- Email: [email protected]
Multiplicity of Nontrivial Solutions of Semilinear Elliptic Equations
Journal of Mathematical Analysis and Applications 249, 289᎐299 Ž2000. doi:10.1006rjmaa.2000.6704, available online at http:rrwww.idealibrary.com on
Multiplicity of Nontrivial Solutions of Semilinear Elliptic Equations 1 Shui-Qiang Liu Department of Mathematics, Shaoyang Teacher’s College, Shaoyang 42200, People’s Republic of China
Chun-Lei Tang Department of Mathematics, Southwest Normal Uni¨ ersity, Chongqing 400715, People’s Republic of China
and Xing-Ping Wu Department of Mathematics, Southwest Normal Uni¨ ersity, Chongqing 400715, People’s Republic of China Submitted by Irena Lasiecka Received February 1, 1999
Two nontrivial solutions are obtained by the reduction method for the nonautonomous semilinear elliptic Dirichlet boundary value problem. Some well-known multiplicity results are generalized. 䊚 2000 Academic Press Key Words: semilinear elliptic equation; Dirichlet boundary value problem; nontrivial solution; reduction method; critical point; ŽPS. condition; Sobolev’s embedding theorem.
1. INTRODUCTION AND MAIN RESULTS Consider the semilinear elliptic Dirichlet boundary problem y⌬u s f Ž x, u . in ⍀ , u s 0 on ⭸ ⍀ , Ž 1. where ⍀ ; R Ž N G 1. is a bounded smooth domain and f : ⍀ = R ª R is a subcritical Caratheodory function; that is, there are positive constants ´ N
1
Project 19871067 supported by National Natural Science Foundation of China. 289 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
290
LIU, TANG, AND WU
C1 , C2 such that f Ž x, t . F C1 < t < py 1 q C2
Ž 2.
for all t g R and a.e. x g ⍀, where p gx2, 2 NrŽ N y 2.w for N G 3, p gx2, q⬁w for N s 1, 2. Let 0 - 1 - 2 - 3 - ⭈⭈⭈ be the sequence of the distinct eigenvalues of the eigenvalue problem y⌬u s u in ⍀ ,
u s 0 on ⭸ ⍀
and let k be a fixed positive integer. With the reduction method, two nontrivial solutions are obtained for the nonresonant or resonant elliptic problem Ž1. Žsee w2, 5᎐8x.. Recall the following multiple existence results. THEOREM A. Suppose that f g C 1 Ž R, R ., f Ž0. s 0, and f X is bounded. Assume that there exists 0 - m F k such that
my 1 - f X Ž 0 . - m ,
k - lim f X Ž t . - kq1 < t <ª⬁
Ž 3.
and such that sup f X Ž t . - kq 1 .
Ž 4.
tgR
Then problem Ž1. has at least two nontrivial solutions. THEOREM B.
Suppose that f Ž t . s k t q g Ž t ., g g C 2 Ž R, R ., lim
< t <ª⬁
gŽ t. t
s 0,
and lim g Ž t . t s q⬁
< t <ª⬁
and that g is bounded or lim inf < t < ª⬁ < g Ž t .< ) 0 holds. Assume that there exists 0 - m F k such that
my 1 F k q
inf
tgR _ 0 4
gŽ t. t
F k q gX Ž 0. - m
and that
k q sup g X Ž t . - kq1 . tgR
Then problem Ž1. has at least two nontrivial solutions.
SEMILINEAR ELLIPTIC EQUATIONS
291
Theorem A is due to Castro and Lazer w6x. Their approach is based on the reduction method and finite dimensional critical point theory. That is, under assumptions Ž3. and Ž4., there exist a function on the finite dimensional space V spanned by the eigenfunctions corresponding to the eigenvalues 1 , . . . , k and a function which is from V to the Sobolev space H01 Ž ⍀ . such that ¨ g V is a critical point of if and only if ¨ q Ž ¨ . is a solution of problem Ž1.. Under assumptions Ž3. and Ž4., one of the nontrivial solutions of problem Ž1. is a critical point of at which attains its maximum. The existence of the second nontrivial solution is deduced from the calculation of the Leray᎐Schauder index of critical points. Theorem B is due to Hirano and Nishimura w7x. Their approach is based on the reduction method and an abstract multiplicity result which is based on the minimax method. In this paper, we obtain some multiplicity results which unify and generalize the results mentioned above. Our approach is based on the reduction method and a three-critical-point theorem due to Brezis and Nirenberg w4x. The main results are the following theorems. THEOREM 1. that
Suppose that Ž2. holds and that there exists a - kq 1 such f Ž x, s . y f Ž x, t . syt
Fa
Ž 5.
