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Uniqueness for small solutions to a superlinear boundary value problem at resonance
Nonlinear
Analysis.
Theory.
Methods
& Applications,
Pergamon
Printed
Vol. 29, NO. 8, pp. 921-935, 1997 651 1997 Elsetier Science Ltd in Grcay Britain. All rights reserved 0362-546X/97 $17.00+0.00
PII: SO362-546X(96)00099-5
UNIQUENESS FOR SMALL SOLUTIONS TO A SUPERLINEAR BOUNDARY VALUE PROBLEM AT RESONANCE LEW LEFTON Department of Mathematics, University of New Orleans, New Orleans, LA 70148, U.S.A. (Received
10 October
1994; received
in revisedform
24 June 1996;
receivedfor
publication
19 September
1996)
Key words and phrases: Boundary value problem at resonance, uniqueness, small solutions, superlinear, eigenvalue estimates, degree theory.
1. INTRODUCTION
Consider the following resonant boundary value problem on the interval (a, b), (1.1) Ly + g(y) = f(x) Bl (y) = 0, B2W = 0 where Bi and 82 are general linear boundary conditions and L. is a selfadjoint linear operator. There has been a great deal of research on the existence theory for problems of this type when g(y)/y has finite limits as y - ?w. Necessary and sufficient conditions for solvability of (1 .l) when g is asymptotically linear are usually established by assuming some type of LandesmanLazer condition or monotonicity condition on g. See [l-7] and many others. The case when g grows faster than linearly at 00 has also been studied by numerous authors. See for example, [8-171 and the references therein. These results, however, still typically assume that the nonlinearity g(y) satisfies a one-sided growth restriction. For other related results in the literature that address semilinear equations with superlinear nonlinearities, we refer to [18-221, the survey article by Rabinowitz [23], and many others. In this paper, we will consider a resonance problem with a particular nonlinearity that is superlinear in both directions, i.e. lim g(Y) -=+w. (1.2)
M-m
y
Moreover, we will focus on small solutions, that is, solutions which are a priori bounded in supremum norm. The existence of such small solutions for a particular case of (1 .l) and a particular g satisfying (1.2) was studied by in [24]. One can show the following theorem. 1. Consider the boundary value problem (1.1) with Ly = y” + p(x)y’ + qb)y and g(y) = ? lyl”-‘y. Assume p, q, and f are in L’ [a, 61 and m > 1. Further assume L is selfadjoint and dim(ker(l)) = 1. If llfll i is sufficiently small then 3 r > 0 such that the boundary value problem has at least one solution ys in B(r) = {y: llyllm < r}.
THEOREM
The above theorem can be proved using the same arguments as in Theorem 2.1 in [24], along with the simplifying observation that if L is selfadjoint then $3 B R(L). See [25] for a generalization of Theorem 1 to nonhomogeneous g(y). We point out that an abstract result of Laloux and Mawhin [26] can also be used to prove Theorem 1. 927
928
LEW
LEFTON
Theorem 1 holds for nonselfadjoint resonance problems provided the range of L satisfies a certain condition. For details on the nonselfadjoint case, see Theorem 2.1 in [24]. Another proof in the nonselfadjoint case, which just uses the implicit function theorem, appears in [27]. The proof of Theorem 1 uses degree theory and hinges on the following a priori bound. If llfll1 is sufficiently small then every solution y from Theorem 1 satisfies
Ml”, 5 mt-Ill.
(1.3)
This bound is established using an argument similar to Lemma 1.4 in [24]. In this paper, we show that the small solutions from Theorem 1 are in fact unique in the class of small solutions. Note that this is only a local uniqueness result. We prove that there is a unique solution in a small ball for f sufficiently small in L’. In general, one cannot expect to extend this result to global uniqueness for the superlinear problem under investigation. Indeed, FuCik and Lovicar [28] (see also [29]) proved that the problem y” + g(y) = f(x) oc1y(O) + 0(2y’(O) = 0, Bl.Y(l) + hY’(l) = 0 has infinitely many solutions for each f E L’ (0, 1) when (1.2) holds. We point out that the uniqueness result for small solutions presented here is related to some recent work by Nabutovsky [30], where he obtains an upper bound on the number of solutions in a given ball to the semilinear problem -Au + g(x, U) = f(x) for IIf II I small. Our main result is proven using a degree theoretic argument and relies on careful eigenvalue estimates which have independent interest. Although topological degree theory is not a standard way to prove uniqueness, it was a natural choice here since the a priori estimates had already been established in [24]. Our method for proving uniqueness using topological degree is to show that the global degree in a ball is + 1 (resp. - 1), hence there is at least one solution in the ball. We then proceed to prove that the local degree at any isolated solution is also +1 (resp. -1). From the excision property of topological degree, we conclude there is exactly one solution in the ball. The precise statement of the main result appears in Section 2. In Section 3 we present the proof, and in Section 4 we give a counterexample for the case when L is nonselfadjoint. Throughout this paper, C denotes an arbitrary positive constant. Note added hJina1 revision: After this paper was submitted, the author became aware of an interesting preprint by Battelli ahd FeEkan [3 l] which provides a deeper analysis of the boundary value problems studied here. Their results include a “Fredholm-like” alternative for abstract operator equations. In particular, they use the classical Lyapunov-Schmidt decomposition and study the case when the bifurcation mapping is flat at the origin. Their work is motivated by an example in [24] which shows that small solutions may not exist for arbitrarily small f. The same example is used in Section 4 of this paper to illustrate nonuniqueness of small solutions for certain f. Battelli and FeEkan generalize this example and show that the results follow from the fact that the operator is equivariant according to linear representations of compact Lie groups. They also “globalize” the results (i.e. they obtain theorems which are not restricted to small solutions), however to do so, they must assume the nonlinearity is globally Lipschitz. This assumption does not hold for superlinear g unless a uniform a priori bound on the solution is established. The reader is referred to [31] for further details. 2. HYPOTHESES
Let p, 4,
AND
RESULTS
f E L’ [a, b]. Define fl = f 1 and consider the nonlinear boundary value problem
929
Superlinear boundary value problem at resonance
fy
=
Ly
+
&(y)
c!g(y)
Bz(y)
=
y” + p(x)y’
q
= oc1yw
+ octy’bd
by(a)
+
+ qwy
+ r?g(y) = f(x),
+ oc3y(b) + WY’(b)
82y’b)
+
83y(b)
+
B‘ly’(b)
(2.1)
= 0, =
0.
