ɛ(w) zdt + A ɛ (w) w]dt [G(t, z + w) + ɛ 2 (z + w)2 ]dt [G(t, z) + ɛ 2 z2 ]dt [G z (t, z) + ɛz w]dt [G(t, z + w) G(t, z) G z (t, z) + ɛ 2 w2 )]dt.

7 By (2.), we have Bounded weak solution for the Hamiltonian system 87 Thus [G(t, z + w) G(t, z) G z (t, z)]dt = O( w R 2n). I(z + w) I(z) DI(z)w = O( w R 2n). Now, we recall the critical point theorem for the indefinite functional (cf. [2]). B r = {u E u r}, B r = {u E u = r}. Theorem 2.. Critical point theorem for the indefinite functional) E be a real Hilbert space with E = E E 2 and E 2 = E. Suppose that I C (E, R), satisfies (P S), and (I) I(u) = (Lu, u) + bu, where Lu = L 2 P u + L 2 P 2 u and L i : E i E i is bounded and self adjoint, i =, 2, (I2) b is compact, and (I3) there exists a subspace Ẽ E and sets S E, Q Ẽ and constants α > ω such that (i) S E and I S α, (ii) Q is bounded and I Q ω, (iii) S and Q link. Then I possesses a critical value c α. 3. Proof of Theorem. We shall show that the functional I(z) satisfies the geometric assumptions of the critical point theorem for indefinite functional. Lemma 3.. Palais-Smale condition) Assume that G satisfies (G) (G3). Then I(z) satisfies the Palais-Smale condition: If for a sequence (z k ), I(z k ) is bounded from above and DI(z k ) as k, then (z k ) has a convergent subsequence.

8 88 Q-Heung Choi and Tacksun Jung Proof. (z k ) be a sequence with (3.) I(z k ) M and (3.2) DI(z k ) = z k A ɛ (G z (t, z k ) + ɛz k ) θ as m, where A ɛ is compact operator and θ = (,, ). We claim that {z k } has a convergent subsequence. Since G z (t, z(t)) + ɛz k is bounded for a small constant ɛ and A ɛ is compact operator, by (3.2), {z k } has a convergent subsequence. Thus we prove the lemma. B ρ = {z E z ρ}, B ρ = {z E z = ρ}, Q = ( B R E ) {re e B E + < r < R}. Lemma 3.2. Assume that G satisfies the conditions (G) (G3). there exist a constant ρ > and sets S E, Q E such that (i) B ρ E + and I Bρ >, (ii) Q is bounded and I Q <, (iii) B ρ and Q link. Proof. (i) z E + E. Since G(x, t, z) is bounded, there exists a constant C > such that I(z) = 2 ( A 2 ɛ z + 2 L 2 + A 2 ɛ M + z (A ɛ ) 2 z 2 L 2 (A ɛ) 2 M z 2 ) G(t, z)dt ɛ 2 z 2 L 2 2 ( A 2 ɛ z + 2 L 2 C ɛ 2 z 2 L 2 for C >. Then there exist a constant ρ > such that if z B ρ E +, then I(z) >. (ii) us choose e B E +. z B r E {re < r}. Then z = w + y, w B r E, y = re. We note that If w B r E, then A ɛ (z) zdt = (A ɛ ) 2 z 2 L 2.

9 Bounded weak solution for the Hamiltonian system 89 By (G3), G(t, w + re) is bounded from below. Thus, there exists a constant C > such that if z = w + re, then we have I(z) = 2 r2 (A ɛ ) 2 z 2 L G(t, w + re)dt ɛ 2 2 z 2 L 2 2 r2 (A ɛ ) 2 z 2 L 2 + C ɛ 2 z 2 L 2. We can choose a constant R > r such that if z = w + re Q = ( B r E ) {re e B E +, < r < R}, then I(z) <. Thus we prove the lemma. Proof of Theorem. By Lemma 2., I(z) is continuous and F réchet differentiable in E and moreover DI C. By Lemma 3., I(z) satisfies the (P.S.) condition. By Lemma 3.2, there exist a constant ρ > and sets B ρ E + with radius ρ >, Q E such that I Bρ >, Q is bounded and I Q <, and B ρ and Q link. By Theorem 2., I(z) possesses a critical value c >. Thus (.) has at least one nontrivial periodic weak solution. Thus we prove Theorem. References [] K. C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhäuser, 993. [2] Benci, V., Rabinowitz, P.H, Critical point theorems for indefinite functionals, Invent. Math. 52 (979), [3] T. Jung and Q. H. Choi, On the number of the periodic solutions of the nonlinear Hamiltonian system, Nonlinear Anal. 7 (2) (29), e-e8. [4] T. Jung and Q. H. Choi, Periodic solutions for the nonlinear Hamiltonian systems, Korean J. Math. 7 (3) (29), [5] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS. Regional Conf. Ser. Math. 65, Amer. Math. Soc. Providence, Rhode Island, 986. Department of Mathematics Education Inha University Incheon 42-75, Korea qheung@inha.ac.kr

10 9 Q-Heung Choi and Tacksun Jung Department of Mathematics Kunsan National University Kunsan 573-7, Korea