Nonlinear multiparameter Sturm-Liouville problems

JOURNAL OF DIFFERENTIAL Nonlinear EQUATIONS 34, 139-146 (1979) Multiparameter Sturm-Liouville PATRICK J. BROWNE* Department B. D. AND Problems SLEEMAN of Mathematics, University of Dundee, Dundee, DDI 4HN, Scotland Received June 6, 1978 We study a linked system of nonlinear Stunt-Liouville equations in which the linking occurs via the spectral parameters. The system is the multiparameter analog of an equation recentiy discussed by R. E. L. Turner. We present an existence theorem for continua of solutions of the system-the solutions being required to have specified nodal properties. 1. INTRODUCTION Many authors have recently investigated nonlinear Sturm-Liouville problems of various types and have given theorems on existence of solutions with certain nodal properties, on bifurcation phenomena, and on completeness of eigenfunctions. Of the many possible references we cite in particular the work of Brown [4], Crandall and Rabinowitz [7], Hartman [9], Rabinowitz [lo], and Turner [12, 131. It is natural to study the analogs of these results for a multiparameter Sturm-Liouville system. To this end, theorems on completeness of eigenfunctions have been given by Browne [5] and Browne and Sleeman [a. Here our motivation is the work of Turner [ 121, and following his lead we use degree theory to generate results concerning solutions with prescribed nodal properties of a nonlinear multiparameter problem. 2. THE NONLINEAR PROBLEM The multiparameter problem to be discussed takes the form -(Pl.(ur %%(O) + W(O) 9 1) 4)’ + Qr(ur , A) + mu, = &+(4 + &4(r) = 0, 1 A) = 0, l<Y<k. (1) (2) Here h = (A1 ,..., A,) E I@’ is a spectral parameter, ur = u,(x,) E C2[0, x], and the boundary conditions (2) are real and nontrivial. Primes denote differentiation * Research supported in part by National Research Council of Canada Grant A9073. 139 0022-0396/79/100139-08$02.00/0 Copyright Q 1979 by Academic Press, Inc. AlI rights of reproduction in any form reserved. 140 BROWNE AND SLREMAN P, , Qr, F, are to take the following with respect to X, E [0, ~1. The functions forms: (i) P& , k) = p,k) + P,‘(u, , h), where (4 P, E CYO, ml, p,(w,) > 0, (b) P,l: C1 x R - Cl is continuous sets to bounded sets), and bounded (i.e., takes bounded (c) P,1(0, A) = 0, (d) for each bounded set in Cr x R”, there exists 6, > 0 such that P,1 > 6, (as a function of X, E [0, ~1) for (q , A) from the given bounded set; QT = (ii) q&d 4x,) + p71(xr , u, ,4, A) + Qrl(ur , G, , where (4 qTECOP,4, (b) qrl: [0, ~1 x R x R x BP + R is continuous on bounded o((5” + 7*Y*) for (6 d near (0,O) uniformly (c) Qr’: Cl x UP - Co is a bounded continuous (4 (iii) QrYO,A) and q,.l(~r , B,T, h) = A-sets, map, = 0; F, = Gaul ~,(~&r)~r(4 + ff,b , u, , 4 , A) + Fi,(u, , ~)W,)) where (4 (b) G* Ecop, 4 f and deth&)> > 0, of the same type as qrl and QI1 above. :S , F:, are functions (3) This is the direct multiparameter analog of the problem considered by Turner [12] and covers a wide range of possibilities. We introduce some notation. E, will denote the collection of functions II, E Cl[O, V] satisfying the boundary conditions (2). This space will be normed by The space R? iI@, u)ll x = El x /I(& ,.-*, e’. x E, will be denoted hk > % ,..