Assume that boundary conditions Br and B2 are linearly independent and that (2.1) defines a selfadjoint problem. Although the proof will work for more general nonlinearities including nonhomogeneous g (see the remark at the end of Section 4), we will, for the sake of clarity, restrict our attention to 1 < m. g(y) = lylm-‘y, (2.2) In particular, we require no growth restriction on the nonlinearity at co. Note however, the proof of existence in Theorem 1 requires a certain local behavior for g near 0 [24,25]. Define BC = {y E C’ [a, b] : y’ is absolutely continuous, B1(y) = 0, B2(y) = O] and observe that BC is a Banach space under the supremum norm II ~11o1 = ~up,~[~,~~ I u(x) I. Observe that L : BC - L’ [a, b] and assume ker(L) is a one dimensional Without
subspace of BC spanned by 4.
(2.3)
loss of generality, l]+]lm = 1.
2. Assume problem (2.1) satisfies (2.2) and (2.3). If IIf]] 1 is sufficiently small then the solution yo E B(r) is the only solution of (2.1) in B(r). Before proving Theorem 2, we establish some notation. Define projection operators
THEOREM
Po:L’
- ker(L) c BC
by
Pof = @of)4
and
E
PI = Z - PO.
Note that&f is real valued and Pof is in the one-dimensional space spanned by 4. L’[u, 61 denote the range of L. One can construct a completely continuous operator PI (BC) such that Gf=y w Ly=f foryEPr(BC), fER(L). Note that since L is selfadjoint, f E R(L) w Pof = 0, hence codim (R(L)) = Fredholm operator of index 0. We observe that the operator d : BC - BC defined
(2.4)
Let R(L) c G : R(L) -
1 and L is a by
-ay = Pay + Po(f - rig(y)) + GPl (f - rig(y)) is compact and has the property that Ay = y if and only if Ly = f. For further details on the construction of 3, see [24]. We point out that this is a different construction than the standard Lyapunov-Schmidt method which reduces to two equations that are solved by substitution. Instead we have reduced to a single equation and we seek fixed points. This equivalence was proved by Mawhin [32] where its relation to classical Lyapunov-Schmidt techniques is carefully described. 3. PROOF OF THEOREM
2.
LEMMA 1. Let yo E B(r) be a solution of (2.1) with IIf]] 1 sufficiently small, but not zero. Then yo is isolated. Proof. Suppose y,, is a sequence of solutions with y, - y. uniformly. 0 = LY” - fyo = Lo(y, - yo) + fuyn - yo)
Then
930
LEW LEFTON
where &(y-yo) 411~ -
yOIIoJ)
= L(y-yo) as Ily -
+qg’(yo)(y-ye)
is the linearization
ofL aboutyo and Q(y-yo)
=
yOlloo- 0. We will show that LO is nonsingular when llfll i is sufficiently
small, and thus lh - yoIlm = IIL;‘R(y, - y~)ll~ = o(lly -yells) which is a contradiction that completes the proof. First, we show that Loy = 0 has only the zero solution when Ilyollm is small. Indeed, suppose y solves (3.1) LY + rlg’(yo)y = 0. By scaling if necessary, we can write y = 4 + w where wl+. We have llwllm = IIWm’(yo)y)lIm
Sinceg’(y)
-Oasy-
5 Cllg’(yo)4ll,
+ cllg’(yo)lh
Ilwllm.
0 we have, for IIyollm sufficiently small, (3.2)
llwllm 5 ag’(yo)4lll.
From (3.1) we observe ,,” ng’ (yo)y+ = 0. Substituting y = 4 + w gives J,” og’ (yo)+2 = s,” -&$(yo)w4. Combining this, (3.2), and the Cauchy-Schwarz inequality we obtain
which is a contradiction for small yo. Thus (3.1) has only the zero solution for small yo. To conclude that Lay = 0 has only the zero solution when llfllr is small, we use the a priori bound (1.3). H The index of an isolated solution yo E B(r) can be found from the well known formula (see [33, P. 281) i(Z - d,yo, 0) = (-l)B, (3.3) where /3 is the number of negative eigenvalues (counting multiplicity) Frechet derivative of A at y = yo. Thus, we consider U - Ao)y = Sy + Pohg’(yo)y)
of Z - A0 and A0 is the
+ GPI hg’(yo)y).
