WITH REPULSIVE SINGULAR FORCES MEIRONG ZHANG. (Communicated by Hal L. Smith) of repulsive type in the sense that G(u)! +1 as u! 0.
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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 2, February 1999, Pages 41{47 S (99)512-5 PERIODIC SOLUTIONS OF DAMPED DIFFERENTIAL SYSTEMS WITH REPULSIVE SINGULAR FORCES MEIRONG ZHANG (Communicated by Hal L. Smith) Abstract. We consider the periodic boundary value problem for the singular dierential system: u +(rf (u)) + rg(u) = h(t) where F 2 C 2 (R N R), G 2 C 1 (R N nfg R), andh 2 L 1 ([ T] R N ). The singular potential G(u) is of repulsive type in the sense that G(u)! +1 as u!. Under Habets- Sanchez's strong force condition on G(u) at the origin, the existence results, obtained by coincidence degree in this paper, have no restriction on the damping forces (rf (u)). Meanwhile, some quadratic growth of the restoring potentials G(u) at innity is allowed. 1. Introduction In this paper, we study existence results to the periodic boundary value problem for the repulsive singular dierential system: (1.1) (1.2) u +(rf (u)) + rg(u)=h(t) u() ; u(t )=u () ; u (T )= where F 2 C 2 (R N R) G2 C 1 (R N nfg R), and h 2 L 1 ([ T] R N ). The potential G(u) is singular (at ) in the sense that G(u) becomes innity whenu tends to. From the physical meaning of Eq. (1.1), the singular potential G(u) is said to be of attractive type (repulsive type, respectively) when G(u)!;1(G(u)! +1, respectively) as u! see Ambrosetti [1]. By a solution u() of (1.1){(1.2), we meanthatu 2 W 2 1 ([ T] R N )such that u(t) 6= for all t 2 [ T] and satises the boundary condition (2.2) and Eq. (2.1) for a.e. t 2 [ T]. When no damping is presented in Eq. (1.1), i.e., F (u), the problem (1.1){ (1.2) is of variational structure and can be handled by critical point theory. Such an approach is initiated by Gordon [3], who introduced a strong force condition on G(u) at to get some compactness for the corresponding action functionals. We Received by the editors September 23, Mathematics Subject Classication. Primary 34C15, 34C25. Key words and phrases. Singular force, strong force condition, damped system, coincidence degree. The author is supported by the National Natural Science Foundation of China and the Tsinghua University Education Foundation. 41 c1999 American Mathematical Society
2 42 MEIRONG ZHANG refer to Ambrosetti [2] for the attractive case and to Coti Zelati [2] and Solimini [1] for the repulsive case. When damping is presented in Eq. (1.1), another approach is also developed by using the Leray-Schauder degree, or its variant, the coincidence degree (see Mawhin [8], [9]). For the one-dimensional systems, we refer to Laser and Solimini [6] and Habets and Sanchez [4]. Of particular interest, when F (u) 1 2 cjuj2 and some necessary assumptions on G(u) at are assumed, Mawhin's result in [9] allows the repulsive potentials G(u) tohave some quadratic growth at innity. The present author has recently generalized this result, in [11], to general dampings F (u) and potentials G(t u), periodic in t. Aninteresting application to the Brillouin electron beam focusing system is also given in [11]. For the higher dimensional systems, Habets and Sanchez [5] introduced a dierent strong force condition on G(u) at the origin from the viewpoint of degree theory. However, the existence results in [5] mainly depend upon a strict damping force condition, i.e., there exists some constant a> such that 2 F (u) 2 ai N for all u 2 R N 2 F 2 (u) ;ai N for all u 2 R N : In this paper we consider higher dimensional systems. Under Habets-Sanchez's strong force condition on G(u) at, the existence results in this paper can be applied to any damping forces without imposing more conditions on singular potentials G(u) than those in Habets and Sanchez [5]. Meanwhile, some quadratic growth of the restoring potentials G(u) at innity isalsoallowed. 2. Main result In the sequel, we use and jjto denote the Euclidean inner product and norm, respectively, inr N. Meanwhile, the L p -norm in the space L ([ T] R p N ), 1 p 1, is denoted by kk p.for h 2 L 1 ([ T] R N ), the mean value is dened by (2.1) (2.2) h = 1 T h(t)dt: As mentioned in Section 1, we consider the following boundary value problem: u +(rf (u)) + rg(u) =h(t) u() ; u(t )=u () ; u (T )= a.e. t 2 [ T] where F 2 C 2 (R N R) G2 C 1 (R N nfg R), and h 2 L 1 ([ T] R N ). We impose the conditions on G as follows: (G1) lim u rg(u) =;1: u! (G2) Habets-Sanchez's Strong Force Condition at [5]: There exists ' 2 C 1 (R N nfg R)such that: (i) lim '(u) =+1 and (ii) u! jr'(u)j2 jurg(u)j + c 1 when < juj 1: Here and henceforth c i represent positive constants. (G3) There exist constants c 2 c 3 such that (2.3) u rg(u) c 2 juj 2 + c 3 8u 2 R N nfg
3 PERIODIC SOLUTIONS OF SINGULAR EQUATIONS 43 and c 2 satises (2.4) c 2 < 2 =T 2 : Theorem 1. Besides the conditions (G1), (G2) and (G3), we assume further that the following (G4) and (G5) are satised: (G4) There exists a constant R 1 > such that, for any solution u() of the problem (2.5) (2.6) u + ; (rf (u)) + rg(u) = h(t) 2 ( 1] u() ; u(t )=u () ; u (T )= one has ju()j <R 1 for some 2 [ T]. (G5) The topological degree deg(rg r R h) 6= for all <r 1 and some suciently large R 1, where r R = fx 2 R N : r< jxj <Rg. Then (2.1){(2.2) has at least one solution. Proof. The conditions (G4) and (G5) are natural from the viewpoint of coincidence degree. (G4) and (G5) are (or implicitly are) assumed in previous works [4], [5]. For more details on our assumptions, see the next section. The proof is based on the following result of Mawhin [8] from coincidence degree theory (see also Mawhin [9]): Let f = f(t x y) :[ T] R R N N! R N beanl 1 -Caratheodory function. Let R N R N beanopen bounded set such that (i) for any solution x of one has (P ) ( x = f(t x x ) 2 ( 1] x() ; x(t )=x () ; x (T )= (x(t) x (t)) 2 8t 2 [ T] =) (x(t) x (t)) 2 8t 2 [ T]: (ii) deg( f \ (R N fg) ) 6= where f(x) = 1 T f(t x )dt: Then the problem (P ) 1 has at least one solution x() such that (x(t) x (t)) 2 for all t 2 [ T]. So the theorem is proved by obtaining a priori bounds for all solutions of (2.5){ (2.6). Let u() be a solution of (2.5){(2.6) for some 2 ( 1]. The estimates for the C 1 -norm of u() are completed in several steps. Step 1. Upper bounds for kuk 1 and ku k 2 follow from (G3) and (G4). Multiplying (2.5) with u, we obtain (2.7) Note that the equalities ;u u = u (rf (u)) + u rg(u) ; h u: ;u u = u u ; d dt u u a.e. t 2 [ T]