-, uk)ll = 11 x 11 + by d and will I/ % /iI + *" be normed + 11 uk by 111 . For a K-tuple of sign factors (I = (q ,..., ak) where each IJ, = f 1 and for a multi-index n = (n, ,..., nk) where each n, is a nonnegative integer, we define Sr(n, o) = {Us E E, ] U, has tl, nodal zeros in (0,~) and U,U, > 0 in a right neighborhood Further, we set ,y(n, u) = f%” x &(n, u) x ‘-’ x Sk(n, u) of x, = O}. NONLINEAR MULTIPARAMETER PROBLEMS 141 and by 9’ we mean the closure in 6’ of the set of nontrivial solutions of (l)trivial solutions being of the form (A, ur ,..., uk) with at least one u, = 0. The linear version of (1) is the by now familiar multiparameter problem subject to boundary conditions (2) and the definiteness hypothesis (3). This problem has a sequence of simple eigenvalues A~ E IFP indexed by the multiindex n in such a way that the corresponding eigenfunctions U&X,) have 71, nodal zeros in (0, m) and are positive (or negative) near X, = 0, 1 < r < k. For details of these results we refer the reader to Binding and Browne [2], Faierman [8], and Sleeman [ll]. 3. LOCAL BIFURCATION LEMMA I. Let Z,(u,) bea linearfunctional dejned on Cl by Z,(u,) = c+,(O) + j?&(O) whereC& - pTar # 0. Considerthe equations -(P,G:) + Q&r , 3 + F&r , A) = 0, ur EEr A C’[O,4 forO<t<l, (4) where p,t = P, + tP,l, Qrt = wr + tqrl + tQ,lu, , F,, = t ~s(a,su,+ tf’,, + t&4. S4 Thenfor any boundedset0 C d there is a constantpr suchthat [I zi, 11,( p,. 1Z,(22,.)/ whenever(h, 22,,..., i&J is a solutionof (4) Zyingin 0. PYOOf. If (A, 22, ,..., z&) E 0 is a solution of (4) for some t, we evaluate P,l, (2’ and Ff8 at (k, zi, ,.,., 22,) to obtain functions of x, lying in fixed bounded srkk.ets of Cl, Co, Co, respectively-these fixed sets arising from the choice of 0 according to the properties of PT1,QT1, and F:, . Further, as in [12, Lemma 2.11 there exists 6, so that p,(x,) + tP:(i& , A) >, 6, independently of the t occurring. The rest of the proof mimics that of [12, Lemma 2.11. Note that only the trivial solution of (4) can have a double zero. For w, E E, with w: absolutely continuous we define in the usual way the selfadjoint operators L,ow, = -(P&q + q,w, 9 142 BROWNE AND SLEEMAN and for a given (x, z+ ,..., ~3 E 8, L,lw, = -(P>(+ If 0 C d is an open bounded set containing CTl= , h)(Xr) w;)‘. (A”, O,..., 0) we let inf min (L,ow, + tL71w, ’ w,) o.,U)EO WI (w, 9 4 OSW where (a, a) denotes the inner product in L2([03vr]). Since P,] is a bounded map, crl > --cx). Weselect(c, ,..., ck) E R? so thatxi=, a,,(x,)c, + c,.l > 1, r = l,..., K. That this is possible follows from results of Atkinson [l]-see also Binding and Browne [3]. Then (LTo + CL, a,.,~, + tL,l)-l will exist for any (X, u) E 0 and any t E [0, l] as a bounded map in L?([O, ~1). Further, 8 1 u,,c,)-~)-~ will have a uniform bound for t E [0, 11. so that (I + tL,.l(L,.O + xk= We now add the quantity --Ci=, ars(x,) c,u, to each side of (4) and use the operator G,’ = Gt’(u, , x)--the Green’s o erator associated with the differential f operator given for w+.E E, by -(L,O + X8=1 a,.,c, + tL,.‘), i.e., by ((P,W + tP71@r 9 G4) 4 + (--~7~(4 - i arsW cs) wT - S=l This enables us to rewrite (4) as II, = Gt’ F,, -j- &” - ; ar8cLPr 9 1 ( s-1 (5) where &rt = t(q,’ + Q,‘). (L,O + & a,,~,)-l is a compact map of L* into E, and so we can construct a continuous map of 0 x [0, l] into a compact subset of El x ... x Ek by using for each coordinate function the expression on the right-hand side of (5) for r = I,..., k. Now fix a multi-index n and a sign factor Q. Define A.