(3.4)
The following lemmas are stated for general linear operators which have this form. LEMMA 2. Suppose X and Y are Banach spaces and let R : X - Y be an operator of the form R = PI + T where T is a bounded linear operator and Pi is a projection with a one-dimensional kernel generated by 4 with II 4 11~= 1. Let PO = Z - Pr and assume the following operator norm estimates hold, IlPoTll I i and IIPlTII I i. (3.5) Then R has precisely one eigenvalue in (-i, i) and no eigenvalues in (-co, il. Moreover, the unique eigenvalue ho E (-i, 4) has multiplicity one. Proof. We begin with a useful observation. IfIhl
(3.6)
Indeed, if Plx = x then (R - h)x = 0 implies TX = Ax - Plx = (A - 1)x. But this would give IlTll 2 IA - 1 I 1 i which contradicts (3.5). Thus, without loss of generality, we assume all eigenvectors x corresponding to I A( < f are scaled so that POX = 4.
931
Superlinear boundary value problem at resonance
Consider F: I’& x Pi (X) - R x PI(X)
given by F(A, w) = (h’, w’) where
A’+ = PoT(c$ + w)
(3.7)
w’=hw-P~T(++W).
Note that (h, w) is a fixed point of F if and only if it is an eigenpair of R. Thus, existence of a unique eigenvalue of R in (- f , i ) will follow by showing F has a unique fixed point in z=((h,W)EWX~l(X):Ihl~~,
IIWllx~~].
Let (h, w) be an arbitrary point in 2. Observe by (3.5) and (3.7) that
IA’1 = lIPoT(dJ + w)I/x 5 ;c1 + Ilwllx) 5 ;t;, -c ;. Also, llw’ll~ I lhlllwll~ + i(l (hz, wz) in C we compute
IIFh
+ IlwllX) I (i)2 + i(i)
= i. Thus F(E) E 2.. For (hi, WI) and
WI) -&AZ, w2)lls = IA; - h;l + lb; - w;llx 5 IIPoTc+ + +llh1w1 -
WI
-
PlT(4
PoT(4
+
+ WI)
s ;llwl - wzllx + llhw1 5 ;llwl - wzllx + flh
-
w2)llx h2W2
hlW2
+ 9T(dJ
+ hlW2
- h2l 5 &hL
+
-
Wl)
w2)IIx
(3.8)
h2W2llX -
u2.
w2)llz.
Thus, F is Lipschitz with constant 2 < 1 and the contraction mapping principle guarantees a unique fixed point. Now suppose 3 h s -i with (R - h)x = 0. Without loss of generality, take 11x11~= 1. Then either 1lPoxll~ 2 i or 11Pixllx 1 i. In either case we can apply projections to Plx + TX = Ax to contradict h I 4. In particular, either i z IlPoTxllx = Ihl IlPoxll~ 1 i Ihl, or i 2 IJP~Txllx = IA - 11 11PiXllX 2 ; lh - 1 I. To show that the unique ho in (-i, i) has multiplicity one, suppose that x1 = al 4 + WI and x2 = a24 + w2 are linearly independent eigenvectors corresponding to ho. From (3.6) we can assume without loss of generality that ai = uz = 1. However, this would mean xi - x2 is an eigenvector in Pi (X) which is impossible. Hence dim(ker(R - ho)) = 1. To show dim(ker(R - ho)2) = 1 we require the following estimate, which will be established later, IA - hoI 5 6(llPoW - A)(+ + w)IIx + lPl(R-
A)(+ + w)llx)>
V(h, w) E 1.
(3.9)
Assuming (3.9) for now, let x1 = ai+ + WI and x2 = u~~#I+ wz be two linearly independent solutions of (R - h~)~x = 0. Without loss of generality, we assume that al = 0. If this isn’t the case, replace x1 by x2 - (u2/ui)xl. From (3.6), Po( (R - &)x1) # 0 so write (R - ho)xl = c$ + wo.
(3.10)
For t E Iw define and
wo(t) = (R - ho)x, - 4 - txl = w. - txl, h(t) = ho -t.
(3.11)
932
LEW
LEFTON
By (3.10) C#J+ wg E ker(R - ho) therefore llwsll~ < i and l&l small t, Ilwo(t)ll~ < i and IA( < i. We calculate (R - h(t))(+
+ we(t))
< i. Hence, for sufficiently (3.12)
= ((R - ho) + t)((R - ho) - f)Xl = -12x1.
Combining (3.9), (3.11) and (3.12) we have ItI = (h(t) - hoI < 6t2 (llP,-,xi IIX + 1lZ’rxi 11~) which is a contradiction for small t > 0. Thus dim(ker(R - ho)2) = 1. It remains to justify (3.9). Using (3.7) we have IIF(A WI - (A w)II,x = IIPoTc+ + WI - Wllx = IIPW - A)(+ + w)llx
+ llP*T(@ + WI - hw + wllx + llPl(R - A)(4 + w)IIx.
(3.13)
Recall that the fixed point (ho, wg) of F in 1 is obtained as n-00 lim F”(h, w) where F” is the nth iterate of F. Using (3.13) and (3.8) we have n-l
IIF”(A
w) - (ho, wo)ll.z I c
($)”
(IIPoW
A)(+
5 6(llPoW
- A)(+
+ w)IIx
+ w)llx
+ IIPIW - A)(+
+ 4x)
k=O
Estimate (3.9) follows.
+ llPl(R - A)(+
+ w)IIx).
n
We observe that the operator Z--A0 in (3.4) clearly has the form described in Lemma 2 with X = BC and Y = L’. We also notice that condition (3.5) will hold if ll~~oll~ is sufficiently small. Also, if ho < 0 then the value of fi in (3.3) is 1 and if ho > 0 then fi = 0. In either case we have (3.14)
i(Z - d,yo, 0) = (-1)s = sgn(h0).
