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On the number of solutions of an elliptic boundary value problem with jumping nonlinearity
Nonlinear Andysw Theory, Merhods & Applicdom. Primed in Great Britain.
0362-s46x/a9 13.00+ .@I @ 1989 Pcrgamon Press plc
Vol. 13. No. 3. pp. 341-351. 1989.
ON THE NUMBER OF SOLUTIONS OF AN ELLIPTIC BOUNDARY VALUE PROBLEM WITH JUMPING NONLINEARITY NGUYEN PHUONG Ck Department
of Mathematics,
The University
of Iowa,
Iowa City, Iowa 52242, U.S.A.
(Received 17 February 1988; received for publication 19 April 1988) Key words and phrases: Eigenvalues,
Leray-Schauder
degree.
1. INTRODUCTION LET Sz be a bounded domain in the Euclidean space RN (N r 2) with smooth boundary aa. We denote by
the sequence of eigenvalues of -A with 0 boundary data, each eigenvalue appears in the sequence as many times as its multiplicity. pi denotes an eigenfunction corresponding to Ai with norm 1 in L2(sZ) (i = 1, 2, . . .); we choose pr > 0 in a. Throughout the paper, ,Ik with a fixed k > 1 is a simple eigenvalue. Letf: IR + IRbe a continuously differentiable function such that liy-f(s)/.s = f* and I,_,
= f(u) + h + rvkin Q
u = 0 on XI,
as ‘5varies. We shall derive this from a detailed discussion of the BVP -Au
= f+u+ -f-u-+h+rp,inR,
u
for various values of r; this discussion is also of independent u-(x) = u+(x) - u(x) for x E Q). For this task, it turns out and C(f_, f,), introduced by Gallouet and Kavian [2] play (&_, , Ak+ 1)2, it is established in [2] that there exist a unique u E Wzs2(sZ) such that -Au
= CYU+- flu- + C(CX,/?)(D~in SI,
=
0 on
asz
(1)
interest (u+(x) = max(O, u(x)), that two numbers, C(f+, f_) a crucial role. Given (Q, /3) E number C(CX,/.?) and a unique
u = 0 on &.2,
(us Q%)= 1 where (4, -) denotes the inner product following.
in L’(Q). Among other results, we shall prove the
THEOREM
1. Suppose that C(f+, f_) = 0, C(f_ ,f+) > 0. Let he E C’(@, ho = 0 on aa, (he, cur) = (h,, pk) = 0 be given. Then for t > 0 sufficiently large, there exist four numbers rf,7,,r2,r,*with-corr:<5,
NGU~N
342
PHUONG CAc
(i) for r E [r,, rz] exactly two solutions, one of which is nonnegative and the other changes its sign on fi; (ii) for 5 E (rr, rt)\[ri, r2] at least two solutions; (iii) for r = r$ at least one solution, and for r > r; no solution; (iv) for r 5 rr, if r: > -co, at least one solution. If r $ [r, , r,], then every solution of (1) changes its sign on n. We mention for comparison that under the assumption C(f+ ,f_) = 0 C(f_ ,f+) > 0, Gallouet and Kavian prove [2] that given any h E L’(Q), h I vlk, there exists a E [Rsuch that if r I a, the BVP (1) has at least one solution. Results of the same nature are given in [5]. On the other hand, the exact number of solutions of the BVP (1) has been obtained by Solimini [6] when r = 0, h = f 9, and the problem is not at resonance, i.e. when C(f+ ,f_)C(f_ ,f+) # 0. However, [6] considers neither the case of semiresonance that we discussed in theorem 1 nor the case of resonance C(f+ ,f_) = C(f_ ,f+) = 0 that we shall also discuss later. These cases involving resonance seem to require some different techniques. Finally, we wish to mention that the cases the interval (f_ ,f+) contains a multiple eigenvalue or more than one eigenvalue are considered by Lazer and McKenna [3, 41. As to be expected, the results obtained are not as precise as in the case of crossing a simple eigenvalue considered here. 2. THE PIECEWISE LINEAR EQUATION In this section we consider the piecewise linear equation (1). We recall a few results established earlier in the literature that we shall need later. Let P denote the projection on 9: in L’(Q). We have the following lemma. LEMMA1 [2]. Given 01,p E (Jk_i, &+,) and h E 9: in L’(0). Then for each s E &?there exists a unique u(s) E W2v2(sZ)such that (u(s), 9J = 0, -Au(s)
= P[cY[u(s) + s9,]+ - p[u(s) + SV)~]-) + h in Q, (2)
u(s) = 0 on an. The mapping s + u(s) is continuous Us) = Ls -
is continuous
*bW !'
from k? to W2s2(S2) and the function + S9klC - P[W
+ ~rp,l-lbDk~,
s E IR,
(3)
on m.
We note that u(s) + ~9~ is a solution of (1) if and only if L(s) = T. Since k > 1, 9k changes its sign on a and we have the following. LEMMA2 [2]. Given a,/3 E &_i, UE w 2~2(sZ)such that -Au
&+i ), there exist a unique number C(CY,/3) and a unique
= CYU+- pu- + C(cr, p)~p~in Sz, (u, 9k) = 1.
u = 0 on &2, (4)
An elliptic
boundary
343
value problem
The function C( *, 0) is continuous from (Ak_, , Ak+,) X (A,_, , A,,,) into R and is strictly decreasing with respect to each variable (the other variable being held fixed). Furthermore, lim LO = C(cr, p), s-m s
lim LO = C(j, 01). s---m S
(3
From now on let
ho E
cva,
h, = 0
on aQ
with (A,, pr) = (h,, e+) = 0
(6)
be given. Then there exists a unique B E WzSp(Q) C C’**(Q) with (ILE (0, l), (6, pr) = (a, ~7~)= 0 satisfying -A6
= f+fi f h, in Sz,
v’ = 0 on an
(7)
Because avr/an < 0 on as2 where n denotes the unit normal to aS2 pointing outward, we can choose t > 0 sufficiently large such that 7i= O+f+tylA,
>OinQ,
2
c 0 on an.
03)
Throughout this section we assume that such a t has been chosen. Let T, < 0 and 7, > 0 be defined as follows: 7vk
s
72 = sup 7> 0
1; +
~ lk
-f+ 7vk
(10)
-f+
Then for 7 E [7r, 721, 7vk
(11)
'+&-f+
is a nonnegative solution of the BVP -Au
= f+u+ -f-u_
+ ho - tp, + ?c& in Q,
u = 0 on an.
(12)
With our choice of t, t > 0 sufficiently large, it is not difficult to see that this is the only soluof (12) which does not change its sign on Sk Since we assume that &_r C f_ < & c f+ < Ak+lr this solution is isolated and its index is (- l)k when considered as a fixed point of the related mapping in L2(sZ). For a function u defined on S2, let tion
[x(n)l(x) = 1
if u(x) > 0,
= 0
if U(X) 5 0,
and 474 = f+2!04 + f-x(-d.
