NEW ZEALAND JOURNAL OF MATHEMATICS Volume 27 (1998), 41-48

L O W E R A N D U P P E R SOLU TIO N S F O R FU N C T IO N A L EQ U ATIO N S OF E LLIPTIC T Y P E

J a m e s G r a h a m - E a g l e

(Received November 1996)

Abstract. Monotone and variational methods have proved extremely powerful in dealing with questions of existence and uniqueness for nonlinear elliptic equations. Such methods appear useless however, when the forcing function depends nonlocally on the dependent variable. Indeed, much of the structure of the solution set present in the local case is absent in the presence of nonlocal operators. In this paper questions of existence and multiplicity of solutions are investigated by topological methods.

1. Introduction

This paper is concerned with nonlinear boundary value problems of the form Lu = /(x ,u , V w ,$ («)) in fl }

f (1) Bu = 0 on dd Jwhere the scalar-valued function w is defined on the closure of the bounded domain Q, in Rn, L is a linear uniformly elliptic second-order operator, B is a linear boundary operator and $ is a (possibly nonlinear) functional. The notation and further assumptions will be given in the next section.

Equations of this form have occurred sporadically in the literature. Bazley and Wake [1] introduced the problem

(2)

—Aw = A exp I ------- r.—77 in f21 + e w 1

u — 0 on dfl

(in which || • || is the supremum norm over fi) as an approximation to the reaction equation with Arrhenius kinetics

-Aw = A exp --------- ) in fl1 1 + e u 1

u — 0 on dfl.In this case the nonlocal problem can be solved explicitly for the simple geometries of the infinite slab and infinite circular cylinder. In a subsequent paper, Bazley and Wake [2] extended their analysis to spherical domains in higher dimensions.

1991 A M S Mathematics Subject Classification: 35J60.K ey words and phrases: Elliptic partial differential equations, functionals, lower and upper solutions, degree theory.

42 JAMES GRAHAM-EAGLE

This idea of approximating intractable problems with nonlocal equations was generalized somewhat by Graham-Eagle [3].

More recently Sisson et al. [4] and Graham-Eagle [5] studied the equation

-Aw = 77 e x p (- l /u ) +A

A + (1 /^ ) fa exp(a/u) dxexp

a — a .

du(3—---- b u — 0 on SCI

on(3)

to model the steady states of an exothermic reaction in which water is conserved. Here the nonlocal behavior appears in the form of an integral over the domain ft.

Existence of solutions to nonlocal equations has been studied by several authors. Sisson et al. prove existence and multiplicity results for (3) using standard monotone techniques (which work in this case because of special monotonicity conditions satisfied by their problem). Bader [6] assumes growth conditions on / in order to apply the Schauder fixed point theorem and uses a contraction argument to obtain a simple uniqueness result. Politjukov [7] and later Bebernes and Ely [8] use (different definitions of) upper and lower solution to investigate parabolic equations, but their approach works also for elliptic equations with little change. However they do not allow / to depend on the derivatives of u.

In this paper degree theory is used to establish existence and multiplicity results along the lines of Amann [9]. We follow closely Bebernes and Ely’s definition of upper and lower solution but modify their proof to account for the dependence of / on Vu.

2. N otation and Assum ptions

Throughout the paper Q denotes a bounded domain in Mn, n > 1. If n > 2 the boundary of ft is assumed to be of class C 2,a for some fixed 0 < a < 1. (In the one dimensional case ft is just an open interval.)

C k(fl) is the space of functions ft —> M with continuous kth derivative on ft and norm

I M U = - y ^ s u p \Du\j— fi

the sum extending over all partial derivatives Du of u up to and including order k. C k,a(Q) is the subspace of Ck(Q) comprising those functions for which

IM Ik ,a = IMIfc + suP { D̂U ( * ) ^7 + «(*). i,j—1 J i= 1

where a^, a*, a E Ca(Q,), i, j = 1 ,2 ,... , n. L is supposed to be uniformly elliptic,i.e. the matrix A — (aij(x)) is symmetric and satisfies, for some c > 0, £TA£ > c|£|2

LOWER AND UPPER SOLUTIONS FOR FUNCTIONAL EQUATIONS 43

for all x G Q and all £ G Mn. The boundary operator B is defined by

duBu = b(x)u + (3 — ,

duwhere d/dv denotes the directional derivative in the direction v. Here v : 80, —>Mn is a nowhere tangent outward (with respect to fi) vector field of class C 1,a, the function 6 : IR is of class C 1,a, and b > 0. Either ( 3 = 1 (Robin or mixed boundary condition), or (3 = 0 in which case 6 = 1 (the Dirichlet boundary condition).

The real-valued function / is assumed locally /i-Holder continuous in all its variables, 0 < /j, < 1. The functional $ is assumed continuous C 1(f2) —>• M, and to map bounded sets with respect to the usual pointwise ordering of functions on w to bounded intervals.