LEMMA 3. Let F : C - X be defined by (3.7) with X = BC, Y = L’, PO and PI defined by (2.4), and T = Po(qg’(yo)y) + GPl (r-/g’(yo)y) where yo is an isolated solution of (2.1). Suppose (P, z) = F(0, 0) and (ho, wg) is the fixed point of F. If Jlyo(lm is small enough, we have sgn(hs) = sgn(p).
Proof. By (3.7) and the definition of p and ho we have /.M$ = POT+ and ho@ = PoT($ Thus it suffices to show Jho - ~1 = IIPoTw,-Jm = o(lpl) as 11~~011~ - 0. Since (ho, wo) is a fixed point of F in C we have +ollm
5 11- hol IIwolloo= llST(+
+ WO).
+ wo)llm I Cllg’(yo)+lh + Cllg’(yoMoll1 5 Cllg’(yoh#Al1 + Cllwollm llg’bo)II1.
Therefore, for ~0 sufficiently small, (3.15)
IIWOllm5 Cllg’(Yob#dll.
Using (3.15) and the Cauchy-Schwarz inequality
we obtain
IIPoTwoIIm = IIPog’(yohoIlm 5 CllPog’(Yo)llm Ilwollm 5 Cllg’(yo)Q,II: 5 Cllg’(yo% But IpI = IIPoT+llm = Cllg’(yo)+2111 which implies
llg’(yoM21h
IIPIITwcII~~ = o(lpl) as lly~ll~ - 0. n
The proof of Theorem 2 now follows from standard arguments. Let llfll i be so small that Theorem 1, Lemma 1, Lemma 2 and Lemma 3 all hold simultaneously. Then Theorem 1 guarantees that 3~0 E B(r) solving (2.1). It is also known [24] that the Leray-Schauder degree &(I - a, B(r), 0) = q.
933
Superlinear boundary value problem at resonance
For each solution yo we have, from Lemma 1, that y. is isolated. By (3.14) and Lemma 3 we deduce that i(Z - Xyo, 0) = sgn(p) where p = Fo(og’(yo)+) = r~CJ,bg’(y~)c$~ dx. Since g’ (yo ) 10 we conclude sgn ( p) = q. Hence, from the excision property of Leray-Schauder degree, there is at most one solution in B(r), and therefore there is exactly one solution in B(r). 4. A COUNTEREXAMPLE
FOR L NONSELFADJOINT
The hypothesis that L is selfadjoint is necessary in Theorem 2. Below we consider an example which is nonselfadjoint and which has infinitely many small solutions for certain small f. See the note at the end of the introduction for recent work related to this example. Consider the problem y” + y3 = f(x)
on -15x51,
y(1) = y(-1),
y’(1) +y’(-1)
= 0.
(4.1) In this example, Ly = y” and $ = 1 spans ker(L). The operator L is a Fredholm operator of index 0. This follows from the closed range theorem [34] after noting that the range of L is a closed, co-dimension 1 subspace of L1 (see Lemma 1.1, [24]). Also observe that $3 = 1 is in R(L) since x2/2 E BC and L(x2/2) = 1. THEOREM
3. The boundary value problem (4.1) has infinitely many solutions for small even f.
Proof. First note that if y is even and y’ is absolutely continuous then y E BC. Also, if y is an even function then so is Ly = y” + y3. We claim that for small positive numbers b and small even functions S, the equation y” + y3 = f has a unique solution satisfying initial conditions y(0) = b > 0
and
y’ (0) = 0.
(4.2)
Let 6 be a positive number, to be determined. Suppose f is an even, integrable function on l-1, 11 such that llfllr < 6. Any solution of y” + y3 = f with initial conditions (4.2) can be written as y(x) = 6 - [0x(x - t)[f (t) - y3(t)] dt. Define ay=6-
I
~(x-1)[f(1)-y3(f)ldf.
We will show that a has a fixed point in D = {y : y’ is absolutely continuous llyllm < 3Sl.Fory~Dwehave IlAyll,
= sup
XE[O.ll I
~6+llflll+
6 -
I
on [0, 1] and
;lx - t)[f 0) - y3(t)l dr 1
x~[o suptl ;IIy311m
I26
+ p3
I 36
where the final inequality holds if and only if d2 I 2/27. Thus, if 62 < 2/27 then 31 maps D into itself. Furthermore, from the mean value theorem, for y and z in D, lly3 - z3 Ilm I 311r(t~211mlly-211m where Ir(r)I < 3bsincer(t) isbetweeny(t) andz(t) forallt.Thus, ]]y3-z3]loo I 27b21]y - z]lm and x IlAy- .AZIIca = sup (x - t)(y3 - z3) dt I 27fi21]y - zllo3. XE[O.I] II 0 Therefore, if S2 < 2/27 then Jt is a contraction on D and by the contraction mapping principle a has a unique fixed point yo in D. Extending yo to [ - 1, 1] by evenness we obtain a solution to n (4.1). Hence, for a fixed f, sufficiently small, there are infinitely many small solutions.