for all s, t g R, s / t, and a.e. x g ⍀. Assume that F Ž x, t . y 12 k t 2 ª q⬁
Ž 6.
as < t < ª ⬁ uniformly for a.e. x g ⍀ and that there exist ␦ ) 0, b ) 0, and 0 - m F k such that 1 2
my 1 t 2 F F Ž x, t . F 12 Ž m y b . t 2
Ž 7.
for all < t < F ␦ and a.e. x g ⍀, where F Ž x, t . s f Ž x, s . ds. Then problem Ž1. has at least two nontri¨ ial solutions in H01 Ž ⍀ .. H0t
THEOREM 2.
Suppose that f g C 1 Ž R, R . is subcritical and sup f X Ž t . - kq 1 .
Ž 8.
tgR
Assume that f Ž t . t y k t 2 ª q⬁
Ž 9.
at < t < ª ⬁ and that there exist ␦ ) 0 and 0 - m F k such that
my 1 F
inf
0- < t <- ␦
f Ž t. t
,
f X Ž 0. - m .
Ž 10 .
292
LIU, TANG, AND WU
Then the problem u s 0 on ⭸ ⍀
y⌬u s f Ž u . in ⍀ , has at least two nontri¨ ial solutions. COROLLARY 1.
Suppose that g g C 1 Ž R, R . and sup g X Ž t . - kq 1 y k . tgR
Assume that g Ž t . t ª q⬁ as < t < ª ⬁ and that there exists ␦ ) 0 and 0 - m F k such that
my 1 y k F
inf
gŽ t.
0- < t
t
gX Ž 0. - m y k .
,
Then the problem y⌬u s k u q g Ž u . in ⍀ ,
u s 0 on ⭸ ⍀
has at least two nontri¨ ial solutions. Remark 1. Theorem A is a consequence of Theorem 2. In fact, lim < t < ª⬁ f X Ž t . ) k implies that f Ž t . t y k t 2 ª q⬁ as < t < ª ⬁ and f X Ž0. - my 1 implies that
my 1 F
inf
f Ž t.
0- < t <- ␦
t
for some ␦ ) 0. There are functions f g C R, R . satisfying our Theorem 2 and not satisfying Theorem A. For example, 1Ž
¡ Ž
f Ž t. s
~
1 2
kq 1
q k . t y 14 Ž kq1 q k y 2 my1 .
t
< t < G 1,
my 1 t q 14 Ž kq1 q k y 2 my1 . < t < t ,
¢
where f X Ž0. s my 1. Remark 2. Corollary 1 generalizes Theorem B. In fact, Corollary 1 has no need of the conditions that lim
< t <ª⬁
gŽ t. t
s0
SEMILINEAR ELLIPTIC EQUATIONS
293
and that g is bounded or lim inf < t < ª⬁ < g Ž t .< ) 0 holds; besides, the condition that
my 1 F k q
gŽ t.
inf
t
tgR _ 0 4
is replaced by the weaker one that
my 1 y k F
inf
gŽ t. t
0- < t
for some ␦ ) 0. There are functions g g C 1 Ž R, R . satisfying our Corollary 1 and not satisfying Theorem B. For example, g Ž t . s f Ž t . y k t , where f is the same as Remark 1 and lim
< t <ª⬁
gŽ t. t
/ 0.
Remark 3. From Remarks 1 and 2, we know that Theorem 2 unifies and generalizes Theorems A and B. Furthermore, Theorem 1 generalizes Theorems A and B to the nonautonomous case; it needs only the basic regularity.
