A Note on Multiple Solutions of Some Semilinear Elliptic Problems

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

211, 626]640 Ž1997.

AY975471

NOTE A Note on Multiple Solutions of Some Semilinear Elliptic Problems E. N. Dancer School of Mathematics and Statistics, Uni¨ ersity of Sydney, Sydney, NSW 2006, Australia

and Yihong Du Department of Mathematics, Statistics and Computing Science, Uni¨ ersity of New England, Armidale, NSW 2351, Australia Submitted by By John La¨ ery Received August 15, 1995

We discuss the existence of multiple solutions of nonlinear elliptic equations by a combination of variational, topological methods and the generalized Conley index theory. We obtain several positive solutions and sign-changing solutions. Our main point is to show the usefulness of the Morse inequalities for Morse decompositions in the generalized Conley index theory. Q 1997 Academic Press

In this note we show how topological and variational methods and the generalized Conley index can be combined to give better multiplicity results for the semilinear elliptic problem yDu s f Ž u . ,

u < ­ D s 0.

Ž 1.

Here D is a bounded domain in R n with regular enough boundary ­ D; f : R1 ª R1 is C 1 , f Ž0. s 0, and < f Ž u.< F C1 q C2 < u < p, 1 F p - n*, n* s Ž n q 2.rŽ n y 2. if n G 3 and n* s ` for n s 1, 2. We make these assumptions throughout this note. 626 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

627

NOTE

It is well known that solutions of Ž1. correspond to Ži. critical points of the functional F Ž u . s < =u < 2r2 y

H

H F Ž u. ,

F Ž u. s

u

H0

f Ž t . dt ;

Žii. fixed points of the operator Au s Ž yD q k .

y1

f Ž u . q ku ,

k G 0 is a constant;

and Žiii. equilibria of the flow generated by the parabolic problem ut y D u s f Ž u. ,

u < ­ D s 0.

Therefore, variational method, topological degree theory, and the generalized Conley index are all applicable to the study of the existence and multiplicity of solutions of Ž1.. We are particularly interested in showing the usefulness of the Morse inequalities for Morse decompositions in the generalized Conley index theory. In w5x, we showed that these Morse inequalities can often give more information than merely the multiplicity of solutions. To be more precise, we used Morse decompositions to prove the existence of a sign-changing solution. In this note we show that combined with other well-established methods, Morse decompositions can sometimes provide much better multiplicity results as well. As before, the basic idea is that when one uses the generalized Conley index theory, one can use the generalized Conley indices of certain solution sets and the Morse inequalities to obtain extra solutions, while one does not have to know all the solutions in the solution sets except their Conley indices. Now let us be more precise. We call u g C 2 Ž D . an upper solution of Ž1. if yDu G f Ž u . , x g D;

u G 0, x g ­ D.

An upper solution is called a strict upper solution if it is not a solution. Lower and strict lower solution are defined by reversing the above inequalities. Let l1 - l 2 - ??? - l n - ??? be the distinct eigenvalues of yDu s l u, u < ­ D s 0. We make the following assumptions on f : ŽH 0 . f 9Ž0. ) l2 ; ŽH ` . lim < u < ª` f Ž u.ru s a g Ž l k , l kq1 ., k G 2; ŽH 1 . Ž1. has a positive strict upper solution u g C 2 Ž D . and a negative strict lower solution u g C 2 Ž D ..