934
LEW LEFTON
We remark that the nonlinearity g(y) in (2.1) can be replaced by a more general function which is structurally similar to lylm-ly for m > 1 in a neighborhood of 0. Although the behavior of g near 0 is crucial for this result on small solutions, we point out that g(y) can have rather arbitrary behavior at co. In particular, homogeneity in y is not necessary (see [25]). Indeed, the result holds for as y - O+, Cl IYI ml-ly + o( Iyl”‘) g(y) = i C21ylmz-‘y + o( Iylmz) as y - oprovided ml > 1 and
m2
> 1. This result is also generalized in [31]. REFERENCES
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Pergamon
Printed
Vol. 29, NO. 8, pp. 921-935, 1997 651 1997 Elsetier Science Ltd in Grcay Britain. All rights reserved 0362-546X/97 $17.00+0.00
PII: SO362-546X(96)00099-5
UNIQUENESS FOR SMALL SOLUTIONS TO A SUPERLINEAR BOUNDARY VALUE PROBLEM AT RESONANCE LEW LEFTON Department of Mathematics, University of New Orleans, New Orleans, LA 70148, U.S.A. (Received
10 October
1994; received
in revisedform
24 June 1996;
receivedfor
publication
19 September
1996)
Key words and phrases: Boundary value problem at resonance, uniqueness, small solutions, superlinear, eigenvalue estimates, degree theory.
1. INTRODUCTION
Consider the following resonant boundary value problem on the interval (a, b), (1.1) Ly + g(y) = f(x) Bl (y) = 0, B2W = 0 where Bi and 82 are general linear boundary conditions and L. is a selfadjoint linear operator. There has been a great deal of research on the existence theory for problems of this type when g(y)/y has finite limits as y - ?w. Necessary and sufficient conditions for solvability of (1 .l) when g is asymptotically linear are usually established by assuming some type of LandesmanLazer condition or monotonicity condition on g. See [l-7] and many others. The case when g grows faster than linearly at 00 has also been studied by numerous authors. See for example, [8-171 and the references therein. These results, however, still typically assume that the nonlinearity g(y) satisfies a one-sided growth restriction. For other related results in the literature that address semilinear equations with superlinear nonlinearities, we refer to [18-221, the survey article by Rabinowitz [23], and many others. In this paper, we will consider a resonance problem with a particular nonlinearity that is superlinear in both directions, i.e. lim g(Y) -=+w. (1.2)
M-m
y
Moreover, we will focus on small solutions, that is, solutions which are a priori bounded in supremum norm. The existence of such small solutions for a particular case of (1 .l) and a particular g satisfying (1.2) was studied by in [24]. One can show the following theorem. 1. Consider the boundary value problem (1.1) with Ly = y” + p(x)y’ + qb)y and g(y) = ? lyl”-‘y. Assume p, q, and f are in L’ [a, 61 and m > 1. Further assume L is selfadjoint and dim(ker(l)) = 1. If llfll i is sufficiently small then 3 r > 0 such that the boundary value problem has at least one solution ys in B(r) = {y: llyllm < r}.
THEOREM
The above theorem can be proved using the same arguments as in Theorem 2.1 in [24], along with the simplifying observation that if L is selfadjoint then $3 B R(L). See [25] for a generalization of Theorem 1 to nonhomogeneous g(y). We point out that an abstract result of Laloux and Mawhin [26] can also be used to prove Theorem 1. 927
928
LEW
LEFTON
Theorem 1 holds for nonselfadjoint resonance problems provided the range of L satisfies a certain condition. For details on the nonselfadjoint case, see Theorem 2.1 in [24]. Another proof in the nonselfadjoint case, which just uses the implicit function theorem, appears in [27]. The proof of Theorem 1 uses degree theory and hinges on the following a priori bound. If llfll1 is sufficiently small then every solution y from Theorem 1 satisfies
Ml”, 5 mt-Ill.
(1.3)
This bound is established using an argument similar to Lemma 1.4 in [24]. In this paper, we show that the small solutions from Theorem 1 are in fact unique in the class of small solutions. Note that this is only a local uniqueness result. We prove that there is a unique solution in a small ball for f sufficiently small in L’. In general, one cannot expect to extend this result to global uniqueness for the superlinear problem under investigation. Indeed, FuCik and Lovicar [28] (see also [29]) proved that the problem y” + g(y) = f(x) oc1y(O) + 0(2y’(O) = 0, Bl.Y(l) + hY’(l) = 0 has infinitely many solutions for each f E L’ (0, 1) when (1.2) holds. We point out that the uniqueness result for small solutions presented here is related to some recent work by Nabutovsky [30], where he obtains an upper bound on the number of solutions in a given ball to the semilinear problem -Au + g(x, U) = f(x) for IIf II I small. Our main result is proven using a degree theoretic argument and relies on careful eigenvalue estimates which have independent interest. Although topological degree theory is not a standard way to prove uniqueness, it was a natural choice here since the a priori estimates had already been established in [24]. Our method for proving uniqueness using topological degree is to show that the global degree in a ball is + 1 (resp. - 1), hence there is at least one solution in the ball. We then proceed to prove that the local degree at any isolated solution is also +1 (resp. -1). From the excision property of topological degree, we conclude there is exactly one solution in the ball. The precise statement of the main result appears in Section 2. In Section 3 we present the proof, and in Section 4 we give a counterexample for the case when L is nonselfadjoint. Throughout this paper, C denotes an arbitrary positive constant. Note added hJina1 revision: After this paper was submitted, the author became aware of an interesting preprint by Battelli ahd FeEkan [3 l] which provides a deeper analysis of the boundary value problems studied here. Their results include a “Fredholm-like” alternative for abstract operator equations. In particular, they use the classical Lyapunov-Schmidt decomposition and study the case when the bifurcation mapping is flat at the origin. Their work is motivated by an example in [24] which shows that small solutions may not exist for arbitrarily small f. The same example is used in Section 4 of this paper to illustrate nonuniqueness of small solutions for certain f. Battelli and FeEkan generalize this example and show that the results follow from the fact that the operator is equivariant according to linear representations of compact Lie groups. They also “globalize” the results (i.e. they obtain theorems which are not restricted to small solutions), however to do so, they must assume the nonlinearity is globally Lipschitz. This assumption does not hold for superlinear g unless a uniform a priori bound on the solution is established. The reader is referred to [31] for further details. 2. HYPOTHESES
Let p, 4,
AND
RESULTS
f E L’ [a, b]. Define fl = f 1 and consider the nonlinear boundary value problem
929
Superlinear boundary value problem at resonance
fy
=
Ly
+
&(y)
c!g(y)
Bz(y)
=
y” + p(x)y’
q
= oc1yw
+ octy’bd
by(a)
+
+ qwy
+ r?g(y) = f(x),
+ oc3y(b) + WY’(b)
82y’b)
+
83y(b)
+
B‘ly’(b)
(2.1)
= 0, =
0.