4 44 MEIRONG ZHANG u (rf (u)) = d (u rf (u) ; F (u)) dt hold. We get,byintegrating (2.7) over [ T] and using the boundary conditions (2.6), (2.8) Z ku T k 2 2 = u rg(u)dt ; h udt (c 2 juj 2 + c 3 )dt + khk 1 kuk 1 = (c 2 kuk khk 1kuk 1 + Tc 3 ) where (2.3) is used. On the one hand, by (G4) and the mean value theorem, we have for all t. Thus ju(t)j = Z t u()+ u (s)ds R 1 + p T ku k 2 (2.9) kuk 1 R 1 + p T ku k 2 : On the other hand, we periodically extend the domain of u() to the whole R and dene v(t) =u(t + ) ; u() where is as in (G4). Then v 2 H 1 ( T) satises the Dirichlet boundary condition Thus which, in turn, implies kvk 2 2 T 2 kuk 2 2 = ku()+vk2 2 v() = v(t )=: 2 kv k 2 2 = T 2 2 ku k 2 2 pt ju()j + kvk2 2 TR p T TR 1 ku k 2 + T 2 (2.1) 2 ku k 2 2: Under the assumption (2.4), it follows from (2.8), (2.9) and (2.1) that there exist some positive constants R 2 R 3 (independent ofu() andof) such that (2.11) (2.12) kuk 1 <R 2 ku k 2 <R 3 : Step 2. Lower bounds for m =minfju(t)j : t 2 [ T]g follow from (G1), (G2) and Step 1. As we assume that (G1) and the strong force condition (G2) hold, the lower bounds for m follow from Lemma 6 of Habets and Sanchez [5] by noticing the estimate (2.11) and the estimate (2.12) (with the factor ). Namely, there is some R 4 > (independent ofu and of ) such that (2.13) minfju(t)j : t 2 [ T]g >R 4 : Step 3. Upper bounds for ku k 1 follow from Steps 1 and 2.
5 PERIODIC SOLUTIONS OF SINGULAR EQUATIONS 45 By (2.11) and (2.13), we have R 4 < ju(t)j <R 2 8t 2 [ T]: Thus there is some c 4 > such that, for all t 2 [ T], and Now we use (2.5) to get jg(u(t))jc 4 j(rf (u(t))) j 2 (u(t))u (t) c 4ju (t)j: ku k 1 = ; (rf (u)) + rg(u) ; h(t) dt (c 4 ju j + c 4 + jh(t)j)dt = c 4 ku k 1 + Tc 4 + khk 1 < p TR 3 c 4 + Tc 4 + khk 1 =: R 5 : For each i =1 ::: N, there exists some t i 2 ( T)suchthatu (t i i)=because u i () = u i (T ). Thus, for any t 2 [ T], Therefore, (2.14) ju i(t)j = ku k 1 = max t2[ T ] Z t u i (s)ds t i ku i k 1 <R 5 : NX i=1! 2 1 ju i(t)j 2 < p NR 5 =: R 6 : Step 4. The existence result follows from (G5) and Steps 1{3. For the completion of the proof, we need only take the domain in Mawhin's result, noting the estimates (2.11), (2.13) and (2.14), as =f(x y) 2 R N R N : R 4 < jxj <R 2 jyj <R 6 g because the degree condition (G5) is assumed. Remarks. 1. When N = 1, a dierent estimating routine is given in [4], [9], [11] because the solutions u() do not cross the singularity. In this case, the strong force condition (G2) is unnecessary. 2. Theorem 1 says that if G(u) satises some strong force condition at the singularity, the existence of periodic solutions to Eq. (2.1) can be obtained provided that the potential G(u) is smaller than the rst eigenvalue of the corresponding Dirichlet problem at innity. Meanwhile, no restriction on the damping term F (u) is imposed. Such a result may compare with the result of Majer [7] for undamped dierential systems with attractive singular forces: (2.15) u + rg(u) =h(t): He used Ljusternik-Schnirelmann theory to obtain the existence of periodic solutions to Eq. (2.15) when G(u) satises some strong force condition at the singularity and G(u) is smaller than the rst eigenvalue of the Dirichlet problem at innity.