628

NOTE

The simplest sufficient condition for ŽH 1 . is ŽHX1 . There exist a - 0 - b such that f Ž a . s f Ž b . s 0. Another simple sufficient condition is from w3x, ŽHY1 . < f Ž u.< - 1 in the interval wyc, c x, where c G eŽ x . in D and e satisfies yD e s 1, e < ­ D s 0. Our main result is the following theorem. THEOREM 1. Suppose that Ž H0 ., Ž H` ., and Ž H1 . are satisfied. Then Ž1. has at least 6 nontri¨ ial solutions. If we assume further that f 9Ž0. f  l k 4 , then Ž1. has at least 7 nontri¨ ial solutions. Remarks. Ž1. It seems unlikely to us that Theorem 1 can be obtained by using the standard variational and topological methods. Under similar but more restrictive conditions, the multiplicity problem for Ž1. was discussed in w3, Theorem 3.8; 9, Theorem 3x, using variational and topological methods, respectively, but less nontrivial solutions were obtained. Ž2. The conditions ŽH 0 . and ŽH ` . can be further weakened. For example, we can use suitable conditions in w6, 5x to replace them. Theorem 1 is a consequence of the following two propositions. PROPOSITION 1. Suppose that ŽH 0 . and ŽH 1 . hold. Then Ž1. has at least 3 nontri¨ ial solutions in the order inter¨ al w u, u x, If further we assume that f 9Ž0. f  l k 4 , then Ž1. has at least 4 nontri¨ ial solutions in this order inter¨ al. Proof. This is only a minor variant of that of Theorem 1 and Corollary 1 in w6x. The result for the case f 9Ž0. f  l k 4 is well known, see, for example, w4, 8x. Remark. As in w6x, we can actually prove a little more than Proposition 1: Under conditions ŽH 0 . and ŽH 1 ., Ž1. has at least one positive solution, one negative solution, and one sign-changing solution in w u, u x. If further f 9Ž0. f  l k 4 , then Ž1. has at least one more sign-changing solution in w u, u x. Moreover, the sign-changing solution are between the minimal positive solution and maximal negative solution of Ž1.. PROPOSITION 2. Suppose that ŽH ` . and ŽH 1 . are satisfied. Then Ž1. has at least 3 nontri¨ ial solutions outside the order inter¨ al w u, u x. Proof. It follows from the standard iteration argument Žsee w1x. that Ž1. has in w u, u x a maximal solution u* and a minimal solution u#. Since 0 is a

629

NOTE

solution, we have u# F 0 F u*. Let f˜Ž x, u . s

½

f Ž u. , f Ž u* Ž x . . ,

if u G u* Ž x . if u - u* Ž x . .

Then f˜ is continuous and for any given finite interval w a , b x, we can find a positive constant k such that f˜Ž x, u. q ku is increasing in u for all x g D and u g w a , b x. Moreover, it is easily seen that u* y «f 1 and u is a pair of strict lower and upper solution for yDu s f˜Ž x, u., u < ­ D s 0. Here « is a sufficiently small positive constant and f 1 is the positive eigenfunction corresponding to l1. It follows then from w2x that the functional

˜ Ž u. s F

H < =u < r2 y H F˜, 2

F˜Ž u . s

u

H0

f˜Ž t . dt,

restricted to E s C01 Ž D . has a local minimum in the order interval ˜ w u* y «f 1 , u x in E, and this local minimum is also a local minimum of F 1Ž . ˜ Ž in H s H0 D . Since clearly u* is the only solution of yDu s f x, u., ˜ u < ­ D s 0 in this interval, it follows that u* is a strict local minimum of F ˜ ˜ Ž . in H. One easily checks that F tf 1 ª y` as t ª q`. Moreover, if F satisfies the P.S. condition, then it is easy to show that for some « ) 0 ˜ Ž u.: 5 u y u* 5 H s « 4 ) F ˜ Ž u*.. ŽNote that it is not necessary to small, inf F have this inequality to be able to use the mountain pass theorem; one can simply use a generalized version of this theorem, for which the fact that u* is a local minimum point is enough, see, for example, w3x.. Hence one can ˜ which, by use the mountain pass theorem to obtain a critical point u1 of F w x Ž the maximum principle, as in 2, 3 , must be a solution of 1. satisfying u1 G u*. Now by the well-known critical group characterization for u1 Žsee, ˜ are isolated, then for example, w10x., if all the critical points of F

˜ , u1 . s d q1G. Cq Ž F

Ž 2.

Since u1 G u* and u1 k u*, it follows from the strong maximum principle that u1 together with a small E-neighborhood is in w u*, q`.. In particular, ˜ < E , u1 . s C#Ž F < E , u1 . by the definitions of the critical groups. But it C#Ž F is well known that C#Ž F < E , u1 . s C#Ž F, u1 . Žsee w3x.. Thus we obtain from Ž2. that C q Ž F , u1 . s d q1G.

Ž 3.

˜ satisfies the P.S. condition. Let u n g H be We still have to show that F such that ˜ Ž u n . F C, F

˜ Ž u n . ª 0. F9

630

NOTE

By standard argument, it suffices to show that  u n4 is bounded. We argue indirectly. Suppose it is unbounded. We may assume that 5 u n 5 H ª `. ˜ Ž u n . ª 0, for any f g H, Since F9

H =u

n

=f y f˜Ž x, u n . f s o Ž 5 f 5 H . .

Ž 4.