Assume that boundary conditions Br and B2 are linearly independent and that (2.1) defines a selfadjoint problem. Although the proof will work for more general nonlinearities including nonhomogeneous g (see the remark at the end of Section 4), we will, for the sake of clarity, restrict our attention to 1 < m. g(y) = lylm-‘y, (2.2) In particular, we require no growth restriction on the nonlinearity at co. Note however, the proof of existence in Theorem 1 requires a certain local behavior for g near 0 [24,25]. Define BC = {y E C’ [a, b] : y’ is absolutely continuous, B1(y) = 0, B2(y) = O] and observe that BC is a Banach space under the supremum norm II ~11o1 = ~up,~[~,~~ I u(x) I. Observe that L : BC - L’ [a, b] and assume ker(L) is a one dimensional Without
subspace of BC spanned by 4.
(2.3)
loss of generality, l]+]lm = 1.
2. Assume problem (2.1) satisfies (2.2) and (2.3). If IIf]] 1 is sufficiently small then the solution yo E B(r) is the only solution of (2.1) in B(r). Before proving Theorem 2, we establish some notation. Define projection operators
THEOREM
Po:L’
- ker(L) c BC
by
Pof = @of)4
and
E
PI = Z - PO.
Note that&f is real valued and Pof is in the one-dimensional space spanned by 4. L’[u, 61 denote the range of L. One can construct a completely continuous operator PI (BC) such that Gf=y w Ly=f foryEPr(BC), fER(L). Note that since L is selfadjoint, f E R(L) w Pof = 0, hence codim (R(L)) = Fredholm operator of index 0. We observe that the operator d : BC - BC defined
(2.4)
Let R(L) c G : R(L) -
1 and L is a by
-ay = Pay + Po(f - rig(y)) + GPl (f - rig(y)) is compact and has the property that Ay = y if and only if Ly = f. For further details on the construction of 3, see [24]. We point out that this is a different construction than the standard Lyapunov-Schmidt method which reduces to two equations that are solved by substitution. Instead we have reduced to a single equation and we seek fixed points. This equivalence was proved by Mawhin [32] where its relation to classical Lyapunov-Schmidt techniques is carefully described. 3. PROOF OF THEOREM
2.
LEMMA 1. Let yo E B(r) be a solution of (2.1) with IIf]] 1 sufficiently small, but not zero. Then yo is isolated. Proof. Suppose y,, is a sequence of solutions with y, - y. uniformly. 0 = LY” - fyo = Lo(y, - yo) + fuyn - yo)
Then
930
LEW LEFTON
where &(y-yo) 411~ -
yOIIoJ)
= L(y-yo) as Ily -
+qg’(yo)(y-ye)
is the linearization
ofL aboutyo and Q(y-yo)
=
yOlloo- 0. We will show that LO is nonsingular when llfll i is sufficiently
small, and thus lh - yoIlm = IIL;‘R(y, - y~)ll~ = o(lly -yells) which is a contradiction that completes the proof. First, we show that Loy = 0 has only the zero solution when Ilyollm is small. Indeed, suppose y solves (3.1) LY + rlg’(yo)y = 0. By scaling if necessary, we can write y = 4 + w where wl+. We have llwllm = IIWm’(yo)y)lIm
Sinceg’(y)
-Oasy-
5 Cllg’(yo)4ll,
+ cllg’(yo)lh
Ilwllm.
0 we have, for IIyollm sufficiently small, (3.2)
llwllm 5 ag’(yo)4lll.
From (3.1) we observe ,,” ng’ (yo)y+ = 0. Substituting y = 4 + w gives J,” og’ (yo)+2 = s,” -&$(yo)w4. Combining this, (3.2), and the Cauchy-Schwarz inequality we obtain
which is a contradiction for small yo. Thus (3.1) has only the zero solution for small yo. To conclude that Lay = 0 has only the zero solution when llfllr is small, we use the a priori bound (1.3). H The index of an isolated solution yo E B(r) can be found from the well known formula (see [33, P. 281) i(Z - d,yo, 0) = (-l)B, (3.3) where /3 is the number of negative eigenvalues (counting multiplicity) Frechet derivative of A at y = yo. Thus, we consider U - Ao)y = Sy + Pohg’(yo)y)
of Z - A0 and A0 is the
+ GPI hg’(yo)y).