x E sz,
PHUONC C.&c
344
NCIN~N
A solution u of (12) is nondegenerate
if and only if [6] the BVP
-Au
= a(u)v in S2,
u = 0 on at2
has only the trivial solution v = 0. Every nondegenerate solution of (12) is isolated by the local inversion theorem [l]. In determining the exact number of solutions of (12) the following result of Solimini [6] plays an important role. LEMMA3. Suppose t > 0 sufficiently large is chosen as above. Then for r E [pi, rz] every solution of (12) that changes its sign on $2 is nondegenerate and its index is (- 1)&-l. In order to handle the cases of semiresonance and resonance, we find it necessary to locate the solutions in relation to pk. For a given [ E R we shall let
E?’ = Iv E L2(Q1(v, ~)k)< 0,
,!?,U’= closure of Ecr' I
E") * * ,- = L.2(D)b!?/n 9Ecr) r = closure of ,I$” respectively. We have the following For [ = 0 we write E,,El,E,,Er for E/O), IT/'), Ej'), ,!TJ'), theorem. THEOREM2. Suppose that ho and t are chosen as in (6) and (8). A. Suppose also that C(f_ ,f+) > 0, C(f+ ,f._) > 0. Then for 7 E [ri, r2] the BVP (12) has exactly three solutions; one solution is nonnegative and two solutions change their signs on a: (i) if T E [r,, 0), two solutions, including the nonnegative one, are located in E,,the other solution is located in E,; (ii) if T = 0, the nonnegative solution is orthogonal to (Pi in L’(Q) and each half space E, and El contains one solution; (iii) if ‘5E (0, r,], two solutions, including the nonnegative one, are located in E,,the other solution is located in E,. B. Suppose that C(J_ ,f+)> 0,C(f+ ,f_)< 0.Then for 5 E [TV, r,] the BVP (12) has exactly two solutions, one solution is nonnegative and the other solution changes its sign on 0: (i) if 5 E [7i, 0), the nonnegative solution is located in E, and the other solution is located in E,; (ii) if T = 0, the nonnegative solution is orthogonal to V)kin L’(Q) and the other solution is located in E,; (iii) if 5 E (0, 721, both solutions are located in E, . C. Suppose that C(f_ ,f+) < 0, C(f+ ,f_) < 0. Then for 5 E [T,, r2] the BVP (12) has exactly one solution which is nonnegative on Sz. If T E [ri, 0) it is located in E,, if 7 = 0 it is orthogonal to V)k, if r E (0, r2] it is located in E,. Proof.
Let K = (- A)-’ be the solution operator.on -Au
= h in Q (h E L2(sZ)),
L’(Q) associated with the BVP v =
0 on an.
Let To: L2(sZ) + L2(n) be defined by T,w := w - K[f+ w+ -f-w_),
w E L?(Q).
(13)
An elliptic boundary
345
value problem
The proof of the exact number of solutions is similar to [6] and therefore we omit it. It makes use of lemma 3 and the fact that because the BVP (12) is nonresonant we can compute (cf. e.g. [2]) deg(T,, BR , K(h, - tyl, + 710~)) where BR is the open ball in L*(Q) centered at the origin and of radius R sufficiently large. To prove the claims about the locations of the solutions we first observe that it follows from the expression (11) of the nonnegative solution that on the interval [7J(& - f+), T,/(& - f+)] the graph of L( a) is a straight segment passing through the origin and of slope & - f+ < 0. The claims are then deduced from (5) and the fact that with u(s) from (2), v(s) + su)&is a solution, of the BVP (11) if and only if L(s) = 7. n We now consider the case of semiresonance C(f_,f+) > 0, C(f+,f_) = 0. We first determine the number of solutions in the half space E, which, because of the assumption C(f_ , f+) > 0, is relatively easier to do. 1. Suppose that h, and t are given as in (6) and (S), suppose f+) > 0. 5, and r2 being defined in (9) and (lo), we have the following.
PROPOSITION
C(f_,
also that
A. For 7 E [0, s2], the BVP (12) has exactly two solutions in the half space I?,, one of which is nonnegative, the other changes its sign on Q. Furthermore there exists a number rf with 7, < 7; < CD such that for 7 E (72, 5;) the BVP (12) has at least two solutions in I?,, for 7 = 7; the BVP (12) has at least one solution; each of these solutions changes its sign on Q and for 5, > 7; the BVP (12) has no solution. B. For 7 E [TV, 0) the BVP (12) has exactly one solution in I?,. For 7 < ‘5, the BVP (12) has at least one solution in I$; each of these solutions changes its sign on 0. Before embarking on the proof, we wish to point out that by this proposition, solutions in the half space I?, does not depend on the behavior of C(f+ , f_).
the number of
Proof. Since C( - , a) is continuous and strictly decreasing in each variable according to lemma 2, we can pick any_ E (f_ , A,),3_ sufficiently close to f_ , such that for every y E [0, 11we have C(f-,f+)
and also (a) if C(f+ ,f_) (b) if C(f+,f_)
2 C(f-
I 0 then C(f+ ,$-) > 0 then C(f+,f-)
+ r(L
< C(f+ ,f-) > C(f+,3_)
-f-),f+)
> 0
(14)
5 0, > 0.
For each y E [0, l] we consider the BVP (P,)
-Au
= f+u+
- [f_ + y(L
- f-)]u-
+ h0 - tq$ + vk
in i&
u = 0 on XL
(15)
For 7 E [51, r2], each of these BVP has a nonnegative solution given by (11). The graph of the corresponding function L(s) on the interval [r,/(& - f+), 7,/(& - f+)] is a straight segment passing through the origin with slope Ak - f+ < 0. Proof of part A. Suppose 7 E [0, r2]. For the reason just given, for each y E [0, l] and E E (0, r,/(Ak - f,)), (P,) has no solution on the boundary of E,@). Furthermore, there exists a constant R > 0 such that if u is a solution of (P,) located in E,‘e’, y E [0, l] then /lull < R (II - 11 denotes the norm in L’(Q)). The proof of this claim proceeds as in [2], making use of (14).
346
NGWZN PHUONG CAc
For y E [0, l] let T,: t’(Q)
--+L’(Q) be defined by
T,w := w - K(f+ w+ - If-
Then by the homotopy
+ Y(f-
invariance of the Leray-Schauder
deg(T, , B, n E,‘“‘, K(h,
w E L2(cJ).
-f-)lw--I,
(16)
degree:
- tq, + SV)~)) = deg( T, , BR n EI(E),K(h,
- tp, + npJ).
(17)
By theorem 2 and lemma 3, in both situations (a) and (b), in BR fl EI(E) the BVP (Pi) has exactly a nonnegative solution with index (- l)k and a solution with changing sign with index (- l)k-‘. Therefore, deg(T,, BR n E,@), K(h,
- tcp, + SV)~)) = 0,
deg(T, , B, fI EI(E),K(h,
- tpl + T(P~)) = 0.
and by (17): (18)
Since the BVP (P,), i.e. (12), has a nonnegative solution with index (- l)k in B, n E,‘E’ and by lemma 3 any solution with changing sign of it has index (- I)&-‘, (18) shows that (12) has exactly one such solution in B, fl E, @). Because E > 0 can be arbitrarily small, we see that the BVP (12) has exactly two solutions in i?,: U, (nonnegative) and u2 (with changing sign). Let si = (ai, (Pi) where tii (i = 1, 2) are solutions of (12) for T = s2, and r; = maxfl(s) 1s2 Is
5 slJ.
(19)
Then r; > r2, otherwise the BVP (12) with r = r2 or T < r2, r close to r2, has more than two solutions, which is a contradiction. For the same reason, L(s) < r! for every s E (- ~0, s,). It is then not difficult to see that all the claims concerning rf are true. Proof of part B. Consider r E [r,, 0). Again, because for each y E [0, 11, the graph of the function L(.) related to the BVP (P,) on the interval [r2/(Ak - f+), r,/(Ak - f,)] is a straight segment passing through the origin with slope & - f+ < 0, for every y E [0, l] the BVP (P,) has no solution on the boundary of E, when r E [ri, 0). As above, using (14). it can be shown that there exists a constant R > 0 such that if u is a solution located in E, of (P,), y E [0, I] then l/u]] < R. Then
deg(G, BR n E,, Wk, - W, +
7vJ)
=
deg(T,
9 BR
n 4,
WQ
-
tv1
+
WA).