By a solution of (1) we mean a classical solution, that is a function u G C 2(Q) satisfying (1) pointwise.

The following fundamental result about uniformly elliptic boundary value problems will be used (see for example Ladyzhenskaya and Ural’tseva [10]). Fix a G (0,1). Then for every g G Ca(Q) the boundary value problem

Lu = g in Q

Bu = 0 on dCl

has a unique solution u G C 2,a(0). Denoting this solution by Kg defines a bounded operator K : Ca(Cl) —*■ C 2,a(fl) which extends (uniquely) to a completely continuous operator C(Q) —> C'1,r(£l), 0 < r < 1.

Rem ark 1. This result requires that a > 0 in O and that a and 6 are not both identically zero. This can be arranged simply by adding an appropriate multiple of u to both sides of the differential equation (1) and redefining L and / .

Suppose now that u € C 1(f2). Then Vw € C(Q) so / (• , u, Vw, ${u)) € C(Q) and the mapping C 1(f2) —> C(Q) : u >—> /(• ,u, Vu, $(u)) is continuous. It follows immediately that the mapping C J(Q) —>■ C 1(Q) : u K / ( - ,« ,V u ,$ (u ) ) is completely continuous. Moreover, if u G C 1(f2) is a fixed point of K f , then u € C'1,T(f2) so / (• , u, Vu, $(w)) 6 CMT(f2) and finally u = K f ( ■, u, Vu, $(w)) € C2(f2). Therefore we have proved:

Lemma 2. The function u G C2(Q) is a solution of (1) if, and only if, u is a fixed point of the completely continuous mapping C 1(fi) —> C 1(0) : u > K f [ •, u, Vu, $ (u )) .

3. Lower and U pper Solutions

In the absence of nonlocal operators lower and upper solutions coupled with monotone schemes have proved extremely effective at proving the existence of solutions of (1) - see for example Keller and Cohen [11] or Sattinger [12]. Variational (Graham-Eagle [13]) and topological (Amann [9]) methods have also been used, the latter yielding information about multiplicity of solutions and extending the results to problems in which / depends on Vu.

44 JAMES GRAHAM-EAGLE

A straightforward generalization of the standard definition of lower and upper solution is the following: v is a lower solution for (1) if v G C 2(Q) and

Lv < f(x ,v ,'V v ,$ (v)') in Q

Bv < 0 on dtt.

Upper solutions have the inequalities reversed. The classical existence theorem asserts that, if v is a lower solution and w is an upper solution with v < w in Q, then (1) has a solution u such that v < u < w in ft. The following example shows that this result can fail if / depends nonlocally on u.

Example 3. — u" = u — 1 — 4u(£) sin 3 ̂d£, u(0) = 0, u(tt) — 0.Simple substitution verifies that v = — 1 is a lower solution and w s 1 is an

upper solution. However, any solution of the given differential equation satisfies

u(x) — A cosx + B sinx + 1 + / 4u(£)sin3£d£Jo

and applying the boundary conditions shows that u(x) — B sinx and 1 + Jq 4u(£) sin3^d^ = 0, a contradiction. Thus the boundary value problem has no solution whatever.

For nonlocal equations the definitions of upper and lower solution are amended to (closely following Bebernes and Ely [8]):

Definition 4. The functions v,w G C 2(Q) are a lower/upper solution pair for (1) if

v < w in 17;

Lv(x) < f (x ,v (x ) , V'u(x), $ (2)) for each x in Q and 2 G Sv(x),Bv < 0 on dtt',

Lw(x) > / ( x , w(x), Vw(x), $ (z )) for each x in and z G Sw(x ),Bw > 0 on dCl

where Su(x) = { z G C(f2) : v < z < w in ft and z(x) = u(x)}. If all the inequalities are strict, then v and w form a strict lower/upper solution pair.

If / does not depend on $ or if /(• , •, •, 3>(z)) is monotone increasing in z (with respect to the standard pointwise ordering of functions on fl) then it is clear that this definition reduces to the classical definition. Indeed in this case the standard existence proofs carry over with no change. This is the case for the problem investigated by Sisson et al. [4], It is straightforward to check that the problem given in the Example 3 fails to satisfy Definition 4.

Henceforth let v,w denote a strict lower/upper solution pair for (1). Define

rw.it) = { z G C'1(fi) : for all x G fJ, v(x) < z(x) < u?(x)}

and write deg(T, for the Leray-Schauder degree of I — T on at 0. In particular, if deg(T, r U)iy) 7̂ 0, then T has a fixed point in r Vjt£;.

LOWER AND UPPER SOLUTIONS FOR FUNCTIONAL EQUATIONS 45

Lem m a 5. Suppose there exists a function h G Ca(Q) (some 0 < cr < 1) such that, for all x G fl and z G r UjU;,

f (x ,v {x ) , 'V v (x )^ (z ) ) < h(x) < f(x ,w (x ),'V w (x ),^ (z )).