(3.4)
The following lemmas are stated for general linear operators which have this form. LEMMA 2. Suppose X and Y are Banach spaces and let R : X - Y be an operator of the form R = PI + T where T is a bounded linear operator and Pi is a projection with a one-dimensional kernel generated by 4 with II 4 11~= 1. Let PO = Z - Pr and assume the following operator norm estimates hold, IlPoTll I i and IIPlTII I i. (3.5) Then R has precisely one eigenvalue in (-i, i) and no eigenvalues in (-co, il. Moreover, the unique eigenvalue ho E (-i, 4) has multiplicity one. Proof. We begin with a useful observation. IfIhl
(3.6)
Indeed, if Plx = x then (R - h)x = 0 implies TX = Ax - Plx = (A - 1)x. But this would give IlTll 2 IA - 1 I 1 i which contradicts (3.5). Thus, without loss of generality, we assume all eigenvectors x corresponding to I A( < f are scaled so that POX = 4.
931
Superlinear boundary value problem at resonance
Consider F: I’& x Pi (X) - R x PI(X)
given by F(A, w) = (h’, w’) where
A’+ = PoT(c$ + w)
(3.7)
w’=hw-P~T(++W).
Note that (h, w) is a fixed point of F if and only if it is an eigenpair of R. Thus, existence of a unique eigenvalue of R in (- f , i ) will follow by showing F has a unique fixed point in z=((h,W)EWX~l(X):Ihl~~,
IIWllx~~].
Let (h, w) be an arbitrary point in 2. Observe by (3.5) and (3.7) that
IA’1 = lIPoT(dJ + w)I/x 5 ;c1 + Ilwllx) 5 ;t;, -c ;. Also, llw’ll~ I lhlllwll~ + i(l (hz, wz) in C we compute
IIFh
+ IlwllX) I (i)2 + i(i)
= i. Thus F(E) E 2.. For (hi, WI) and
WI) -&AZ, w2)lls = IA; - h;l + lb; - w;llx 5 IIPoTc+ + +llh1w1 -
WI
-
PlT(4
PoT(4
+
+ WI)
s ;llwl - wzllx + llhw1 5 ;llwl - wzllx + flh
-
w2)llx h2W2
hlW2
+ 9T(dJ
+ hlW2
- h2l 5 &hL
+
-
Wl)
w2)IIx
(3.8)
h2W2llX -
u2.
w2)llz.
Thus, F is Lipschitz with constant 2 < 1 and the contraction mapping principle guarantees a unique fixed point. Now suppose 3 h s -i with (R - h)x = 0. Without loss of generality, take 11x11~= 1. Then either 1lPoxll~ 2 i or 11Pixllx 1 i. In either case we can apply projections to Plx + TX = Ax to contradict h I 4. In particular, either i z IlPoTxllx = Ihl IlPoxll~ 1 i Ihl, or i 2 IJP~Txllx = IA - 11 11PiXllX 2 ; lh - 1 I. To show that the unique ho in (-i, i) has multiplicity one, suppose that x1 = al 4 + WI and x2 = a24 + w2 are linearly independent eigenvectors corresponding to ho. From (3.6) we can assume without loss of generality that ai = uz = 1. However, this would mean xi - x2 is an eigenvector in Pi (X) which is impossible. Hence dim(ker(R - ho)) = 1. To show dim(ker(R - ho)2) = 1 we require the following estimate, which will be established later, IA - hoI 5 6(llPoW - A)(+ + w)IIx + lPl(R-
A)(+ + w)llx)>
V(h, w) E 1.
(3.9)
Assuming (3.9) for now, let x1 = ai+ + WI and x2 = u~~#I+ wz be two linearly independent solutions of (R - h~)~x = 0. Without loss of generality, we assume that al = 0. If this isn’t the case, replace x1 by x2 - (u2/ui)xl. From (3.6), Po( (R - &)x1) # 0 so write (R - ho)xl = c$ + wo.
(3.10)
For t E Iw define and
wo(t) = (R - ho)x, - 4 - txl = w. - txl, h(t) = ho -t.
(3.11)
932
LEW
LEFTON
By (3.10) C#J+ wg E ker(R - ho) therefore llwsll~ < i and l&l small t, Ilwo(t)ll~ < i and IA( < i. We calculate (R - h(t))(+
+ we(t))
< i. Hence, for sufficiently (3.12)
= ((R - ho) + t)((R - ho) - f)Xl = -12x1.
Combining (3.9), (3.11) and (3.12) we have ItI = (h(t) - hoI < 6t2 (llP,-,xi IIX + 1lZ’rxi 11~) which is a contradiction for small t > 0. Thus dim(ker(R - ho)2) = 1. It remains to justify (3.9). Using (3.7) we have IIF(A WI - (A w)II,x = IIPoTc+ + WI - Wllx = IIPW - A)(+ + w)llx
+ llP*T(@ + WI - hw + wllx + llPl(R - A)(4 + w)IIx.
(3.13)
Recall that the fixed point (ho, wg) of F in 1 is obtained as n-00 lim F”(h, w) where F” is the nth iterate of F. Using (3.13) and (3.8) we have n-l
IIF”(A
w) - (ho, wo)ll.z I c
($)”
(IIPoW
A)(+
5 6(llPoW
- A)(+
+ w)IIx
+ w)llx
+ IIPIW - A)(+
+ 4x)
k=O
Estimate (3.9) follows.
+ llPl(R - A)(+
+ w)IIx).
n
We observe that the operator Z--A0 in (3.4) clearly has the form described in Lemma 2 with X = BC and Y = L’. We also notice that condition (3.5) will hold if ll~~oll~ is sufficiently small. Also, if ho < 0 then the value of fi in (3.3) is 1 and if ho > 0 then fi = 0. In either case we have (3.14)
i(Z - d,yo, 0) = (-1)s = sgn(h0).