But deg(T,, BR i7 E,, K(h,
- ta, + ~9~)) = (- l)k-l,
(20)
because in both situations (a) and (b), by theorem 2, the BVP (Pi) has exactly one solution with changing sign located in E, with index (- l)k-l when T E [ri, 0). Since any solution of (P,), i.e. the BVP (12), that is located in E, must change its sign (for T E [pi. 0) the solution of (12) of constant sign is located in E,), its index is (- l)k-l by lemma 3. It then follows from (20) that there is exactly one such solution of the BVP (12) in El. Since lim L(s)/s = C(f_ , f+) > 0 and s-.---m L(a) is continuous, it is clear that for r < T, the BVP (12) has at least one solution. n The proof in [6] can be adapted to show that for r < 0, 151large, the BVP (12) has exactly one solution in E, under the hypothesis C( f_ , f+) > 0. In [6] the proof is carried out for the case ho = 0, r = 0.
Remark.
An elliptic
boundary
337
value problem
Using the same method we can prove the following. PROPOSITION2. Suppose that h, and t are given as in (6) and (8), suppose also that C(f_ ,f+) c 0. 7, being defined as in (lo), we have the following. A. For 7 E [O, 7J the BVP (12) has exactly one solution in the half space E,, it is nonnegative. For 7 > rz the BVP (12) has at least one solution, it changes its sign on R. B. For 7 < 0 the BVP (12) has no solution in the half space Z?!. We now determine the number of solutions of the BVP (12) in the half space E, when it is at semiresonance, i.e. when C(f_ ,f+) > 0, C(f+ ,f_) = 0, for example. PROPOSITION3. Suppose that he and t are given as in (6) and (8); suppose also that C(f_ ,f+) > 0 and C(f+ ,f_) = 0. ti being defined as in (9), we have the following. A. For 7 L 0 the BVP (12) has no solution in E,. B. For r E [T,, 0) the BVP (12) has exactly one solution in E,, it is nonnegative. Furthermore, there exists r: with - CQzz 7f < T, such that for T E (7?, ti) the BVP (12) has at least one solution in E,,it changes its sign on a; if - m < 7; then the BVP (12) has no solution in E,for T < 7;. Proof. As in the proof of proposition f_, such that for each y E (0, l]
1, we choose anK
E (f- , A,), f_ sufficiently close to
Y(3-- s-hf+>zf+ 0, w-+ ,A. + YCL-f-N < at-+ 7.f.J= 0. C(f- J,)
> 0%
+
(21)
For each y E IO, I] we consider the BVP (P,) of (I>). Proof of part A. Suppose 7 1 0. Because the function L.(a) is continuous on IR, to show that the BVP (12) has no solution in E,for 7 2 0, it suffices to show that it has no solution in Erfor T E [0, 7.J where 7, is defined by (10). As we have noted in the proof of proposition 1, with 7 E IO, rJ, y E [0, 1) (P,) in (15) has no solution u with (u, @)k)= E where E is any number in (0*7*4& - f,)). We first observe that for 0 < y s 1, because of (21), we deduce from theorem 2, Part B (ii), (iii), that (15) has no solution in E,w where E E (0, t,/(& - f,)). For 7 E [0, 7J, the only solution of the BVP (P,) in (15) that does not change its sign on Q is nonnegative and given by (ll)? its index is (- l)k. All other solutions of (P,) change their signs on R, are nondegenerate, isolated and each has index (- I)*-’ by lemma 3. We deduce from this that any bounded subset of E,?,E E (0, T,/(&- f,)) can contain only a finite number of solutions of (P,). In fact, suppose by contradiction 3 R > 0 such that BR n E,? contains an infinite sequence of solutions (v,) of (P,). It then follows without too much difficulty from the theory of elliptic boundary value problems that we can extract a subsequence, still denoted by (u,], converging in w’~‘(SJ) to u and u is also a solution of (P,). But then v is not isolated. Suppose now, by contradiction, that the set of solutions of (PO) in .!ZF) is not empty. Then we can find a bounded open set U of E,'" which contains n solutions of (P,) with n > 1 but the boundary aU of it does not contain any solution of (P,), 0 s y I 1. In fact, we can choose R > 0 large enough so that BR flEj&'contains n solutions u,, u2, . . . , u, (n 1 1) of (P,). If aB, 17i?,?contains no solutions of (P,) then we take U = BR I? E,?.If aBR tl Eyj contains
NGUYEN PHUONG Clc
348
some solutions (necessarily finite in number), say wi, ..., w,, we can find an open set V containing w,, w2, . . . , w, such that P contains none of the Vi (i = 1, . . ., n). We then take U = (BR fl E,‘“‘)\t? We recall that by previous argument (P,), y E (0, l] has no solutions in 0. Thus with the mapping T,: L2(sZ) + L’(Q) defined as in (16), it follows from the homotopy invariance of Leray-Schauder degree that deg(T,, u, K(h, - t9, + ~9~)) = deg(T,, U, K(& - t9, + ~9~)) = 0. But since the index of each ui (i = 1, 2, . . . , n; n 2 1) contained in U is the same, this is not possible. Since E is arbitrary in (0, ~l/(Ak - f,)), part A is proved. Proof of part B. Suppose T E [Al, 0). For each y E [0, I] the BVP (P,) of (15) has a nonnegative solution (0 given by (7)) ~ u, = 0 + f+ty,
+
-
T9k
lk -f+’
located in E,. The index of u1 is (- l)k. There is no other solution of (P,) that does not change its sign on a besides ut. We are going to show that in E, there are no solutions of (P,) with changing sign either. Above we have shown that a bounded set of E, can only contain a finite number of solutions of (P,) (we note that obviously u1 is an isolated solution of (P,)). Furthermore, by theorem 2, part B (i), for y E (0, 11, (P,) has exactly two solutions, one of which is ut , the other one is located in E,. Therefore if (P,) has other solutions in the half space E, besides ut, then, as above, we can find a bounded open set U of E, containing the solutions Ul, u2, . . . . u, (00 > n L 2) of (P,) but aCJ contains no solutions of (P,), 0 i y 5 1. The homotopy invariance of degree gives deg(G, u,K(ho
- t9, +
T9k)) = deg(T,
US&~,
+ f91 + T9k)) = (- I)k.
But this is impossible, because index U, = (- l)k and index Ui = (- l)k-’ for 2 I i 5 n. Concerning other claims in part- B, consider the function L( *) on the interval b,/(Ak - f+), ‘=‘)a If J!,(S) is not bounded from below on this interval, we take rr = - 00. Suppose then that " -lS
We have tf < ~1. If not, then either L(e) is constant on the interval [rt/& - f,), co) and the BVP (P,) of (15) for T = 51 has an infinity of solutions or it has at least two solutions for T > ~1, T close to '51,contradicting what we have established above. Because L(e) is continuous, for any T E (~7, TV) we can find s such that L(s) = T and therefore the BVP (P,) has at least one n solution. By definition of TV this solution must change its sign on Q. Remark. If the BVP (12) is at semiresonance with C(f+ , f_) = 0 and C(f_ ,f+) C 0 instead of C(f_ , f+) > 0 as in proposition 3, then proposition 3 still holds. The proof is similar, making
use of theorem 2, part C, instead. Proof of theorem 1. Theorem
1 is just a consequence of proposition
1 and proposition
3.
n
An elliptic boundary value problem
349
We now consider the case when the BVP (12) is at resonance. THEOREM 3.