Choosing f s u n and using < f˜Ž x, u.< F C1 q C2 < u <, we have 5 u n 5 2H F C1X q C2X 5 u n 5 2L2 q o Ž 5 u n 5 H . . This implies that 5 u n 5 L2 ª ` and that ¨ n s u nr5 u n 5 L2 is bounded in H. We may assume that ¨ n ª ¨ weakly in H and strongly in L2 Ž D .. Now dividing Ž4. by 5 u n 5 L2 and then passing to the limit we obtain

H =¨ =f y a¨

q

f s 0,

;f g H.

Here ¨ qs max ¨ , 04 . This implies that ¨ solves yD¨ s a¨ q,

¨ < ­ D s 0.

But this is impossible as a ) l1 and ¨ / 0. This contradiction proves that the P.S. condition is satisfied. In a similar way, we can show that Ž1. has a solution u 2 - u#, and if every solution of Ž1. outside w u#, u*x is isolated, then C q Ž F , u 2 . s d q1G.

Ž 5.

Next we use the generalized Conley index to show that Ž1. has at least one more solution outside w u#, u*x. As in w5x, we choose p ) n, a g Ž1r2 q nrŽ2 p ., 1., X s L p Ž D ., and let X a be the fractional power space induced by yD. By our choices of p and a , X a imbeds continuously into E s C 1 Ž D .. Now let p Ž t .: uŽ0, x . ª uŽ t, x . be the local semiflow on X a generated by the solution uŽ t, x . of the parabolic problem

½

ut y D u s f Ž u. ,

t ) 0, x g D

u Ž t , x . s 0,

t ) 0, x g ­ D.

Ž 6.

Let K denote the set of all the bounded full solutions of Ž6.. Then, by ŽH ` ., it follows from Lemma 4.1 and Proposition 2.2 in w5x that Hq Ž h Ž p , K . . s 0

for q s 0, 1.

Ž 7.

NOTE

631

Let K 0 be the set of full solutions of Ž6. lying inside w u, u x. Then a similar argument to that of Lemma 3.5 in w5x gives Hq Ž h Ž p , K 0 . . s d q 0 G.

Ž 8.

Now we are ready to show that Ž1. has a third solution outside w u#, u*x. We prove this by a contradiction argument. Suppose that u1 and u 2 are the only such solutions of Ž1.. Denote K 1 s  u14 and K 2 s  u 2 4 . We show that  K 0 , K 1 , K 2 4 is a Morse decomposition of K. It suffices to show that for any u g K, either Ži. u g K i for some 0 F i F 2 or Žii. a Ž u. g K j and v Ž u. g K i for some 0 F i - j F 2. Let u g K and suppose that Ži. does not occur. We show that Žii. occurs. Since Ž6. has a Lyapunov functional, a Ž u. consists of solutions of Ž1.. If a Ž u. l w u#, u*x / B, then there exists t n ª y` such that u F uŽ t n , x . F u. Hence it follows from the parabolic maximum principle that u F uŽ t, x . F u for all t G t n . This implies that u g K 0 , contradicting our assumption that Ži. does not occur. Hence a Ž u. is outside w u#, u*x and we necessarily have a Ž u. s K i , i s 1 or 2, as a Ž u. is connected. If a Ž u. s K 1 , then we can find t n ª y` such that uŽ t n , ? . ª u1 in X a and hence uŽ t n , x . G u* for all large n. It then follows from the parabolic maximum principle that uŽ t, x . G u* for all t ) t n . This implies that uŽ t, x . G u* for all t g Žy`, `.. In particular, v Ž u. g w u*, `.. Since u / K 1 , and v Ž u. consists of solutions of Ž1., and u* is the only solution of Ž1. which is different from u1 in w u*, `., we must have v Ž u. s  u*4 ; K 0 . Similarly we can show that if a Ž u. s K 2 then v Ž u. s  u#4 ; K 0 . This proves that  K 0 , K 1 , K 2 4 forms a Morse decomposition for K. Now we use the Morse inequalities for this Morse decomposition Ž7., Ž8., Ž3., and Ž5. above, and Proposition 2.1 of w5x. We arrive at the same contradiction as in the proof of Theorem 4.1 in w5x. This completes the proof of Proposition 2. Remark. In the above proof, one can also use Theorem 1 ŽC. of w8x to obtain u1 and u 2 . But one needs an extra condition, f Ž u. q ku is increasing for all u g Žy`, `., to be able to use w8x directly. In the rest of this note, we give several improvements and variants of Theorem 1, many of which have independent interests. With a little more effort, we can improve Proposition 2 slightly. Namely PROPOSITION 29. Under conditions ŽH ` . and ŽH 1 ., Ž1. has, outside the inter¨ al w u, u x, at least one positi¨ e solution greater than u* s v Ž u., one negati¨ e solution less than u# s v Ž u., and one solution not comparable with at least one of u* and u#. Here, and in what follows, we say that u and ¨ are comparable if either u F ¨ or ¨ F u holds.