LEMMA 3. Let F : C - X be defined by (3.7) with X = BC, Y = L’, PO and PI defined by (2.4), and T = Po(qg’(yo)y) + GPl (r-/g’(yo)y) where yo is an isolated solution of (2.1). Suppose (P, z) = F(0, 0) and (ho, wg) is the fixed point of F. If Jlyo(lm is small enough, we have sgn(hs) = sgn(p).
Proof. By (3.7) and the definition of p and ho we have /.M$ = POT+ and ho@ = PoT($ Thus it suffices to show Jho - ~1 = IIPoTw,-Jm = o(lpl) as 11~~011~ - 0. Since (ho, wo) is a fixed point of F in C we have +ollm
5 11- hol IIwolloo= llST(+
+ WO).
+ wo)llm I Cllg’(yo)+lh + Cllg’(yoMoll1 5 Cllg’(yoh#Al1 + Cllwollm llg’bo)II1.
Therefore, for ~0 sufficiently small, (3.15)
IIWOllm5 Cllg’(Yob#dll.
Using (3.15) and the Cauchy-Schwarz inequality
we obtain
IIPoTwoIIm = IIPog’(yohoIlm 5 CllPog’(Yo)llm Ilwollm 5 Cllg’(yo)Q,II: 5 Cllg’(yo% But IpI = IIPoT+llm = Cllg’(yo)+2111 which implies
llg’(yoM21h
IIPIITwcII~~ = o(lpl) as lly~ll~ - 0. n
The proof of Theorem 2 now follows from standard arguments. Let llfll i be so small that Theorem 1, Lemma 1, Lemma 2 and Lemma 3 all hold simultaneously. Then Theorem 1 guarantees that 3~0 E B(r) solving (2.1). It is also known [24] that the Leray-Schauder degree &(I - a, B(r), 0) = q.
933
Superlinear boundary value problem at resonance
For each solution yo we have, from Lemma 1, that y. is isolated. By (3.14) and Lemma 3 we deduce that i(Z - Xyo, 0) = sgn(p) where p = Fo(og’(yo)+) = r~CJ,bg’(y~)c$~ dx. Since g’ (yo ) 10 we conclude sgn ( p) = q. Hence, from the excision property of Leray-Schauder degree, there is at most one solution in B(r), and therefore there is exactly one solution in B(r). 4. A COUNTEREXAMPLE
FOR L NONSELFADJOINT
The hypothesis that L is selfadjoint is necessary in Theorem 2. Below we consider an example which is nonselfadjoint and which has infinitely many small solutions for certain small f. See the note at the end of the introduction for recent work related to this example. Consider the problem y” + y3 = f(x)
on -15x51,
y(1) = y(-1),
y’(1) +y’(-1)
= 0.
(4.1) In this example, Ly = y” and $ = 1 spans ker(L). The operator L is a Fredholm operator of index 0. This follows from the closed range theorem [34] after noting that the range of L is a closed, co-dimension 1 subspace of L1 (see Lemma 1.1, [24]). Also observe that $3 = 1 is in R(L) since x2/2 E BC and L(x2/2) = 1. THEOREM
3. The boundary value problem (4.1) has infinitely many solutions for small even f.
Proof. First note that if y is even and y’ is absolutely continuous then y E BC. Also, if y is an even function then so is Ly = y” + y3. We claim that for small positive numbers b and small even functions S, the equation y” + y3 = f has a unique solution satisfying initial conditions y(0) = b > 0
and
y’ (0) = 0.
(4.2)
Let 6 be a positive number, to be determined. Suppose f is an even, integrable function on l-1, 11 such that llfllr < 6. Any solution of y” + y3 = f with initial conditions (4.2) can be written as y(x) = 6 - [0x(x - t)[f (t) - y3(t)] dt. Define ay=6-
I
~(x-1)[f(1)-y3(f)ldf.
We will show that a has a fixed point in D = {y : y’ is absolutely continuous llyllm < 3Sl.Fory~Dwehave IlAyll,
= sup
XE[O.ll I
~6+llflll+
6 -
I
on [0, 1] and
;lx - t)[f 0) - y3(t)l dr 1
x~[o suptl ;IIy311m
I26
+ p3
I 36
where the final inequality holds if and only if d2 I 2/27. Thus, if 62 < 2/27 then 31 maps D into itself. Furthermore, from the mean value theorem, for y and z in D, lly3 - z3 Ilm I 311r(t~211mlly-211m where Ir(r)I < 3bsincer(t) isbetweeny(t) andz(t) forallt.Thus, ]]y3-z3]loo I 27b21]y - z]lm and x IlAy- .AZIIca = sup (x - t)(y3 - z3) dt I 27fi21]y - zllo3. XE[O.I] II 0 Therefore, if S2 < 2/27 then Jt is a contraction on D and by the contraction mapping principle a has a unique fixed point yo in D. Extending yo to [ - 1, 1] by evenness we obtain a solution to n (4.1). Hence, for a fixed f, sufficiently small, there are infinitely many small solutions.
934
LEW LEFTON
We remark that the nonlinearity g(y) in (2.1) can be replaced by a more general function which is structurally similar to lylm-ly for m > 1 in a neighborhood of 0. Although the behavior of g near 0 is crucial for this result on small solutions, we point out that g(y) can have rather arbitrary behavior at co. In particular, homogeneity in y is not necessary (see [25]). Indeed, the result holds for as y - O+, Cl IYI ml-ly + o( Iyl”‘) g(y) = i C21ylmz-‘y + o( Iylmz) as y - oprovided ml > 1 and
m2
> 1. This result is also generalized in [31]. REFERENCES
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