(n, =, 4 = infIll u, II1 I u, E Wn, 4 and (x, u,) is a solution of (4) for some t E [0, l]}, NONLINEAR MULTIPARAMETER PROBLEMS 143 for 1 < r < k. For definiteness we take inf 4 = co. We claim that each /$(n, Q, A) is positive and lower semicontinuous on EP - {An}. To see this let (ti , pi, uli ,..., uki), i = 1, 2 ,..., be a sequence in [0, l] x (UP - {An}) x &(n, 4 x ..* x S,(n, o) such that pi--t )i # hn, 11uf II1 < ,8,(n, o, $) + l/i, and each (ti , pi, uIi ,..., uki) satisfies (4). We aim to prove &.(n, 0, A) < h$f B,(n, 0, 14 l<r<R. We can take a subsequence and assume that as i --) co, lim Pr(n, Q, pLi) exists and is finite. From (5) we obtain 4424 i 1 so that uri lies in a compact set in E, and so again passing to a subsequence necessary we may assume ~4,~-+ 24,E S,(n, o), (6) if tj + t E [O, 11. Then passing to the limit in (6) we obtain u, = G,’ 1 a,sc,u, , s=1 i.e., we have shown that (A, u) is a solution of (4). Now and so the result follows. To see that ,$.(n, Q, A) > 0 we take pi = h, i = 1, 2 ,..., and note that the above analysis could be repeated after first dividing both sides of (6) by 11uri jlr . We then obtain that u,i/ll uf II1 is convergent to say a, # 0. However, if 11u,i Ii1 -+ 0, then the limiting version of (6) would give which leads to which is impossible since X f An. The argument presented here is based on the proof of [7, Lemma 2.151 and leads to the fact that the nonlinear problem (1) can have solutions (x, u) in Y(n, o) with Ii I(,. /II small only if X is close to A’. 144 BROWNE AND SLEEMAN THEOREM 1. Let n be a multi-index and Q a S;gn factor k-tuple. any open bounded set in 8 containing (An, O,..., 0) we have Then if 0 is Here 20 denotes the boundary of p. Proof. Let I,, 1 < T ,< k, be the linear functionals that E,(u,) > 0 on eigenvectors u,. of the linear problem of Lemma 1 so chosen in S,(n, u). Set cq = ((A,u) E8 j I&,.) = 7, 1 < Y,( k}. &7,, is a subspace of d with finite codimension. By taking 7 > 0 sufficiently small we can ensure that none of the problems (5) for 0 < t < 1 has a solution in g,, n a0 n Y(n, a). To see this, note that a solution (A, u) E i%Vn -9’(n, KS)with each 1,(q) = 7 small must be such that each I/ U, II1 is small-this follows from Lemma 1. By our remarks above A can be made as close to A” as we wish by taking each /I u,. //r sufficiently small. On the other hand, however, (h, u) cannot be arbitrarily close to (An, 0) since (h, II) is supposed to be in X’. For the eigenvalue Xn of the linear problem let (uIo,..., +O) E Sr(n, o) x .a. x S,(n, o) be the corresponding eigenvectors with &.(u,O) = 1. We now define maps !P&): So--f B. (0 < t < 1) in stages as follows. First we map (h, u) E go to (A, u”) E a,, by defining urn = u,. + q1c,O.Now (h, LP) is mapped to wn, where w,.” = u,” - Gtr(u,n, n) F,.&,.~, A) + &‘(u and finally we map this point to (l,(w,q,..., zk(w,q, Win - z~(wl~)z410,...,Wkn - z~(wm)u,o) E %J . A zero of the map Yt(7) corresponds to a solution (A, u) of (6) with Z,(u,) = 7. We shall apply Yt(v) to points (h, IP) in gn = {(A, u) E LB0/ (A, un) E 0 n Y(n, 0)). Note that (h, u”) E gn . We take 17 > 0 so that (6) has no solution in a,, n 30 n 9(n, IS) for any t E [0, I]. Any point u, E AS,(n, (I) must have a double zero and so we have no nontrivial solutions of (6) in a,, n 0 n S(n, 0). Then working within the Banach space go we see that Y,(q) has no zeros on 29,, (the boundary to be taken with respect to go), 0 < t < 1. Then using the homotopic invariance