2. PROOFS OF THEOREMS Define the functional on the Sobolev space H01 Ž ⍀ . by
Ž u . s y 12 5 u 5 2 q
u g H01 Ž ⍀ . ,
H⍀ F Ž x, u . dx,
where F Ž x, t . s H0t f Ž x, s . ds, 5 u 5 s Ž H⍀ < ⵜu < 2 dx .1r2 is the usual norm in H01 Ž ⍀ .. Then is continuously differentiable and
² X Ž u . , ¨: s yH ⵜu ⵜ¨ dx q H f Ž x, u . ¨ dx ⍀
⍀
for u, ¨ g H01 Ž ⍀ .. It is well known that u g H01 Ž ⍀ . is a solution of problem Ž1. if and only if u is a critical point of . Let V s E Ž 1 . q ⭈⭈⭈ qE Ž k .
294
LIU, TANG, AND WU
and W s V H , where EŽ i . stands for the eigenspace corresponding to i , i.e., the finite dimensional space spanned by the eigenfunctions corresponding to i . Define the functional
Ž ¨ . s sup Ž ¨ q w . ,
¨ gV
wgW
It follows easily from Theorem 2.3 of Amann w1x that LEMMA 1. Suppose that Ž2. and Ž5. hold. Then : V ª R is continuously differentiable and
X Ž ¨ . s PV X Ž ¨ q Ž ¨ . . ,
¨ g V,
where PV : H01 Ž ⍀ . ª V is the corresponding projection onto V along W, : V ª W and is a continuous mapping satisfying that
Ž¨. s Ž¨ q Ž¨.. for e¨ ery ¨ g V. It is a simple corollary that ¨ q Ž ¨ . is a critical point of if ¨ is a critical point of . LEMMA 2.
Suppose that Ž2. and Ž6. hold. Then is coerci¨ e on V, i.e., as 5 ¨ 5 ª ⬁ in V .
Ž¨. q ⬁ Thus is coerci¨ e.
Proof. By Ž6. there exists M ) 0 such that F Ž x, t . y 12 k t 2 G 0 for all < t < G M and a.e. x g ⍀. It follows from Ž2. that F Ž x, t . F
C1 p
M py1 q C2 M
for all < t < F M and a.e. x g ⍀. Hence we have F Ž x, t . G
1 2
k t 2 y
1 2
k M 2 y
C1 p
M p y C2 M
for all t g R and a.e. x g ⍀. If is not coercive, there exist M0 ) 0 and a sequence Ž ¨ n . in V such that 5 ¨ n 5 ª ⬁ and Ž ¨ n . F M0 . Let ¨ n s a n q bn , a n g EŽ 1 . q ⭈⭈⭈ q EŽ ky 1 ., and bn g EŽ k .. Then 5 ¨ n 5 2 s 5 a n 5 2 q 5 bn 5 2 .
SEMILINEAR ELLIPTIC EQUATIONS
295
In the case that Ž a n . has a subsequence Ž a n i . such that 5 an 5 ª ⬁ i
as i ª ⬁
one has 1 1 Ž ¨ n i . G y 5 a n i 5 2 q k 5 a n i 5 2L2 y C3 2 2 G
1 2
ž
k ky 1
y 1 5 a n i 5 2 y C3
/
ª q⬁ as i ª q⬁, where C3 s Ž 12 k M 2 q Ž C1rp . M p q C2 M . meas ⍀. This is a contradiction. In the case that Ž5 a n 5. is bounded by C4 , one has 5 bn 5 ª ⬁ as n ª ⬁. It follows from the first part of the proof of Lemma 3.2 in w3x that for every ␣ ) 0 there exists m␣ ) 0 such that meas x g ⍀ ¬ ¨ Ž x . - m␣ 5 ¨ 5 4 - ␣ for all ¨ g EŽ k .. Let A n s x g ⍀ ¬ bn Ž x . G m␣ 5 bn 5 4 . Then one has measŽ ⍀ _ A n . - ␣ . By the finite dimensionality of V, there exists C5 ) 0 such that sup a n Ž x . ¬ x g ⍀ 4 F C5 for all n. For every  ) 0, there exists M ) 0 such that F Ž x, t . y 12 k t 2 G  for all < t < G M and a.e. x g ⍀ by Ž6.. Let Bn s x g ⍀ ¬ ¨ n Ž x . G M 4 . For x g A n , one has ¨ n Ž x . G bn Ž x . y a n Ž x . G m␣ 5 bn 5 y C5 ,
296
LIU, TANG, AND WU
which implies A n ; Bn for large n. Now we have
H⍀
ž
F Ž x, ¨ n . y
1 2
k < ¨ n < 2 dx
/
G  meas Bn y G  meas A n y
ž ž
1
C1
k M 2 q
2 1 2
p C1
k M 2 q
G  Ž meas ⍀ y ␣ . y
1
ž
2
p
M p q C2 M meas Ž ⍀ _ Bn .