Suppose that ho and t are given as in (6) and (8). Suppose also that C(f_ ,f+) = C(f+ ,f_) = 0. Then for r E [rr, r,], where rt and r2 are defined in (9) and (lo), the BVP (12) has exactly one solution which is nonnegative. Furthermore, there exist rr < r1 and rf > r, such that for r E (57, rf)\[rl, rz] the BVP (12) has at least one solution. Proof. Choose an arbitrary 3_ E (f_ , A,) and for y E [0, I] we again consider the BVP (P-J
-Au
= f+u+ - [f- + y(j;_ - f_)lu-
+ h, - tv, + rp, in S2,
u = 0 on f3R.
(22)
ForOcyzzlwehavebylemmal: C(f-
+ r(j’- -f-M-+)
C(f+ ,f- +
< C(f- ,I-+) = 0
vt3- - f-1) < w-+ J-1 = 0.
therefore by theorem 2, part C, the BVP (P,) of (22) has for r E [rr, r2] exactly one solution which is nonnegative and located in E, if r, I T < 0, is orthogonal to $Jkif r = 0 and is located in E, if 0 < r I r,. Using this, we can prove the claims for the case r E [r,, rz] by the same method used for proposition 3. The existence of rf and rz with the claimed properties can be seen as in the proof of part B, proposition 3. H 3. THE
NONLINEAR
EQUATION
Let f: R -+ R be of class C’ such that lim f’(c) r-
= fk
with
fm
g(c)
=
f(C) - f+[’
+ f_[-
lk-l
Ak
lk+l,
is a bounded function.
We have the following. THEOREM 4.
Suppose that C(f_, f+) > 0, C(f+, f-)
= 0. Then given he E C’(n),
h0 = 0 on
6X2,the BVP -Au
= f(u) + ho - tfp, + tr’pk in Q
u = 0 on
asz,
(23)
for t > 0 sufficiently large and ]r’] adequately small has exactly two solutions. The proof is based on theorem 1 and it will be clear that similar results concerning the BVP (23) can be proved under other situations of nonresonance, semiresonance or resonance, making use of theorem 3, for example in the case of resonance. We use the method of proof of Solimini [6], who considers the case /I, = 0, r’ = 0 and the BVP is not at resonance. We shall have to make modifications in his proof to allow for h, # 0 and for semiresonance. The proof is achieved in three stages. PROPOSITION 4. Under the hypotheses of theorem 4, the BVP (23) has at least two solutions for any given h, E C’(a), ho = 0 on asZ, t > 0 sufficiently large and ]r’] adequately small.
350
Cic
NGUYEN PHUONG
Proof. Using its positive homogeneity,
we deduce
from theorem
1 that for T’ with 15’1 small,
the BVP -Au
= f+n’
-f-u-
- q7, + s’yl, in S2,
u = 0 on
has exactly two solutions ut, u2 and they are nondegenerate. homogeneous boundary condition, let
T,u = u - K(f+u+
u E aUi = dist.(T,u, Choose
-A
having
u E L’(Q).
we can find disjoint
K(-cp,
With K = (-A)-‘,
(24)
-f-u-),
Tu = u - Kj-(u), Because Ui (i = 1,2) are nondegenerate stant c > 0 s.t.
asz,
neighborhoods
+ ~‘9~)) > c in c(Q),
t, > 0 such that if t > t, then we have the following
Vi of Ui and a con-
i = 1,2.
three properties:
for u E aUi, i = 1,2;
i=1,2; and, for later use; (iii) the BVP = f+u+ -f-u-
-Au has exactly
two solutions.
ho + t - p, + r’fpk in Q,
This is possible t2 =
zIlKI
by theorem [me&91
u =
0 on
ac2,
1. Let
I’* supkl,
and H(y, u) = T,u + y(Tu - T,u), Then with t > max(t,, dist(H(y,
t2), we have for u E a(tu,),
w), K(h,
Y E [O, 11.
u E L*(Q),
i = 1,2 and y E [0, I]:
- tcp, + ~T’v)~)) > dist(T,u,
K(h,
- tp, + ~T’v)~)) - T
rCt_ct,>o. 2
Then the homotopy
invariance
deg(T, tUi, K(h,
of the Leray-Schauder
- 19, + t~‘qk)) = deg(T,,
= *1,
2
degree gives tUi, K(h,
i= 1,2,
- tpl - tr’pk))
351
An elliptic boundary value problem
using the positive homogeneity of To and (ii) above. Since t& (i = 1,2) are disjoint, proposition 4 is proved. n PROPOSITION 5. Suppose that C(f_ ,f+) > 0, C(f+ ,f_) = 0 and h, E C’(n), h, = 0 on XX For T’ sufficiently small, let U be a bounded open set in L’(Q) containing ali solutions of the BVP (24), then there exists to f IRsuch that for t > to the BVP (23) has no solutions in L2(Q)\tiJ. Proof, A similar result is proved in [6] for h, = 0, 7’ = 0 under the nonresonance assumption # 0. Suppose by contradiction that for each n = 1,2, . . . there exists I, with t, -+ ~0 and u,, E L’(Z))\~ satisfying the BVP: C(f+ , f_)
-Au
= f+u+ - fu_u-
g(t,u) + 7
n
u = 0 on i3Q.
t,
Since by theorem f with (7’1 small the BVP are nondegenerate and for large n the above it follows from proposition 2.2 in [6] that (i = 1,2). Therefore there exists a constant /u,/[ cc
+ &I - - q1 + s’q+ in 52,
(24) has exactly two solutions u,, u2 both of which BVP also has exactly two nondegenerate solutions, the local inversion theorem [I] is applicable at ui c such that
v n = I,2, ..*
in L2(sZ).
By the theory of elliptic boundary value problems, we can then extract from {u,) a subsequence, still denoted by {u,), converging to u in W2’2(fz) with u being a solution of the BVP (24) and u E L’(Q)\U. This is impossible because cf contains all solutions of the BVP (24). P~oo~o~~~eo~e~ 4, From proposition 5 and the proof of proposition 4, in particular (iii) in that proof, we see that for t > 0 large, 17’1small, the BVP - A.u = f+u’
h0
- f_ u- + -
t
- p, + 7’~~ in fz,
u =
0 on X&
has exactly two nondegenerate solutions Us, u2 and we can find in L2(52) bounded open neighborhoods Vi (i = I, 2) of ui that are disjoint and such that in each tUi there is at least one solution of the BVP (23) and no solution in _L’(Q)\t(LJ, U V,). It can now be proved as in [6], theorem 1, that taking t larger still if necessary, there is in each tiJi exactly one solution of the BVP (23). q REFERENCES 1. AMBROSEW A. & PRODSG., On the inversion of some differentiable mappings with singularities between Banach spaces, Annati Mar. pura appl. 93, 231-247 (1973). 2. GALLOU~T TH. & KAVIA?JO., RCsuitars d’existence et de non existence pour certains problemes demi-Iiniaires B l’infini, Annali Fat. Sci. Toulouse 3, 201-246 (1981). 3. LAZERA. C. & MCKENNAP. J., Multiplicity results for a semilinear boundary value problem with the nonlinearity crossing higher eigenvalues, Nor&fear Arzaiysis 9, 335-349 (1985). 4. LAZERA. C. & MCKENNAP. J., Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues I and II, Communs PartiaZ diff: Eqns 10, 107-150 (1985) and (1987). 5. RUF B., On nonlinear elliptic boundary value problems with jumping nonlinearities, Annali Mat. pur4 appt. 128, 133-151 (1980). 6. SOL~INI S., Some remarks on the number of solutions of some nonlinear elliptic equations, Analyse Non LinPaire, Inst. Henri PoincarP 2, 143-156 (1985)
0362-s46x/a9 13.00+ .@I @ 1989 Pcrgamon Press plc
Vol. 13. No. 3. pp. 341-351. 1989.