632

NOTE

Proof. This is a modification of the proof of Proposition 2. We will be sketchy. The idea is again to make use of the Morse inequalities for suitable Morse decompositions. Suppose that u*, u#, K 0 , and K are the same as in the proof of Proposition 2. Now instead of letting K 1 s  u14 and K 2 s  u 2 4 we define K 1 to be the set of full solutions u of Ž6. satisfying uŽ t, x . ) u*Ž x . and u* f v Ž u., K 2 to be the set of full solutions u of Ž6. satisfying u - u#, and u# f v Ž u.. If Ž1. has no solution incomparable with u* and u#, then it is easy to show that  K 0 , K 1 , K 2 4 forms a Morse decomposition of K. Therefore, if we can show that Hq Ž h Ž p , K 1 . . s Hq Ž h Ž p , K 2 . . s d 1 q G,

Ž 9.

then, using the Morse inequalities for this new Morse decomposition of K, we arrive at a contradiction as before. Thus it suffices to prove Ž9.. We use the continuation property of the generalized Conley index Žsee w11x.. We consider K 1 only. The proof for K 2 is similar. First we make a change of variables: u s u* q ¨ . Then u g K 1 if and only if ¨ g K 1X where K 1X denotes the set of full solutions of ¨ t y D¨ s g Ž x, ¨ . ' f Ž u* Ž x . q ¨ . y f Ž u* Ž x . . ,

¨ < ­ D s 0,

Ž 10 .

which satisfy ¨ ) 0 and 0 f v Ž ¨ .. Let p 9 denote the local semiflow generated by Ž10. in X a. Then one easily sees that h Ž p , K 1 . s h Ž p 9, K 1X . . Now choose K ) 0 such that Ku q f Ž u. is increasing for 0 F u F 5 u 5 ` . Then let e 0 ) 0 be small enough such that

Ž 1 y e 0 . a y e 0 K ) l1 . It is easy to check that ¨ ' u y u* ) 0 is an upper solution for yD¨ s ge Ž x, ¨ . ' Ž 1 y e . g Ž x, ¨ . y e K¨ ,

¨ < ­ D s 0, 0 F e F e 0 .

Moreover, the iteration yD¨ n q K¨ n s ge Ž x, ¨ ny1 . q K¨ ny1 ,

¨ n < ­ D s 0, ¨ 0 s ¨

satisfies ¨ n F ¨ ny1 , ¨ n ª 0. Using this fact and lim ge Ž x, ¨ . r¨ s Ž 1 y e . a y e K G Ž 1 y e 0 . a y e 0 K ) l1 ,

¨ ªq`

for all e g w0, e 0 x, one can show, by the continuation property of the generalized Conley index, that hŽp 9Ž e ., K 1X Ž e .. does not depend on e g w0, e 0 x, where p 9Ž e . denotes the local semiflow induced by ¨ t y D¨ s ge Ž x, ¨ . ,

¨ < ­ D s 0,

633

NOTE

and K 1X Ž e . is the set of full solutions of p 9Ž e . which satisfy ¨ ) 0 and 0 f v Ž ¨ .. The key point here is that, for large R ) 0, N s  ¨ G ¨ : 5 ¨ 5 X a F R4 is a common isolating neighbourhood for K 1X Ž e ., e g w0, e 0 x. Similar Žbut not exactly the same. arguments were used in w5x. Now let h Ž ¨ . s l0¨ q Ž l` y l0 .

¨3

1 q¨2

,

l0 g Ž 0, l1 . , l` g Ž l1 , l2 . ,

and consider the following homotopy ¨ t y D¨ s ht Ž x, ¨ . ' t ge 0Ž x, ¨ . q Ž 1 y t . h Ž ¨ . ,

¨ < ­ D s 0, t g w 0, 1 x .

Ž 11 . We have lim ht Ž x, ¨ . r¨ s t Ž 1 y e 0 . a y e 0 K q Ž 1 y t . l` ) l1

¨ ªq`

uniformly for t g w0, 1x and x g D, and lim ht Ž x, ¨ . r¨ s lt Ž x . ' t Ž 1 y e 0 . f 9 Ž u* Ž x . . y e 0 K q Ž 1 y t . l 0 .