/ /
M p q C2 M meas Ž ⍀ _ A n .
k M 2 q
C1 p
M p q C2 M ␣
/
for large n. Hence one has lim inf nª⬁
H⍀
ž
1
F Ž x, ¨ n . y
2
k < ¨ n < 2 dx
/
G  Ž meas ⍀ y ␣ . y
ž
1 2
k M 2 q
C1 p
M p q C2 M ␣ .
/
Letting ␣ ª 0, we obtain lim inf nª⬁
H⍀ Ž F Ž x, ¨
n
. y 12 k < ¨ n < 2 . dx G  meas ⍀ .
By the arbitrariness of  we have lim inf nª⬁
H⍀ Ž F Ž x, ¨
n
. y 12 k < ¨ n < 2 . dx s q⬁.
Hence one has
Ž ¨n . G
H⍀ Ž F Ž x, ¨
n
. y 12 k < ¨ n < 2 . dx ª q⬁
as n ª ⬁, which is a contradiction. Therefore is coercive on V. Thus is coercive. LEMMA 3. Let V1 s EŽ 1 . q ⭈⭈⭈ qEŽ my1 ., V2 s EŽ m . q ⭈⭈⭈ qEŽ k .. Suppose that Ž7. holds. Then there exists ␦ 0 ) 0 such that
Ž¨. G 0
for ¨ g V1
and
5¨ 5 F ␦0
Ž¨. F 0
for ¨ g V2
and
5¨ 5 F ␦0 .
and
297
SEMILINEAR ELLIPTIC EQUATIONS
Proof. By the finite dimensionality of V1 , there exists C6 ) 0 such that sup ¨ Ž x . ¬ x g ⍀ 4 F C6 5 ¨ 5 for all ¨ g V1. Let ␦ 1 s Ž1rC6 . ␦ . Then for ¨ g V1 , 5 ¨ 5 F ␦ 1 , one has
Ž ¨ . G Ž ¨ . G y 12 5 ¨ 5 2 q 12 my1 5 ¨ 5 2L2 G 0. On the other hand, it follows from Ž7. that f Ž x, 0. s 0 for a.e. x g ⍀. Associating with Ž5. we have f Ž x, t . t F at 2 for all t g R and a.e. x g ⍀, which implies that F Ž x, t . F 12 at 2 for all t g R and a.e. x g ⍀ by the fact that F Ž x, t . s H01 f Ž x, ts . t ds. Hence one has F Ž x, t . F 12 Ž m y b . t 2 q 12 Ž a q b y m . t 2 F 12 Ž m y b . t 2 q 12 < a q b y m < ␦ 2yp < t < p for all < t < G ␦ and a.e. x g ⍀. It follows from Ž7. that F Ž x, t . F 12 Ž m y b . t 2 q C 7 < t < p
Ž 11 .
for all t g T, a.e. x g ⍀, and some C 7 ) 21 < a q b y m < ␦ 2y p. By Sobolev’s embedding theorem, there exists C8 ) 0 such that 5 u 5 L p F C8 5 u 5 for all u g H01 Ž ⍀ .. It follows from the continuity of that there exists ␦ 0 gx0, ␦ 1w such that ¨ q Ž¨. F
b
ž
2 m C 7 C8p
1r Ž py2 .
