Periodic Solutions of Nonlinear Wave Equations and Hamiltonian Systems

We consider the nonlinear vibrating string equation u,,-uxx + h(u) = 0 under Dirichiet boundary conditions on a finite interval. We assume that h is nondecreasing, h(O) = 0 and limjuj_I[h(u)/u] = 0. We prove that for T sufficiently large, there is a nontrivial T-periodic solution. A similar result holds for Hamiltonian systems. 0. Introduction. Consider the following nonlinear wave equation: u,,-u,,,, + h(u)=O O < x < 7r, t E R. (1) under the boundary conditions: u(O, t) = u( 7r, t) = 0, (2) where h: R R is a continuous nondecreasing function such that h(O) = 0. We assume: lim h(u) =0 (3) I uI-o U There exists a constant R such that h(u) ? 0 for I u ? R. (4) We seek nontrivial solutions of (1), (2) which are T-periodic (in t). By "nontrivial" we mean that h(u(x, t)) ? 0 on a set (x, t) of positive measure; in particular, u(x, t) ? 0 on that set. In Section 1 we prove the following THEOREM 1. There exists To > 0 such that for every T : To, with T/7r rational, Problem (1), (2) admits a nontrivial T-periodic (weak) solution u E Lw. Manuscript received May 15, 1980. Aniericani Journail of Mathemiiatics, Vol. 103, No. 3, pp. 559-570 0002-9327/81/1033-0559 $01.50 Copyright ? 1981 by The Johns Hopkins University Press 559 560 HAIM BREZIS AND JEAN-MICHEL CORON By a result of [4], weak solutions are in fact smooth if h is smooth and strictly increasing. The existence of nontrivial solutions for (1), (2) has been considered by several authors under assumptions which differ from ours (see [1, 4, 5, 7, 8]). In Section 2 we discuss a comparable result for Hamiltonian systems. Our investigation has been stimulated by the results of [3] (Section 4). Our technique relies on a duality device used in [6] for Hamiltonian systems and subsequently in [5] for the wave equation. We thank P. Rabinowitz for helpful discussions. Proof of Theorem 1. The proof is divided into five steps. Step 1 Generalities about Au = ut uxx Step 2 Determination of To. Step 3 Existence of a nontrivial solution for Au + h(u) + cu = 0 (c > 0 small). Step 4 Estimates. Step 5 Passage to the limit as c 0. Step 1 Generalities about Au = ut ux Since T/7r E Q we may write T = 27rb/a where a and b are coprime integers. Let H = L2(Q) with Q (0, 7r) X (0, T). In H we consider the operator Al = U,,acting on functions satisfying (2) and which are T-periodic in t. We summarize some of the main properties of A which we shall use (see e.g. [4] and the references in [4]): i) A* =A ii) N(A) consists of functions of the form 2 ir T T/b ) N(A) = {p(t + x)-p(t-x), wherep has period -= -and P = ) a b Oi iii) R(A) is closed and R(A) = N(A)'; whenever u E H we shall write U = III + u2with uI E R(A), u2 E N(A). PERIODIC SOLUTIONS AND HAMILTONIAN SYSTI EMS 561 iv) The eigenvalues of A arej2 [(27r/T)k]2, j = 1, 2, 3, ... and k = 0, 1, 2, .... The corresponding eigenfunctions are sin jx sin(2J kt) and sin jx cos2( kt). We denote by X 1(T) the first negative eigenvalue. Note that X-1(T) 0 as T oo. Indeed, let ,u = j2 [(2 7r/T)k]2 with j = 1 and k = [T/2 7r] + 1. We have 1[1 + (2 7r/T)]2 < ,u < 0 and so X1l(T)l sH ?y1 s4 j 1 + r v) Givenf E R(A), there exists a unique u E R(A) n C(Q) such that Au =1. We set i/ =Kf -(A ')J We have KfHLOO s CIHIIILI Vf E R(A), IKf HI s C IHIfL2 Vf E R(A). K is a compact self-adjoint operator in R(A). Step 2 Determination of To We set