¨ ª0

Since u* is isolated from above as a solution of Ž1., and Ž1. has an upper solution u ) u*, we must have

l1 Ž yf 9 Ž u* . . G 0. Here l1Ž q . denotes the first eigenvalue of the problem yDu q qu s l u,

u < ­ D s 0.

Using the property that q ª l1Ž q . is concave, we easily obtain

l1 Ž ylt . G tl1 Ž Ž 1 y e 0 . Ž yf 9 Ž u* . . q e 0 K . . q Ž 1 y t . l1 Ž yl0 . G t Ž 1 y e 0 . l1 Ž yf 9 Ž u* . . q e 0 l1 Ž I . q Ž 1 y t . Ž l1 y l 0 . G te 0 Ž l1 q K . q Ž 1 y t . Ž l1 y l 0 . G s 0 ) 0. This implies that for some d 0 ) 0 small, Ž11. has no stationary positive solution satisfying ¨ / 0 and 5 ¨ 5 ` F d 0 . Now a similar argument to that in the proof of Lemma 4.2 in w5x shows that h Ž p 9 Ž e 0 . , K 1 Ž e 0 . . s h Ž p 0 , K 1Y . , where p 0 denotes the local semiflow induced by Ž11. with t s 0, and K 1Y denotes the set of full solutions of Ž11. with t s 0 that satisfy ¨ ) 0 and

634

NOTE

0 f v Ž ¨ .. As in w5x, it follows from the choice of hŽ ¨ . that K 1Y s  ¨ 0 4 where ¨ 0 is the unique positive solution of yD¨ s hŽ ¨ ., ¨ < ­ D s 0. Since ¨ 0 is a mountain pass solution, Hq Ž h Ž p 0 , K 1Y . . s d q1G. This proves Ž9.. The proof is complete. It would be interesting to know whether there is a sign-changing solution under the conditions of Proposition 29. This is answered in the following result. PROPOSITION 20. Under the conditions of Proposition 2, Ž1. has at least one sign-changing solution. Proof. This is a variant of the proof of Proposition 29. However, we need the following technical result which has its own interest. LEMMA. Under the condition ŽH 1 . and ŽH ` ., Ž1. has a non-negati¨ e solution u* ˜ - u such that it is comparable with any positi¨ e solution of Ž1. and that there is a strict upper solution u ˜ F u of Ž1. with the following property: The iteration yDu n q Ku n s f Ž u ny1 . q Ku ny1 ,

u n < ­ D s 0,k u 0 s u ˜

con¨ erges to u*, ˜ where K ) 0 is defined as in the proof of Proposition 29. Ž Note that this property is equi¨ alent to v Ž u ˜. s u*. ˜ . There is an analogous non-positive solution. Assuming this Lemma, then the result follows directly from Proposition 29 and its proof by replacing u and u* by u ˜ and u*, ˜ respectively, and replacing u and u# by their analogues too. Thus it suffices to prove the above Lemma. To this end, we define U s  u g C 2 Ž D . : u F u is a non-negative strict upper solution of Ž 1 . 4 . Clearly u g U. Define A: C 2 Ž D . ª C 2 Ž D . by Au s Ž yD q K .

y1

Ž f Ž u . q Ku . .

Then it is well known that u is a solution of Ž1. if and only if it is a solution for Au s u, and that if u is a strict upper Žlower. solution of Ž1. then it is a strict upper Žlower. solution of Au s u; conversely, if u is a strict upper Žlower. solution of Au s u, then Au is a strict upper Žlower. solution of Ž1.. Moreover, A is completely continuous, and it can be extended to a continuous, strongly order-preserving map from C a Ž D . to C 2, a Ž D ., 0 a - 1, and from C Ž D . to C 1 Ž D .. Here we have used several terminologies