ON THE NUMBER OF SOLUTIONS OF AN ELLIPTIC BOUNDARY VALUE PROBLEM WITH JUMPING NONLINEARITY NGUYEN PHUONG Ck Department
of Mathematics,
The University
of Iowa,
Iowa City, Iowa 52242, U.S.A.
(Received 17 February 1988; received for publication 19 April 1988) Key words and phrases: Eigenvalues,
Leray-Schauder
degree.
1. INTRODUCTION LET Sz be a bounded domain in the Euclidean space RN (N r 2) with smooth boundary aa. We denote by
the sequence of eigenvalues of -A with 0 boundary data, each eigenvalue appears in the sequence as many times as its multiplicity. pi denotes an eigenfunction corresponding to Ai with norm 1 in L2(sZ) (i = 1, 2, . . .); we choose pr > 0 in a. Throughout the paper, ,Ik with a fixed k > 1 is a simple eigenvalue. Letf: IR + IRbe a continuously differentiable function such that liy-f(s)/.s = f* and I,_,
= f(u) + h + rvkin Q
u = 0 on XI,
as ‘5varies. We shall derive this from a detailed discussion of the BVP -Au
= f+u+ -f-u-+h+rp,inR,
u
for various values of r; this discussion is also of independent u-(x) = u+(x) - u(x) for x E Q). For this task, it turns out and C(f_, f,), introduced by Gallouet and Kavian [2] play (&_, , Ak+ 1)2, it is established in [2] that there exist a unique u E Wzs2(sZ) such that -Au
= CYU+- flu- + C(CX,/?)(D~in SI,
=
0 on
asz
(1)
interest (u+(x) = max(O, u(x)), that two numbers, C(f+, f_) a crucial role. Given (Q, /3) E number C(CX,/.?) and a unique
u = 0 on &.2,
(us Q%)= 1 where (4, -) denotes the inner product following.
in L’(Q). Among other results, we shall prove the
THEOREM
1. Suppose that C(f+, f_) = 0, C(f_ ,f+) > 0. Let he E C’(@, ho = 0 on aa, (he, cur) = (h,, pk) = 0 be given. Then for t > 0 sufficiently large, there exist four numbers rf,7,,r2,r,*with-corr:<5,
NGU~N
342
PHUONG CAc
(i) for r E [r,, rz] exactly two solutions, one of which is nonnegative and the other changes its sign on fi; (ii) for 5 E (rr, rt)\[ri, r2] at least two solutions; (iii) for r = r$ at least one solution, and for r > r; no solution; (iv) for r 5 rr, if r: > -co, at least one solution. If r $ [r, , r,], then every solution of (1) changes its sign on n. We mention for comparison that under the assumption C(f+ ,f_) = 0 C(f_ ,f+) > 0, Gallouet and Kavian prove [2] that given any h E L’(Q), h I vlk, there exists a E [Rsuch that if r I a, the BVP (1) has at least one solution. Results of the same nature are given in [5]. On the other hand, the exact number of solutions of the BVP (1) has been obtained by Solimini [6] when r = 0, h = f 9, and the problem is not at resonance, i.e. when C(f+ ,f_)C(f_ ,f+) # 0. However, [6] considers neither the case of semiresonance that we discussed in theorem 1 nor the case of resonance C(f+ ,f_) = C(f_ ,f+) = 0 that we shall also discuss later. These cases involving resonance seem to require some different techniques. Finally, we wish to mention that the cases the interval (f_ ,f+) contains a multiple eigenvalue or more than one eigenvalue are considered by Lazer and McKenna [3, 41. As to be expected, the results obtained are not as precise as in the case of crossing a simple eigenvalue considered here. 2. THE PIECEWISE LINEAR EQUATION In this section we consider the piecewise linear equation (1). We recall a few results established earlier in the literature that we shall need later. Let P denote the projection on 9: in L’(Q). We have the following lemma. LEMMA1 [2]. Given 01,p E (Jk_i, &+,) and h E 9: in L’(0). Then for each s E &?there exists a unique u(s) E W2v2(sZ)such that (u(s), 9J = 0, -Au(s)
= P[cY[u(s) + s9,]+ - p[u(s) + SV)~]-) + h in Q, (2)
u(s) = 0 on an. The mapping s + u(s) is continuous Us) = Ls -
is continuous
*bW !'
from k? to W2s2(S2) and the function + S9klC - P[W
+ ~rp,l-lbDk~,
s E IR,
(3)
on m.
We note that u(s) + ~9~ is a solution of (1) if and only if L(s) = T. Since k > 1, 9k changes its sign on a and we have the following. LEMMA2 [2]. Given a,/3 E &_i, UE w 2~2(sZ)such that -Au
&+i ), there exist a unique number C(CY,/3) and a unique
= CYU+- pu- + C(cr, p)~p~in Sz, (u, 9k) = 1.
u = 0 on &2, (4)
An elliptic
boundary
343
value problem
The function C( *, 0) is continuous from (Ak_, , Ak+,) X (A,_, , A,,,) into R and is strictly decreasing with respect to each variable (the other variable being held fixed). Furthermore, lim LO = C(cr, p), s-m s
lim LO = C(j, 01). s---m S
(3
From now on let
ho E
cva,
h, = 0
on aQ
with (A,, pr) = (h,, e+) = 0
(6)
be given. Then there exists a unique B E WzSp(Q) C C’**(Q) with (ILE (0, l), (6, pr) = (a, ~7~)= 0 satisfying -A6
= f+fi f h, in Sz,
v’ = 0 on an
(7)
Because avr/an < 0 on as2 where n denotes the unit normal to aS2 pointing outward, we can choose t > 0 sufficiently large such that 7i= O+f+tylA,
>OinQ,
2
c 0 on an.
03)
Throughout this section we assume that such a t has been chosen. Let T, < 0 and 7, > 0 be defined as follows: 7vk
s
72 = sup 7> 0
1; +
~ lk
-f+ 7vk
(10)
-f+
Then for 7 E [7r, 721, 7vk
(11)
'+&-f+
is a nonnegative solution of the BVP -Au
= f+u+ -f-u_
+ ho - tp, + ?c& in Q,
u = 0 on an.
(12)
With our choice of t, t > 0 sufficiently large, it is not difficult to see that this is the only soluof (12) which does not change its sign on Sk Since we assume that &_r C f_ < & c f+ < Ak+lr this solution is isolated and its index is (- l)k when considered as a fixed point of the related mapping in L2(sZ). For a function u defined on S2, let tion
[x(n)l(x) = 1
if u(x) > 0,
= 0
if U(X) 5 0,
and 474 = f+2!04 + f-x(-d.