of degree (with respect to the zero vector in go) we have d&y1 The map Y, is derived ,q,,) = deg(yo, from the linearized p,,). problem and has (h’, 0) as its only NONLINEAR zero in 9, . The derivative MULTIPARAMETER 145 PROBLEMS of Y,, at (h”, 0) takes (I*, z1 ,..., ZJ E a,, to the point (e4lS,Sk ; (4 + &4) %%3sr)~ where This map is of the form “identity plus compact” so we may check its nonsingularity by showing that (0, 0) is its only null vector. Thus if each zui = 0, 1 ,( Y < k, we obtain (p,.q’ - q,z, - and after forming which implies i a,.,c,z,. s=1 f (h,” - c,) UrsZ, + q 2 ~,U&.O A-=1 the inner product of each side with u,O we see that TV = 0 since det((a,,zl,O, -(p&c)’ = 0, s=~l + q,z, + f &%A. ~,.a)} + 0. Now it follows that = 0, 1 <r<k, 5=1 so that Z, = /$ur6 for some constants p,. . However, lr(z,) = 0 since (IL, z) E go . Thus fl, = 0 and hence z,. = 0, 1 < I’ < k. We now claim that deg(Y, , S’,,) # 0 and so by the homotopic invariance of degree, deg(Y/, ,9,) # 0. Consider deg(Y,(t), gE) where 6 E [q, 001. If we assume that Y n X’ n 9(n, o) = P then Y,(t) will have no zero on 29, for PE) any 5 3 7. In that case it follows from degree theory that deg(Yi([), will be constant in 5. However for [ large enough, 9”E = 0 and so deg(Y/,([), 9”c) = O-a contradiction. This proves the theorem. COROLLARY. The problem connecting (Xn, 0) to 00. (I), (2) has a continuum of solutions (A, u) E .9(n, cr) Proof. Let 6 C L be an open bounded set containing (A”, 0). Then K = Y n 6 n P(n, a) is a compact set in g-this depends essentially on the compactness of the maps Gir-see (5) and the remarks immediately following. Further, if A = {(I’, 0)) then our theorem implies A C K. Set B = X n K. Then A and B are disjoint closed subsets of K. If there is no subcontinuum of K joining A and B then we can find disjoint compact sets KA , K, containing d, B, respectively, so that K = KA u KB ([ll, Lemma 1.331, c.f., [14, Chap. I]). K.4 , K, are compact subsets of F so we may find an open neighborhood .Q of K.4 for which Q C 6, .Q n FP = J, and 8 n KS == cr. Now (In, 0) E Q, and 146 BROWNE AND SLEEMAN thus it follows from the theorem that there is a solution (h, u) E Y n X2 n 9(n, 0). Hence (A, u) E K but not in KA u KB . This contradiction shows that there is a continuum of solutions connecting (X0, 0) to FQ in .Y n -Y(n, 0); but @ is arbitrary and so our result is established. REFERENCES 1. F. V. ATKINSON, “Multiparameter Eigenvalue Problems,” Vol. I: “Matrices and Compact Operators,” Academic Press, New York, 1972. 2. P. A. BINDING AND P. J. BROWNE, A variational approach to multiparameter eigenvalue problems in hilbert space, SZ.4M J. Math. Anal. 9 (1978), 1054-1067. 3. P. A. BINDING AND P. J. 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RABINOWITZ, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Comm. Pure -4~~1. Math. 23 (1970), 939-961. 11, B. D. SLEEMAN, Left-definite multiparameter eigenvalue problems, in “Proceedings, Symposium on Spectral Theory and Differential Equations,” pp. 307-321, Lecture Notes in Mathematics, No. 488, Springer-Verlag, Berlin, 1974. 12. R. E. L. TURNER, Nonlinear Sturm-Liouville problems, J. Differential Equations 10 (1971). 141-146. 13. R. E. L. TL~RNER, Superlinear (1973), 157-171. 14. G. T. WHYBURN, 1964. View publication stats “Topological Sturm-Liouville Analysis,” problems, Princeton J. Di’erential Univ. Equations 13 Press, Princeton, N. J.,