Then deg(K f ,T v>w) = 1.

P roof. Define ip = Kh. We begin by showing ip G TVjW. Indeed, for x G Cl

L(w — ip){x) > f ( x , w(x), S7w(x), $(w )) — h(x) > 0 and B(w — ip )> 0

so ip < w in Cl follows immediately from the maximum principle. Similarly ip < v in fI.

Now define, for 0 < t < 1, the family of operators Tt : C(r2) —» C(f2) by

(Ttu)(x) = (1 - t)(K fu )(x ) + tip(x).

Since K f is completely continuous, so clearly is Tt for each t. Since ip G(Tv,w it is immediate that deg(Ti, r^)U;) = 1. To establish deg(T0, r U;W) = 1 by using the homotopy invariance of degree, we must show that Tt has no fixed point on dTVjW, 0 < t < 1. Suppose to the contrary that such a fixed point does exist with u(x') = w(x') at some point x' G Q. If x' G O, then applying L to the identity u{x) = (1 — t) (K fu )(x ) + tip(x) yields

Lu(x) = (1 — t ) f (x , u(x), Vu(x), ^*(u)) + th{x).

In particular at x'

Lu(x') = (1 — t)f(x ',w (x ') ,V w (x ') ,$ (u ))+ th (x ') .

Subtracting this from

Lw(x') > f(x ',w (x '),'V w (x '),$ (u ))

gives the’inequality

L[w — u](x') > t[f(x ',w (x '),\ 7w (x '),$ (u ))—h(x')] > 0.

By the uniform ellipticity condition on L, this implies the Hessian matrix of w — u is negative at x' which contradicts the fact that w — u has a minimum there. On the other hand, if x' G dfl, then at x' it is clear that dwfdu < du/dv and so by the boundary conditions on u and w, u(x') < w(x'). A similar argument shows that no fixed point of Tt can touch v in Cl. □

Lemma 6 . If v and w form a strict lower/upper solution pair for (1), then deg (K f,T VjW) = 1.

P roof. Consider the modified problem

Lu = / ( x , « , V m ,$(«)) in Cl

Bu = 0 on dfl

where

Lu = Lu + cu and / (x ,y , V u ,$ (u )) = / ( x , u, Vu,

46 JAMES GRAHAM-EAGLE

with c > 0 chosen so that, for all x € and z e r W;W

hi(x) = sup / (x , v(x), Vv(x), $ (2:)) < inf / (# , w(x), Vw(x), $ (2)) = h2(x). zerv,w zerVjW

Such a c exists because v < w in the compact set Q and is a compactsubset of E. Clearly hi and h2 are continuous. Set h = \{hi -f h2). Then hi(x) < h(x) < h2(x) for x € f2 and if h is not of class Ca(Cl) it can be approximated by a function which is (e.g. a polynomial) while retaining this inequality. Applying Lemma 5 shows that d eg (K f,T VyW) — 1 (where K is the inverse of L) and by the translation invariance of degree this shows that deg (if/, r U)lt>) = 1. □

An immediate corollary of this result is:

Theorem 7. If v and w form a strict lower/upper solution pair for (1), then (1) has a solution u which satisfies v < u < w in ft.

Rem ark 8 . It is a simple matter to relax the strict inequality in the boundary conditions for the lower/upper solution pair. Indeed, it is easy to see that, for sufficiently small e > 0, v — e and w + e will form a strict lower/upper solution pair so Theorem 7 guarantees a solution u£ lying between v — e and w + e with {u£} precompact in C 1(S1).

Rem ark 9. This leads quickly to a proof of Theorem 7 of Bader [6]. Indeed, under the assumptions of that theorem it is straightforward to check that, if 0 is the solution of

L

LOWER AND UPPER SOLUTIONS FOR FUNCTIONAL EQUATIONS 47

Rem ark 12. This generalizes the result of Amann [9].

Rem ark 13. The above results can be extended easily to the case of nonhomoge- neous boundary conditions. Indeed, if g : dfl —> 0)

-A u = A exp ( ------U — ■■■■ | in Cl+ I (4)

u — 0 on A exp ( -VI +e

48 JAMES GRAHAM-EAGLE

9. H. Amann, On the number of solutions of nonlinear equations in ordered Banach spaces, J. Functional Anal. 11 no. 3 (1972), 346.

10. O.A. Ladyzhenskaya and N.N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.

11. H.B. Keller and D.S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1976), 1361.

12. D.H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J. 21 (1972), 979.

13. J. Graham-Eagle, A variational approach to upper and lower solutions, IMA J. Appl. Math. 44 (1990), 181.

James Graham-EagleDepartment of Mathematical SciencesUniversity of Massachusetts LowellLowell, M A 01854U.S.A.james_grahameagle@uml.edu

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mailto:james_grahameagle@uml.edu