x E sz,
PHUONC C.&c
344
NCIN~N
A solution u of (12) is nondegenerate
if and only if [6] the BVP
-Au
= a(u)v in S2,
u = 0 on at2
has only the trivial solution v = 0. Every nondegenerate solution of (12) is isolated by the local inversion theorem [l]. In determining the exact number of solutions of (12) the following result of Solimini [6] plays an important role. LEMMA3. Suppose t > 0 sufficiently large is chosen as above. Then for r E [pi, rz] every solution of (12) that changes its sign on $2 is nondegenerate and its index is (- 1)&-l. In order to handle the cases of semiresonance and resonance, we find it necessary to locate the solutions in relation to pk. For a given [ E R we shall let
E?’ = Iv E L2(Q1(v, ~)k)< 0,
,!?,U’= closure of Ecr' I
E") * * ,- = L.2(D)b!?/n 9Ecr) r = closure of ,I$” respectively. We have the following For [ = 0 we write E,,El,E,,Er for E/O), IT/'), Ej'), ,!TJ'), theorem. THEOREM2. Suppose that ho and t are chosen as in (6) and (8). A. Suppose also that C(f_ ,f+) > 0, C(f+ ,f._) > 0. Then for 7 E [ri, r2] the BVP (12) has exactly three solutions; one solution is nonnegative and two solutions change their signs on a: (i) if T E [r,, 0), two solutions, including the nonnegative one, are located in E,,the other solution is located in E,; (ii) if T = 0, the nonnegative solution is orthogonal to (Pi in L’(Q) and each half space E, and El contains one solution; (iii) if ‘5E (0, r,], two solutions, including the nonnegative one, are located in E,,the other solution is located in E,. B. Suppose that C(J_ ,f+)> 0,C(f+ ,f_)< 0.Then for 5 E [TV, r,] the BVP (12) has exactly two solutions, one solution is nonnegative and the other solution changes its sign on 0: (i) if 5 E [7i, 0), the nonnegative solution is located in E, and the other solution is located in E,; (ii) if T = 0, the nonnegative solution is orthogonal to V)kin L’(Q) and the other solution is located in E,; (iii) if 5 E (0, 721, both solutions are located in E, . C. Suppose that C(f_ ,f+) < 0, C(f+ ,f_) < 0. Then for 5 E [T,, r2] the BVP (12) has exactly one solution which is nonnegative on Sz. If T E [ri, 0) it is located in E,, if 7 = 0 it is orthogonal to V)k, if r E (0, r2] it is located in E,. Proof.
Let K = (- A)-’ be the solution operator.on -Au
= h in Q (h E L2(sZ)),
L’(Q) associated with the BVP v =
0 on an.
Let To: L2(sZ) + L2(n) be defined by T,w := w - K[f+ w+ -f-w_),
w E L?(Q).
(13)
An elliptic boundary
345
value problem
The proof of the exact number of solutions is similar to [6] and therefore we omit it. It makes use of lemma 3 and the fact that because the BVP (12) is nonresonant we can compute (cf. e.g. [2]) deg(T,, BR , K(h, - tyl, + 710~)) where BR is the open ball in L*(Q) centered at the origin and of radius R sufficiently large. To prove the claims about the locations of the solutions we first observe that it follows from the expression (11) of the nonnegative solution that on the interval [7J(& - f+), T,/(& - f+)] the graph of L( a) is a straight segment passing through the origin and of slope & - f+ < 0. The claims are then deduced from (5) and the fact that with u(s) from (2), v(s) + su)&is a solution, of the BVP (11) if and only if L(s) = 7. n We now consider the case of semiresonance C(f_,f+) > 0, C(f+,f_) = 0. We first determine the number of solutions in the half space E, which, because of the assumption C(f_ , f+) > 0, is relatively easier to do. 1. Suppose that h, and t are given as in (6) and (S), suppose f+) > 0. 5, and r2 being defined in (9) and (lo), we have the following.
PROPOSITION
C(f_,
also that
A. For 7 E [0, s2], the BVP (12) has exactly two solutions in the half space I?,, one of which is nonnegative, the other changes its sign on Q. Furthermore there exists a number rf with 7, < 7; < CD such that for 7 E (72, 5;) the BVP (12) has at least two solutions in I?,, for 7 = 7; the BVP (12) has at least one solution; each of these solutions changes its sign on Q and for 5, > 7; the BVP (12) has no solution. B. For 7 E [TV, 0) the BVP (12) has exactly one solution in I?,. For 7 < ‘5, the BVP (12) has at least one solution in I$; each of these solutions changes its sign on 0. Before embarking on the proof, we wish to point out that by this proposition, solutions in the half space I?, does not depend on the behavior of C(f+ , f_).
the number of
Proof. Since C( - , a) is continuous and strictly decreasing in each variable according to lemma 2, we can pick any_ E (f_ , A,),3_ sufficiently close to f_ , such that for every y E [0, 11we have C(f-,f+)
and also (a) if C(f+ ,f_) (b) if C(f+,f_)
2 C(f-
I 0 then C(f+ ,$-) > 0 then C(f+,f-)
+ r(L
< C(f+ ,f-) > C(f+,3_)
-f-),f+)
> 0
(14)
5 0, > 0.
For each y E [0, l] we consider the BVP (P,)
-Au
= f+u+
- [f_ + y(L
- f-)]u-
+ h0 - tq$ + vk
in i&
u = 0 on XL
(15)
For 7 E [51, r2], each of these BVP has a nonnegative solution given by (11). The graph of the corresponding function L(s) on the interval [r,/(& - f+), 7,/(& - f+)] is a straight segment passing through the origin with slope Ak - f+ < 0. Proof of part A. Suppose 7 E [0, r2]. For the reason just given, for each y E [0, l] and E E (0, r,/(Ak - f,)), (P,) has no solution on the boundary of E,@). Furthermore, there exists a constant R > 0 such that if u is a solution of (P,) located in E,‘e’, y E [0, l] then /lull < R (II - 11 denotes the norm in L’(Q)). The proof of this claim proceeds as in [2], making use of (14).
346
NGWZN PHUONG CAc
For y E [0, l] let T,: t’(Q)
--+L’(Q) be defined by
T,w := w - K(f+ w+ - If-
Then by the homotopy
+ Y(f-
invariance of the Leray-Schauder
deg(T, , B, n E,‘“‘, K(h,
w E L2(cJ).
-f-)lw--I,
(16)
degree:
- tq, + SV)~)) = deg( T, , BR n EI(E),K(h,
- tp, + npJ).
(17)
By theorem 2 and lemma 3, in both situations (a) and (b), in BR fl EI(E) the BVP (Pi) has exactly a nonnegative solution with index (- l)k and a solution with changing sign with index (- l)k-‘. Therefore, deg(T,, BR n E,@), K(h,
- tcp, + SV)~)) = 0,
deg(T, , B, fI EI(E),K(h,
- tpl + T(P~)) = 0.
and by (17): (18)
Since the BVP (P,), i.e. (12), has a nonnegative solution with index (- l)k in B, n E,‘E’ and by lemma 3 any solution with changing sign of it has index (- I)&-‘, (18) shows that (12) has exactly one such solution in B, fl E, @). Because E > 0 can be arbitrarily small, we see that the BVP (12) has exactly two solutions in i?,: U, (nonnegative) and u2 (with changing sign). Let si = (ai, (Pi) where tii (i = 1, 2) are solutions of (12) for T = s2, and r; = maxfl(s) 1s2 Is
5 slJ.
(19)
Then r; > r2, otherwise the BVP (12) with r = r2 or T < r2, r close to r2, has more than two solutions, which is a contradiction. For the same reason, L(s) < r! for every s E (- ~0, s,). It is then not difficult to see that all the claims concerning rf are true. Proof of part B. Consider r E [r,, 0). Again, because for each y E [0, 11, the graph of the function L(.) related to the BVP (P,) on the interval [r2/(Ak - f+), r,/(Ak - f,)] is a straight segment passing through the origin with slope & - f+ < 0, for every y E [0, l] the BVP (P,) has no solution on the boundary of E, when r E [ri, 0). As above, using (14). it can be shown that there exists a constant R > 0 such that if u is a solution located in E, of (P,), y E [0, I] then l/u]] < R. Then
deg(G, BR n E,, Wk, - W, +
7vJ)
=
deg(T,
9 BR
n 4,
WQ
-
tv1
+
WA).