NOTE

635

from w7x, and we use the natural order in these spaces. Let G s A2 Ž U . , where U denotes the closure of U in C 2 Ž D .. It is evident that any u g U is either a strict upper solution or a solution of Ž1.. Thus, for any u g U, u, and hence Ak u, k G 1, is either a solution of Ž1. or a strict upper solution of Ž1.. It follows that G ; U. Next we show that there exists u ˆ g G such that u G uˆ for any u g G. We first use Zorn’s lemma to show that G has a minimal element in the natural order in C 2 Ž D .. It suffices to show that any totally ordered set M ; G has a lower bound in G. By its definition, and the properties of A, G is precompact. Thus M is compact. Therefore it has a minimal element z, which is clearly a lower bound of M and is contained in U. Hence A2 z g G and satisfies A2 z F Az F z. This shows that A2 z is a lower bound of M in G. Now we can use Zorn’s lemma to conclude that G has a minimal element u. ˆ It is easy to see that uˆ must be a solution of Ž1.. We show that u G u ˆ for any u g G. In fact, if for some u 0 g G, u 0 h u, ˆ then u ˆ k 0 and u1Ž x . s min uˆŽ x ., u 0 Ž x .4 satisfies u1 g C a Ž D ., u1 - u, ˆ u1 - u 0 . Using the strongly order-preserving property of A we obtain Au1 < Au 0 F u 0 , Au1 < Au ˆ F u. ˆ This implies that Au1 - u1 and hence ¨ 1 s Au1 g U. But A2 ¨ 1 g G and A2 ¨ 1 - u1 F u. ˆ This contradicts the definition of u. ˆ The above argument also shows that any positive solution u 0 of Ž1. is comparable with u ˆ for otherwise we can construct a nonnegative strict upper solution less than u ˆ as above. Since u ˆ g G is a solution of Ž1., we must have uˆ g U R U. Hence there exists wn g U such that wn ª u. ˆ Let u n s lim k ª` Ak wn . If uˆ s u n for some n, then we choose u* ˜ s u n and we are done. Now suppose that un / u ˆ for all n. We must have u n G uˆ and u n ª uˆ since u n g G, u n F wn , and  u n4 is precompact. By the definition of u n , as in the proof of Proposition 29, we deduce l1Žyf 9Ž u n .. G 0. Thus

l1 Ž yf 9 Ž u ˆ. . s lim l1 Ž yf 9 Ž u n . . G 0. Since u ˆ is not an isolated solution of Ž1., we must have l1Žyf 9Ž uˆ.. s 0. Let Br Ž u. s  ¨ : 5 ¨ y u 5 C 2 F r 4 and let S denote the set of positive solutions of Ž1.. It follows from Ž H` . that S is precompact. We claim that for some e 0 ) 0 small, Be 0Ž u ˆ. l S is totally ordered. This follows from an easy indirect argument. Or one can use a classical result of Crandall and Rabinowitz to show that any solution of Ž1. close to u ˆ can be expressed in the form usu ˆ q th q w Ž t . ,

636

NOTE

where t g R1 is small, h is a sign-preserving eigenfunction corresponding to l1Žyf 9Ž u ˆ.. s 0, yD h y f 9 Ž u ˆ. h s 0,

h < ­ D s 0,

and 5 w Ž t .5 C 2 s oŽ t . as t ª 0. Finally we show that there exists N ) 0 large such that for all n ) N, u n is comparable with all the positive solution of Ž1.. Clearly this would finish our proof Žby choosing u* ˜ s u n for some n ) N .. We argue indirectly. Let ¨ n g S be such that u n and ¨ n are not comparable for all n. Then we must have ¨ n ) u ˆ and 5 ¨ n y uˆ5 C 2 ) e 0 . Since S is precompact, we may assume that ¨ n ª ¨ . Then ¨ / u ˆ and ¨ G u. ˆ By the strong maximum principle, there is a small C 1 neighbourhood N1 of ¨ and a small C 1-neighbourhood N2 of u ˆ such that for any ¨ 9 g N1 and any u9 g N2 , ¨ 9 G u9. But this implies that ¨ n G u n for all large n. This contradiction completes the proof. Note that it follows from the above proof that the sign-changing solution in Proposition 20 is different from that in the Remark following Proposition 1 when ŽH 0 . is satisfied. Thus we have the following improvement of Theorem 1. THEOREM 19. Under the conditions ŽH 0 ., ŽH ` ., and ŽH 1 ., Ž1. has at least two positi¨ e solutions, two negati¨ e solutions, and two sign-changing solutions. If further, f 9Ž0. f  l k 4 , then Ž1. has at least one more sign-changing solution. In the case that f is odd, f Žyu. s yf Ž u., Proposition 1 can be improved considerably. PROPOSITION 3. Suppose that f is odd and that ŽH 0 . and ŽH 1 . hold. Then Ž1. has at least 4 nontri¨ ial solutions in w u, u x, with one pair sign-preser¨ ing, one pair sign-changing. If further f 9Ž0. ) l3 , then Ž1. has at least 6 nontri¨ ial solutions in w u, u x, with one pair sign-preser¨ ing, and two pairs sign-changing. Proof. Again we will be sketchy. The first part follows directly from Proposition 1 and the Remark following it. Nevertheless we sketch its proof as we need it later for the case f 9Ž0. ) l3 . As in w6x, we obtain a minimal positive solution u 0 and a maximal negative solution u 0 of Ž1. in w u, u x. ŽHere, we must have u 0 s yu 0 as f is odd.. One can show that they are local minima of F restricted on w u 0 , u 0 x. It then follows from the mountain pass theorem on w u 0 , u 0 x Žsee w8x or w3x. that Ž1. has a mountain pass solution u1 in this order interval. ŽNote that one could also prove this as at the beginning of the proof of Proposition 2, using the modification trick to f and then the classical mountain pass theorem.. u1 / 0 as, due to ŽH 0 ., the critical groups for 0 and u1 are different Žhere we assume that Ž1. has only isolated solutions in w u, u x for otherwise we are done by the