6 46 MEIRONG ZHANG 3. Concluding remarks In this section we will give a discussion on our assumptions (G1){(G5) and derive some results from Theorem The condition (G1) implies, by a conventional trick, that, for any m> one has G(u) >c 5 ; m log juj < juj1: Thus, lim u! G(u) =+1, which means that G(u) is of repulsive type. 2. Condition (G3) is concerned with the growth of G(u) at innity, because (G1) implies that (2.3) holds when juj is small. 3. The potentials G(u) =juj ; satisfy (G1) and (G3) for all >, while Habets-Sanchez's strong force condition (G2) holds only when (G4) can be implied by each of the following conditions (independent of F ): (G4) There is some constant R 7 > such that, for any C 1 T -periodic function u satisfying (3.1) one has (3.2) Z 1 T rg(u(t))dt = h T ju()j <R 7 for some 2 [ T]: (G4) There is some constant R 7 > suchthat (3.3) jrg(u)j < j hj 8jujR 7 : It is obvious that (G4) implies (G4). Assume now that (G4) holds. Then, for any solution u to (2.5){(2.6), the integration of (2.5) over [ T] yields (3.1). So the condition (G4) holds by (3.2). Under the condition (G4),wehave the following result as a corollary of Theorem 1. Theorem 2. Assume that G and h satisfy (G1), (G2) and (G4). Then (2.1){(2.2) has at least one solution. Proof. (G1) and (G4) imply that (G3) holds for any <c 2 1. On the other hand, the degree condition (G5) can be obtained by introducing the following homotopy rg(u)+(1; )rg (u) 2 [ 1] where G (u) =juj ;, >. As h 6= by (3.3), we can compute that deg(rg r R h) 6= : Remarks. 1. Theorem 2 shows that the assumption (A-1), i.e., the dissipative condition (1.3) or (1.4), in [5, Theorem 1] for the repulsive case is unnecessary and h 2 L 1 is also allowed. 2. When F (u), Theorem 2 is just Theorem 3 in [5].
7 PERIODIC SOLUTIONS OF SINGULAR EQUATIONS 47 References [1] A. Ambrosetti, Critical Points and Nonlinear Variational Problems, Mem. Soc. Math. France, 49 (1992). MR 93g:5824 [2] V. Coti Zelati, Dynamical systems with eective-like potentials, Nonlinear Anal. 12 (1988), 29{222. MR 89c:5834 [3] W. B. Gordon, Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc. 24 (1975), 113{135. MR 51:14152 [4] P. Habets and L. Sanchez, Periodic solutions of some Lienard equations with singularities, Proc. Amer. Math. Soc. 19 (199), 135{144. MR 9k:3449 [5] P. Habets and L. Sanchez, Periodic solutions of dissipative dynamical systems with singular potentials, Dierential Integral Equations 3 (199), 1139{1149. MR92a:344 [6] A. C. Lazer and S. Solimini, On periodic solutions of nonlinear dierential equations with singularities, Proc. Amer. Math. Soc. 99 (1987), 19{114. MR 87k:3464 [7] P. Majer, Ljusternik-Schnirelmann theory with local Palais-Smale conditions and singular dynamical systems, Ann. Inst. H. Poincare Anal. Non Lineaire 8 (1991), 459{476. MR 93e:5824 [8] J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS{ Regional Conf. Math. no. 4, Amer. Math. Soc., Providence, RI, MR 8c:4755 [9] J. Mawhin, Topological degree andboundary value problems for nonlinear dierential equations, intopological Methods for Ordinary Dierential Equations (M. Furi and P. Zecca, Eds.), pp. 74{142, Lecture Notes in Math. vol 1537, Springer-Verlag, New York/Berlin, MR 94h:47121 [1] S. Solimini, On forced dynamical systems with a singularity of repulsive type, Nonlinear Anal. 14 (199), 489{5. MR 91d:5841 [11] M. Zhang, Periodic solutions of Lienard equations with singular forces of repulsive type, J. Math. Anal. Appl. 23 (1996), 254{269. MR 97m:3479 Department of Applied Mathematics, Tsinghua University, Beijing 184, People's Republic of China address: mzhang@math.tsinghua.edu.cn
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