But deg(T,, BR i7 E,, K(h,
- ta, + ~9~)) = (- l)k-l,
(20)
because in both situations (a) and (b), by theorem 2, the BVP (Pi) has exactly one solution with changing sign located in E, with index (- l)k-l when T E [ri, 0). Since any solution of (P,), i.e. the BVP (12), that is located in E, must change its sign (for T E [pi. 0) the solution of (12) of constant sign is located in E,), its index is (- l)k-l by lemma 3. It then follows from (20) that there is exactly one such solution of the BVP (12) in El. Since lim L(s)/s = C(f_ , f+) > 0 and s-.---m L(a) is continuous, it is clear that for r < T, the BVP (12) has at least one solution. n The proof in [6] can be adapted to show that for r < 0, 151large, the BVP (12) has exactly one solution in E, under the hypothesis C( f_ , f+) > 0. In [6] the proof is carried out for the case ho = 0, r = 0.
Remark.
An elliptic
boundary
337
value problem
Using the same method we can prove the following. PROPOSITION2. Suppose that h, and t are given as in (6) and (8), suppose also that C(f_ ,f+) c 0. 7, being defined as in (lo), we have the following. A. For 7 E [O, 7J the BVP (12) has exactly one solution in the half space E,, it is nonnegative. For 7 > rz the BVP (12) has at least one solution, it changes its sign on R. B. For 7 < 0 the BVP (12) has no solution in the half space Z?!. We now determine the number of solutions of the BVP (12) in the half space E, when it is at semiresonance, i.e. when C(f_ ,f+) > 0, C(f+ ,f_) = 0, for example. PROPOSITION3. Suppose that he and t are given as in (6) and (8); suppose also that C(f_ ,f+) > 0 and C(f+ ,f_) = 0. ti being defined as in (9), we have the following. A. For 7 L 0 the BVP (12) has no solution in E,. B. For r E [T,, 0) the BVP (12) has exactly one solution in E,, it is nonnegative. Furthermore, there exists r: with - CQzz 7f < T, such that for T E (7?, ti) the BVP (12) has at least one solution in E,,it changes its sign on a; if - m < 7; then the BVP (12) has no solution in E,for T < 7;. Proof. As in the proof of proposition f_, such that for each y E (0, l]
1, we choose anK
E (f- , A,), f_ sufficiently close to
Y(3-- s-hf+>zf+ 0, w-+ ,A. + YCL-f-N < at-+ 7.f.J= 0. C(f- J,)
> 0%
+
(21)
For each y E IO, I] we consider the BVP (P,) of (I>). Proof of part A. Suppose 7 1 0. Because the function L.(a) is continuous on IR, to show that the BVP (12) has no solution in E,for 7 2 0, it suffices to show that it has no solution in Erfor T E [0, 7.J where 7, is defined by (10). As we have noted in the proof of proposition 1, with 7 E IO, rJ, y E [0, 1) (P,) in (15) has no solution u with (u, @)k)= E where E is any number in (0*7*4& - f,)). We first observe that for 0 < y s 1, because of (21), we deduce from theorem 2, Part B (ii), (iii), that (15) has no solution in E,w where E E (0, t,/(& - f,)). For 7 E [0, 7J, the only solution of the BVP (P,) in (15) that does not change its sign on Q is nonnegative and given by (ll)? its index is (- l)k. All other solutions of (P,) change their signs on R, are nondegenerate, isolated and each has index (- I)*-’ by lemma 3. We deduce from this that any bounded subset of E,?,E E (0, T,/(&- f,)) can contain only a finite number of solutions of (P,). In fact, suppose by contradiction 3 R > 0 such that BR n E,? contains an infinite sequence of solutions (v,) of (P,). It then follows without too much difficulty from the theory of elliptic boundary value problems that we can extract a subsequence, still denoted by (u,], converging in w’~‘(SJ) to u and u is also a solution of (P,). But then v is not isolated. Suppose now, by contradiction, that the set of solutions of (PO) in .!ZF) is not empty. Then we can find a bounded open set U of E,'" which contains n solutions of (P,) with n > 1 but the boundary aU of it does not contain any solution of (P,), 0 s y I 1. In fact, we can choose R > 0 large enough so that BR flEj&'contains n solutions u,, u2, . . . , u, (n 1 1) of (P,). If aB, 17i?,?contains no solutions of (P,) then we take U = BR I? E,?.If aBR tl Eyj contains
NGUYEN PHUONG Clc
348
some solutions (necessarily finite in number), say wi, ..., w,, we can find an open set V containing w,, w2, . . . , w, such that P contains none of the Vi (i = 1, . . ., n). We then take U = (BR fl E,‘“‘)\t? We recall that by previous argument (P,), y E (0, l] has no solutions in 0. Thus with the mapping T,: L2(sZ) + L’(Q) defined as in (16), it follows from the homotopy invariance of Leray-Schauder degree that deg(T,, u, K(h, - t9, + ~9~)) = deg(T,, U, K(& - t9, + ~9~)) = 0. But since the index of each ui (i = 1, 2, . . . , n; n 2 1) contained in U is the same, this is not possible. Since E is arbitrary in (0, ~l/(Ak - f,)), part A is proved. Proof of part B. Suppose T E [Al, 0). For each y E [0, I] the BVP (P,) of (15) has a nonnegative solution (0 given by (7)) ~ u, = 0 + f+ty,
+
-
T9k
lk -f+’
located in E,. The index of u1 is (- l)k. There is no other solution of (P,) that does not change its sign on a besides ut. We are going to show that in E, there are no solutions of (P,) with changing sign either. Above we have shown that a bounded set of E, can only contain a finite number of solutions of (P,) (we note that obviously u1 is an isolated solution of (P,)). Furthermore, by theorem 2, part B (i), for y E (0, 11, (P,) has exactly two solutions, one of which is ut , the other one is located in E,. Therefore if (P,) has other solutions in the half space E, besides ut, then, as above, we can find a bounded open set U of E, containing the solutions Ul, u2, . . . . u, (00 > n L 2) of (P,) but aCJ contains no solutions of (P,), 0 i y 5 1. The homotopy invariance of degree gives deg(G, u,K(ho
- t9, +
T9k)) = deg(T,
US&~,
+ f91 + T9k)) = (- I)k.
But this is impossible, because index U, = (- l)k and index Ui = (- l)k-’ for 2 I i 5 n. Concerning other claims in part- B, consider the function L( *) on the interval b,/(Ak - f+), ‘=‘)a If J!,(S) is not bounded from below on this interval, we take rr = - 00. Suppose then that " -lS
We have tf < ~1. If not, then either L(e) is constant on the interval [rt/& - f,), co) and the BVP (P,) of (15) for T = 51 has an infinity of solutions or it has at least two solutions for T > ~1, T close to '51,contradicting what we have established above. Because L(e) is continuous, for any T E (~7, TV) we can find s such that L(s) = T and therefore the BVP (P,) has at least one n solution. By definition of TV this solution must change its sign on Q. Remark. If the BVP (12) is at semiresonance with C(f+ , f_) = 0 and C(f_ ,f+) C 0 instead of C(f_ , f+) > 0 as in proposition 3, then proposition 3 still holds. The proof is similar, making
use of theorem 2, part C, instead. Proof of theorem 1. Theorem
1 is just a consequence of proposition
1 and proposition
3.
n
An elliptic boundary value problem
349
We now consider the case when the BVP (12) is at resonance. THEOREM 3.