637

NOTE

following observation.. u1 must be sign-changing due to the definitions of u 0 and u 0 . Hence we have at least 4 nontrivial solution u 0 u 0 , u1 , and yu1 in w u, u x with one pair sign-preserving, one pair sign-changing. Now assume further that f 9Ž0. ) l3 and that Ž1. has no more signchanging solutions in w u, u x. We want to derive a contradiction. First, since u1 and yu1 are mountain pass solutions, C q Ž F , u1 . s C q Ž F , yu1 . s d q1G.

Ž 12 .

Using f 9Ž0. ) l3 and the shifting theorem Žsee, for example, w10x., we have C q Ž F , 0 . s 0,

q s 0, 1, 2.

Ž 13.

Let K be the set of full solutions of Ž6. satisfying u F u F u, K 1 be the set of full solutions of Ž6. with u 0 F u F u, and K 2 be the set of full solutions of Ž6. enjoying u F u F u 0 . Define also K 3 s  u14 , K 4 s  yu14 , and K 0 s  04 . Then it can be checked that, according to whether F Ž0. ) F Ž u1 . or F Ž0. F F Ž u1 .,  K 1 , K 2 , K 3 , K 4 , K 0 4 or  K 1 , K 2 , K 0 , K 3 , K 4 4 is a Morse decomposition of K. Here one uses the parabolic maximum principle much as before and the fact that F decrease along any solution of Ž6. as t increases. By Ž12. and Ž13., Hq Ž h Ž p , K 3 . . s Hq Ž h Ž p , K 4 . . s d q1G, Hq Ž h Ž p , K 0 . . s 0,

q s 0, 1, 2.

It is also not hard to show that Hq Ž h Ž p , K . . s Hq Ž h Ž p , K 1 . . s Hq Ž h Ž p , K 2 . . s d q 0 G. Then an application of the Morse inequality for this Morse decomposition give a contradiction. The proof is complete. Remarks. Ž1. One can also use proper Morse decompositions for the flow restricted on the invariant order interval w u 0 , u 0 x to prove Proposition 3. Ž2. If we drop the information on the sign of the solutions, Proposition 3 can be proved by a well-known method for even functionals. One could also use the theory on even functionals to obtain more solutions if f 9Ž0. ) l k , k ) 3. Here one uses the fact that w u 0 , u 0 x is an invariant set of the gradient flow of F.

638

NOTE

Combining Proposition 20 and 3, we obtain the following result. THEOREM 2. If ŽH 0 ., ŽH 1 ., and ŽH ` . are satisfied and f is odd, then Ž1. has at least 8 nontri¨ ial solutions with two pairs sign-preser¨ ing, and two pairs sign-changing. If further f 9Ž0. ) l3 , then there is at least one more pair of sign-changing solutions. To end this note, we show that Theorem 1 is still true if we replace the asymptotically linear condition ŽH ` . by the following superlinear condition, ŽHX` . For some M ) 0 and u ) 2, 0 - u F Ž u. F f Ž u. u for all < u < G M. THEOREM 3. The conclusions of Theorem 1 remain true if the condition ŽH ` . is replaced by ŽHX` .. Proof. Clearly we need only show that Proposition 2 is still true. We obtain u# and u* as in the proof of Proposition 2. Then we again obtain u1 and u 2 by the mountain pass theorem. This is possible because ŽHX` . guarantees that the P.S. condition is satisfied by the modified func˜ tional F. Now suppose that u1 and u 2 are the only solutions of Ž1. outside w u#, u*x. Then it follows from Proposition 1 of w6x that for all large R, deg H Ž I y A, BR , 0 . s x Ž H , FyR . , where Au s u y F9Ž u. s ŽyD .y1 f Ž u., BR s  u g H: 5 u 5 - R4 , and Fa s  u g H: F Ž u. F a4 . It follows from ŽHX` . and w12x that x Ž H, FyR . s 0. Hence deg H Ž I y A, BR , 0. s 0. Using the commutativity property of the fixed point index, the regularity of ŽyD .y1 , and Sobolev imbeddings as in w6x, we have deg H Ž I y A, BR , 0 . s deg E Ž I y A, B9, 0 . , where B9 denotes a large ball in which contains all the fixed points of A. Therefore deg E Ž I y A, B9, 0 . s 0.

