Stuart, C. A.

Toland, J. F.

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A PROPERTY OF SOLUTIONS OFELLIPTIC DIFFERENTIAL EQUATIONS WITHDISCONTINUOUS NONLINEARITIESC. A. STUART AND J. F. TOLAND1. IntroductionLet Q be a bounded domain in U" which has a boundary dQ. of class C2 if n ^ 2 oran open interval when n = 1. We shall follow the notation of [1] for spaces of real-valued functions on Q.Let L denote the linear, second order uniformly elliptic operator (see [1] fordefinition) which is formally defined byLu(x) = - X * 0 M ^ P + Z « , - (* )^ +a(x)u(x)i,7= l OXiOXj i = l OXiwhere au e C0>1(Q) and atj = ajh at e L°°(fi) (1 ^ i,j ^ n) and a e L°°(Q).Let BN denote the set of all functions / : U -• U which have bounded variation oncompact intervals and which have the property that f(t) lies between f{t + ) and f(t —)for aWteU. The functions in BN need not be continuous but they can only suffer pointsof jump discontinuity. It is convenient to introduce the following notation. If/ e BN,= {t€U:f(t-)<f(t + )},In this paper we are concerned with solutions u of the Dirichlet problemLu(x) = f{u(xj) + p(x), xeQ )(11)u{x) = 0, xedQ }when p e £(Q), fe BN. However as in earlier work[2,3,4] we shall replace (1.1) by theless specific problem (1.2) below for which an adequate existence theory obtains:Lu{x) G/(M(X)) 4- p(x) a.e. in Q \(1-2)u e W2'r(Q) n W^'2(fi). JHere( {f{t)} if t*D(f)f(t) ={ (t-)-] if te D(f).Received 5 February, 1979.[J. LONDON MATH. SOC. (2), 21 (1980), 329-335]330 C. A. STUART AND J. F. TOLANDWe will only consider the case where r ^ 2. Let us recall that W2'"{Q) £ C°(Q) ifr > n/2 and that in this case.W2'r{Q) n Wj'r(Q) = {ue W2'r{Q): u{x) = 0 for x e dfy .This paper is a sequel to [3] where we used a variational approach to study (1.2) inthe case where L = — A and r = 2. We showed that by assuming certain growthrestrictions on / it is possible to prove the existence of a solution u* of (1.2) which is inW2<2(Q) n Wl'2(Q) n L°°(Q) and which minimises the extended-real-valued functionalassociated with (1.1), (1.2), namely:= {i\Vu(x)\2-F(u(x))-p(x)u(x)}dxwhere F is a primitive of / (see [3] for further details).This minimiser u* was seen to have the rather unexpected property thatm{xeQ:u*{x)eU{f)} = 0 (*)where m denotes n-dimensional Lebesgue measure.In this paper we are specifically concerned with elaborating circumstances underwhich we can assert that a solution of (1.2) enjoys property (*). In Section 2 we give anexample which shows that (*) is not a general property of solutions of (1.2).The existence question for (1.2) has been solved using the theory of increasing mapsin an ordered Banach space [2,4] and we may ask whether this method invariablyyields solutions which satisfy (*). In [2,4] it is shown that problem (1.2) has a greatestand at least solution in [0 , ^] = {u e W2''{ti): 0 ^ u ^ \j/} provided that ^ and(j) e W2r{Cl) are super- and sub-solutions respectively, and r > n/2.In general it is not true that these greatest and least solutions enjoy (*), butnonetheless we shall show that only in exceptional (and in a sense degenerate)circumstances do they fail to have this property. As a consequence it will follow that inmany cases the existence of a solution failing to satisfy (*) ensures the existence ofseveral other solutions which do have property (*).We illustrate this idea in Section 3 with the following special case:(1.3)u(0) = u(l) = 0where fe BN. Under quite general conditions on / we will show that (1.3) has aninfinite number of distinct solutions each of which enjoys property (*).2. Solutions having property (*)Throughout this section we suppose that / e B N and that peE(Q) for somer > n/2. An element (f) e W2-r(Q) is called a sub-solution of (1.2) ifL(f){x) ^ max {t: t e/(0(x))} +p(x) a.e. on Q,(f>(x) ^ 0 on dQ,A PROPERTY OF SOLUTIONS OF ELLIPTIC DIFFERENTIAL EQUATIONS 3 3 1and i/f e W2r{Q) is called a super-solution of (1.2) ifL\j/(x) ^  min {t: t e/(^(x))} + p(x) a.e. on Q\Jj(x) ^  0 on aO.Clearly any solution of (1.2) is both a sub- and a super-solution of (1.2). In [4] thefollowing result is established.THEOREM 2.1. Suppose that (1.2) has a sub-solution $ and a super-solutionif/ in W2'r(Q) such that <j> ^ if/ on Q. Then, in the order interval[0} ^ ] = {w G W2r(Q): (f) ^ u ^ ifr}, the problem (1.2) has a greatest solution u anda least solution u.In the following example it is clear that such greatest and least solutions u, u, neednot have property (*).Example. Consider the special case of (1.2) with L = — A, p = 0 and f = fe where- 1 if t < 0 ,fit) = e if t = 0 ,+ 1 if t > 0 .If 8e [ - 1 , 1 ] , then/ £ eBN.By the arguments of [2, 3, 4] there exist solutions of (1.2) and by [3] at least onewhich has property (*). Moreover it is easy to see that for all e e [ — 1,1] there exist twosolutions u and u such that u{x) < 0 < u(x) for all x e Q. (Let u be the unique solutionof the problem — Au(x) = 1 in Q, u = 0 on dCl, and let u = —u. The maximumprinciple says that u > 0 > u on Q and so u and u solve (1.2) for any e G [-1,1] .)However in studying the qualitative features of solutions of (1.2) we mustdistinguish two cases.I. e = 0. If e = 0, then u = 0 is a solution of (1.2) which does not satisfy (*). If(f> = ij/ = 0 are respectively sub- and super-solutions, then u = u = 0 are respectivelythe greatest and least solutions of (1.2) in [0, ij/'] in this case.II. e =^ 0. Ife =^ 0, then every solution of (1.2) has property (*). To see this we simplynote that Au(x) = 0 a.e. on {x e Q: u(x) = 0} if u G W2'r(Q) (Lemma 7.7 of [1]).It is clear that this example displays many features which are typical of the moregeneral situation, and we are lead to make the following definition.A function / , e BN is called a selection of / G BN iffi(t)' =f{t) foraUte[/(r-),/(r + )] iovteU(f).Clearly Jx{t) = J{t) for t^U(f).332 C. A. STUART AND J. F. TOLANDA function fe BN is said to be non-critical with respect to p e L2(Q) ifm{xeQ:a(x)t =f{t) + p{x)} = 0for all t G U(f). (Recall that a is the lowest order coefficient in the expression for L.)The significance of these definitions lies in the next result.LEMMA 2.2. Suppose thatf e BN is non-critical with respect tops E(Q). Then everysolution o/(1.2) has property (*).Proof. Let u be a solution of (1.2). Then for each t e U,-— and -—-— = 0 a.e. onOXf OXiOXj{xefl : u{x) = t], for 1 ^ i,j ^ n, [1; Lemma 7.7]. Hencea(x)t = f(t) + p(x) a.e. on {x e Q : u(x) = t} .Since / is non-critical with respect to p, m{x e ft: u(x) = t] = 0 if t e (/(/), and since[/(/) is denumerable,ffi{xefi: u(x) G t/(/)} = 0 .This completes the proof of the lemma.THEOREM 2.3. Let the hypotheses of Theorem 2.1 hold.(a) Suppose that f has a selection fx such that(1) /x is non-critical with respect to p,(2)fl(t)^f(t)forallteU(f),(3) if/ is a super-solution of (1.2) for fx.Then u (of Theorem 2.1) has property (*) (and consequently is a solution of(\.2)for anyselection off). Furthermore in [0, i//], u is the greatest solution of (1.2) for any selectionof f which satisfies (1), (2), (3) above.(b) Suppose that f has a selection f2 such that(4) f2 is non-critical with respect to p,(5)f2(t)^f(t)foralltsU(f),(6) 0 is a sub-solution of (1.2) for f2.Then u has property (*) and in [0, i/f] is the least solution of (1.2) for any selectionwhich satisfies (4), (5), (6) above.Remark 1. A sub-(super-) solution (f>(\p) of (1.2) for any selection of / is a sub-(super-) solution of (1.2) for any other selection of/ provided (f}(\jj) satisfies (*). Inapplications this is often the case and (3) and (6) do not restrict the applicability of thisresult.Remark 2. The question of whether a selection of / is critical or not is easilyresolved in the case where p and a are both constant functions on Q. Then / has a non-A PROPERTY OF SOLUTIONS OF ELLIPTIC DIFFERENTIAL EQUATIONS 333critical selection fl{f2) with f{ ^ f{f2 ^ / ) provided that when teU{f) andf(t) - at-p thenat-p<f{t + ) {at-p> f{t-)).Proof of Theorem 2.3. We need only prove part (a), since the proof of (b) is similar.Clearly u (of Theorem 2.1) is a sub-solution for fx, and so, by Theorem 2.1 thereexists in [«, i/'] a greatest solution v and a least solution v for (1.2) with / , . By theassumption that j \ is non-critical with respect to p it follows from Lemma 2.2 that bothv and v satisfy (*). Hence v and v solve (1.2) for / . But in \_(j), i//], u is the greatestsolution of (1.2) with / and xjj^-v^-v^-u^fy. Therefore v = v = u, and hence usatisfies (*) and is the greatest solution of (1.2) for fY in [</>, i//].As a consequence, if/ has a non-critical selection for p, then one can infer a non-uniqueness result from a priori knowledge of the existence of a solution for which (*)fails to hold.In the next section we examine the implications of this observation in the case of anautonomous ordinary differential equation where its consequences are most striking.3. An autonomous ordinary differential equationWe consider a one-dimensional version of (1.2), namely-u"(x)e/(u(x))a.e. on [0,1], )(3-1)u(0) = u(l) = 0, u E W2'2(l0,1]), jwhere / e BN. Since all solutions of (3.1) are continuously differentiable on [0,1] and /is bounded on bounded sets, all solutions are in W2co([0, 1]). Our main result is that if/ is odd and 0 e U(f), then (3.1) may have an infinite number of distinct solutions.THEOREM 3.1. Let / e BN be such that0) OeU(f),(ii) f(t) = -J(-t)for allteU, and(iii) either(a) f{t + ) > Ofor all t e U{f) such that f(t) = 0, and (3.1) has a super-solutioni// > 0 which satisfies (*), or(b) f{t-) < Ofor all t e £/(/) such that f{t) = 0 and (3.1) has a sub-solution0 < 0 which satisfies (*).Then (3.1) has an infinite number of distinct solutions, each of which has property (*).COROLLARY. If fe BN is odd, 0 e U{f) and lim sup (f(t)/t) < n2 then (3.1) has an«-»ooinfinite number of distinct solutions with property (*) if f(t + ) > Ofor all t e U(f) suchthat f{t) = 0 {or iff(t-) < Ofor all t e U(f) such that f{t) = 0).334 C. A. STUART AND J. F. TOLANDProof of Theorem 3.1. Suppose that (i), (ii), and (iii)(a) hold. Since0 e U{f), 7(0) = {/(0)}, and so /(0) = 0 (by (ii)). Hence u = 0 is a solution of (3.1),which does not have property (*).By Theorem 2.3 the greatest solution u of (3.1) in [0, ^ ] has property (*).Let \j/2(x) = u(x) for x e [0, j]. Then i/f2 is a super-solution for the problem:-u"(x)e/(u(x))a.e. on [0,£], \(3.2)JSince i)/2 has property (*), the greatest solution u2 of problem (3.2) in [0, ^ 2 ] hasproperty (*) (by Theorem 2.3). Letu2(x) i f O ^ xW2(X) =[ - u 2 ( l - x ) i f i < xThen w2 e C^EO, 1]) and w2(0) = w2(|) = w2(l) = 0.Furthermore, for any h e C^([0,1]),'2(x)h'(x)dx = u2(x)h\x)dx+ \ u'2(l-x)h\x)dxJ0 0 i= - u'2'(x)h{x)dx+ u'2'(l-x)h{x)dx.0 iThus w'2' e Lw([0,1]) and-w2'(x) = -u2ej{u2{x)) =/(w2(x)) if 0 ^ x ^ i-w2'(x) = M ' 2 ( 1 - X ) 6 -J(u2{l-x)) =/(w2(x)) if f ^ x < 1.Thus we see that w2 is a solution of (3.1) which enjoys property (*). Furthermore w2 ^ usince vv2 < 0 on [y, 1]. We can now use w2 as a super-solution on [0, £] of(3-3)We then use the oddness of / to generate a new solution of (3.1) on [0,1].It is clear that, by continuing in this way, the result may be established by induction.The proof when (i), (ii) and (iii)(b) hold is similar.Proof of Corollary. This follows from the theorem once it is shown that the sub-A PROPERTY OF SOLUTIONS OF ELLIPTIC DIFFERENTIAL EQUATIONS 335and super- solutions (j) and \j/ exist. To see this let a, V be such that f(t) ^ (n2 -a)t + Vfor alH ^ 0, and let \j/ be the solution of the problem:- u"{x) = {n2 - a)u(x) + T for x e (0,1),M(0) = M(l) = 0 .Then if/ > 0 on (0, 1) and t]/ has property (*). Put <f> = — \\i.References1. D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order (Springer,Berlin, 1977).2. C. A. Stuart, "Maximal and minimal solutions of elliptic differential equations with discontinuous non-linearities", Math. Z., 163 (1978), 239-249.3. C. A. Stuart, and J. F. Toland, "A variational method for boundary value problems with discontinuousnonlinearities", J. London Math. Soc. (2), 21 (1980), 319-328.4. C. A. Stuart, "Elliptic boundary value problems with discontinuous non-linearities", to appear.Departement de Mathematique, Department of Mathematics,Ecole Polytechnique Federale de Lausanne, University College,6, av. de Cour, Gower Street,Lausanne CH-1007, London WC1E 6BT.Switzerland.

A Property of Solutions of Elliptic Differential Equations with Discontinuous Nonlinearities

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