Suppose that ho and t are given as in (6) and (8). Suppose also that C(f_ ,f+) = C(f+ ,f_) = 0. Then for r E [rr, r,], where rt and r2 are defined in (9) and (lo), the BVP (12) has exactly one solution which is nonnegative. Furthermore, there exist rr < r1 and rf > r, such that for r E (57, rf)\[rl, rz] the BVP (12) has at least one solution. Proof. Choose an arbitrary 3_ E (f_ , A,) and for y E [0, I] we again consider the BVP (P-J
-Au
= f+u+ - [f- + y(j;_ - f_)lu-
+ h, - tv, + rp, in S2,
u = 0 on f3R.
(22)
ForOcyzzlwehavebylemmal: C(f-
+ r(j’- -f-M-+)
C(f+ ,f- +
< C(f- ,I-+) = 0
vt3- - f-1) < w-+ J-1 = 0.
therefore by theorem 2, part C, the BVP (P,) of (22) has for r E [rr, r2] exactly one solution which is nonnegative and located in E, if r, I T < 0, is orthogonal to $Jkif r = 0 and is located in E, if 0 < r I r,. Using this, we can prove the claims for the case r E [r,, rz] by the same method used for proposition 3. The existence of rf and rz with the claimed properties can be seen as in the proof of part B, proposition 3. H 3. THE
NONLINEAR
EQUATION
Let f: R -+ R be of class C’ such that lim f’(c) r-
= fk
with
fm
g(c)
=
f(C) - f+[’
+ f_[-
lk-l
Ak
lk+l,
is a bounded function.
We have the following. THEOREM 4.
Suppose that C(f_, f+) > 0, C(f+, f-)
= 0. Then given he E C’(n),
h0 = 0 on
6X2,the BVP -Au
= f(u) + ho - tfp, + tr’pk in Q
u = 0 on
asz,
(23)
for t > 0 sufficiently large and ]r’] adequately small has exactly two solutions. The proof is based on theorem 1 and it will be clear that similar results concerning the BVP (23) can be proved under other situations of nonresonance, semiresonance or resonance, making use of theorem 3, for example in the case of resonance. We use the method of proof of Solimini [6], who considers the case /I, = 0, r’ = 0 and the BVP is not at resonance. We shall have to make modifications in his proof to allow for h, # 0 and for semiresonance. The proof is achieved in three stages. PROPOSITION 4. Under the hypotheses of theorem 4, the BVP (23) has at least two solutions for any given h, E C’(a), ho = 0 on asZ, t > 0 sufficiently large and ]r’] adequately small.
350
Cic
NGUYEN PHUONG
Proof. Using its positive homogeneity,
we deduce
from theorem
1 that for T’ with 15’1 small,
the BVP -Au
= f+n’
-f-u-
- q7, + s’yl, in S2,
u = 0 on
has exactly two solutions ut, u2 and they are nondegenerate. homogeneous boundary condition, let
T,u = u - K(f+u+
u E aUi = dist.(T,u, Choose
-A
having
u E L’(Q).
we can find disjoint
K(-cp,
With K = (-A)-‘,
(24)
-f-u-),
Tu = u - Kj-(u), Because Ui (i = 1,2) are nondegenerate stant c > 0 s.t.
asz,
neighborhoods
+ ~‘9~)) > c in c(Q),
t, > 0 such that if t > t, then we have the following
Vi of Ui and a con-
i = 1,2.
three properties:
for u E aUi, i = 1,2;
i=1,2; and, for later use; (iii) the BVP = f+u+ -f-u-
-Au has exactly
two solutions.
ho + t - p, + r’fpk in Q,
This is possible t2 =
zIlKI
by theorem [me&91
u =
0 on
ac2,
1. Let
I’* supkl,
and H(y, u) = T,u + y(Tu - T,u), Then with t > max(t,, dist(H(y,
t2), we have for u E a(tu,),
w), K(h,
Y E [O, 11.
u E L*(Q),
i = 1,2 and y E [0, I]:
- tcp, + ~T’v)~)) > dist(T,u,
K(h,
- tp, + ~T’v)~)) - T
rCt_ct,>o. 2
Then the homotopy
invariance
deg(T, tUi, K(h,
of the Leray-Schauder
- 19, + t~‘qk)) = deg(T,,
= *1,
2
degree gives tUi, K(h,
i= 1,2,
- tpl - tr’pk))
351
An elliptic boundary value problem
using the positive homogeneity of To and (ii) above. Since t& (i = 1,2) are disjoint, proposition 4 is proved. n PROPOSITION 5. Suppose that C(f_ ,f+) > 0, C(f+ ,f_) = 0 and h, E C’(n), h, = 0 on XX For T’ sufficiently small, let U be a bounded open set in L’(Q) containing ali solutions of the BVP (24), then there exists to f IRsuch that for t > to the BVP (23) has no solutions in L2(Q)\tiJ. Proof, A similar result is proved in [6] for h, = 0, 7’ = 0 under the nonresonance assumption # 0. Suppose by contradiction that for each n = 1,2, . . . there exists I, with t, -+ ~0 and u,, E L’(Z))\~ satisfying the BVP: C(f+ , f_)
-Au
= f+u+ - fu_u-
g(t,u) + 7
n
u = 0 on i3Q.
t,
Since by theorem f with (7’1 small the BVP are nondegenerate and for large n the above it follows from proposition 2.2 in [6] that (i = 1,2). Therefore there exists a constant /u,/[ cc
+ &I - - q1 + s’q+ in 52,
(24) has exactly two solutions u,, u2 both of which BVP also has exactly two nondegenerate solutions, the local inversion theorem [I] is applicable at ui c such that
v n = I,2, ..*
in L2(sZ).
By the theory of elliptic boundary value problems, we can then extract from {u,) a subsequence, still denoted by {u,), converging to u in W2’2(fz) with u being a solution of the BVP (24) and u E L’(Q)\U. This is impossible because cf contains all solutions of the BVP (24). P~oo~o~~~eo~e~ 4, From proposition 5 and the proof of proposition 4, in particular (iii) in that proof, we see that for t > 0 large, 17’1small, the BVP - A.u = f+u’
h0
- f_ u- + -
t
- p, + 7’~~ in fz,
u =
0 on X&
has exactly two nondegenerate solutions Us, u2 and we can find in L2(52) bounded open neighborhoods Vi (i = I, 2) of ui that are disjoint and such that in each tUi there is at least one solution of the BVP (23) and no solution in _L’(Q)\t(LJ, U V,). It can now be proved as in [6], theorem 1, that taking t larger still if necessary, there is in each tiJi exactly one solution of the BVP (23). q REFERENCES 1. AMBROSEW A. & PRODSG., On the inversion of some differentiable mappings with singularities between Banach spaces, Annati Mar. pura appl. 93, 231-247 (1973). 2. GALLOU~T TH. & KAVIA?JO., RCsuitars d’existence et de non existence pour certains problemes demi-Iiniaires B l’infini, Annali Fat. Sci. Toulouse 3, 201-246 (1981). 3. LAZERA. C. & MCKENNAP. J., Multiplicity results for a semilinear boundary value problem with the nonlinearity crossing higher eigenvalues, Nor&fear Arzaiysis 9, 335-349 (1985). 4. LAZERA. C. & MCKENNAP. J., Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues I and II, Communs PartiaZ diff: Eqns 10, 107-150 (1985) and (1987). 5. RUF B., On nonlinear elliptic boundary value problems with jumping nonlinearities, Annali Mat. pur4 appt. 128, 133-151 (1980). 6. SOL~INI S., Some remarks on the number of solutions of some nonlinear elliptic equations, Analyse Non LinPaire, Inst. Henri PoincarP 2, 143-156 (1985)