Ž 14 .

Now u1 and u 2 have their critical groups again characterized by Ž3. and Ž5.. Hence for all small r ) 0, deg E Ž I y A, BrX Ž u i . , 0 . s y1,

i s 1, 2.

Ž 15 .

Here BrX Ž ¨ . s  u g E: 5 u y ¨ 5 E - r 4 . We can choose r small enough such that BrX Ž u i . ; B9 and BrX Ž u i . l w u, u x s B.

NOTE

639

It is easy to see that any solution of yDu s tf Ž u . ,

u < ­ D s 0, 0 F t F 1

is in the interior of the order interval w u, u x in E. Hence by the homotopy invariance of the degree, one easily obtains deg E Ž I y A, inf u, u , 0 . s 1.

Ž 16 .

Now by the additivity of the degree, deg E Ž I y A, B9, 0 . s deg E Ž I y A, BrX Ž u1 . , 0 . q deg E Ž I y A, BrX Ž u 2 . , 0 . q deg E Ž I y A, int u, u , 0 . . But this clearly contradicts Ž14. ] Ž16.. The proof is complete. Remarks. Ž1. In Theorem 3, if f is odd, then it is well known that Ž1. has infinitely many nontrivial solutions. Ž2. We can combine ideas here with that in w6x to show that Ž1. has at least one sign-changing solution outside the upper and lower solution interval under the conditions of Proposition 2, but with ŽH ` . replaced by a suitable superlinear condition, which is slightly more restrictive than ŽHX` ., so that the positive and negative solutions of Ž1. are a priori bounded. Ž3. After this paper was submitted, we learned of several additional articles that treat related topics, namely Ža. T. Bartsch and Z. Q. Wang, On the existence of sign-changing solution for semi-linear Dirichlet Problems, Topological Method Nonlinear Anal., in press. Žb. Zhaoli Liu, ‘‘Multiple Solution Problems for Differential Equations,’’ Ph.D. thesis, Shandong Univ., 1992. wIn Chinesex Žc. A. Castro, J . Cossio, and J. M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., in press. If one uses the flow generated by the gradient of the energy functional and the order structure as in Ža. or Žb., one can prove what is claimed in Remark Ž2. above under only ŽHX` .. An idea in Žb. can be used to give an alternative proof of our Proposition 20. Our Theorem 19 asserts the existence of at least two sign-changing solutions, but Ža. and Žc. deal with the existence of one sign-changing solution only.

640

NOTE

REFERENCES 1. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Re¨ . 18 Ž1976., 620]709. 2. H. Brezis and L. Nirenberg, H 1 versus C 1 local minimizers, C. R. Acad. Sci. Paris 317 Ž1993., 465]477. 3. K. C. Chang, ‘‘Infinite Dimensional Morse Theory and Multiple Solution Problems,’’ Birkhauser, Boston, 1993. ¨ 4. E. N. Dancer, Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Angew. Math. 350 Ž1984., 1]22. 5. E. N. Dancer and Y. Du, Multiple solutions of some semilinear elliptic equations via the generalised Conley index, J. Math. Anal. Appl. 189 Ž1995., 848]871. 6. E. N. Dancer and Y. Du, On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal. 56 Ž1995., 193]206. 7. E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math. 419 Ž1991., 125]139. 8. H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 Ž1982., 493]514. 9. Z. Liu, On Dancer’s conjecture and multiple solutions of elliptic partial differential equations, Northeast. Math. J. 9 Ž1993., 388]394. 10. J. Mawhin and M. Willem, ‘‘Critical Point Theory and Hamiltonian Systems,’’ SpringerVerlag, New YorkrBerlin, 1989. 11. K. P. Rybakowski, ‘‘The Homotopy Index and Partial Differential Equations,’’ SpringerVerlag, New YorkrBerlin, 1987. 12. Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincare´ Anal. Non Lineaire, 8 Ž1991., 43]58. ´