/
for all ¨ g V2 with 5 ¨ 5 F ␦ 0 . Thus Ž11. implies that
Ž¨. s Ž¨ q Ž¨.. Fy
1
¨ q Ž¨.
2
q C7 ¨ q Ž ¨ . Fy
b 2 m
2
q
for all ¨ g V2 with 5 ¨ 5 F ␦ 0 .
2
Ž m y b . ¨ q Ž ¨ .
p Lp
¨ q Ž¨.
F0
1
2
q C 7 C8p ¨ q Ž ¨ .
p
2 L2
298
LIU, TANG, AND WU
Proof of Theorem 1. By the finite dimensionality of V and Lemma 1 we know that satisfies the ŽPS. condition. In the case that inf ¨ g V Ž ¨ . s 0, all ¨ g V2 with 5 ¨ 5 F ␦ 0 are minima of by Lemma 3, which implies that has infinite critical points. In the case that inf ¨ g V Ž ¨ . - 0, from the proof of Lemma 2 one obtains inf Ž ¨ . ) y⬁.
¨ gV
It follows from Theorem 4 in w4x that has at least two nonzero critical points. Hence has at least two nonzero critical points. Thus problem Ž1. has at least two nontrivial solutions in H01 Ž ⍀ .. Proof of Theorem 2. Let a s sup t g R f X Ž t .. It follows from the mean value theorem and Ž8. that f Ž s. y f Ž t . syt
s fXŽ . F a
for all s, t g R, s / t. Hence Ž5. holds. For every  ) 0, there exists M ) 0 such that f Ž t . t y k t 2 G  for every < t < G M by Ž9.. Hence one has f Ž t . y k t G
 t
for all t G M, which implies that F Ž t . y F Ž M . y 12 k t 2 q 12 k M 2 G  ln t y  ln M. Thus we have F Ž t . y 12 k t 2 ª q⬁ as t ª q⬁. In a similar way we obtain F Ž t . y 12 k t 2 ª q⬁ as t ª y⬁. Hence Ž6. holds. Let b s 12 Ž m y f X Ž0... From the first part of Ž10. and the continuity of f, one obtains that f Ž0. s 0. By Ž10., there exists ␦ 0 gx0, ␦ w such that f Ž t. t
F m y b
SEMILINEAR ELLIPTIC EQUATIONS
299
for all 0 - < t < - ␦ 0 . Hence we have
my 1 t 2 F f Ž t . t F Ž m y b . t 2 for all < t < - ␦ 0 . Thus one has 1 2
my 1 t 2 F F Ž t . F 12 Ž m y b . t 2 .
for all < t < - ␦ 0 . Therefore Ž7. is proved. Now Theorem 2 follows from Theorem 1.
REFERENCES 1. H. Amann, Saddle points and multiple solutions of differential equations, Math. Z. 169 Ž1979., 127᎐166. 2. D. Arcoya and D. G. Costa, Nontrivial solutions for a strongly resonant problem, Differential Integral Equations 8, No. 1 Ž1995., 151᎐159. 3. P. Bartolo, V. Benci, and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal. 7, No. 7 Ž1983., 981᎐1012. 4. H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 Ž1991., 939᎐963. 5. N. P. Cac, On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl. 132 Ž1988., 473᎐483. 6. A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Math. 18 Ž1977., 113᎐137. 7. N. Hirano and T. Nishimura, Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities, J. Math. Anal. Appl. 180 Ž1993., 566᎐586. 8. K. Thews, A reduction method for some nonlinear Dirichlet problems, Nonlinear Anal. 3 Ž1979., 795᎐813.
Multiplicity of Nontrivial Solutions of Semilinear Elliptic Equations 1 Shui-Qiang Liu Department of Mathematics, Shaoyang Teacher’s College, Shaoyang 42200, People’s Republic of China
Chun-Lei Tang Department of Mathematics, Southwest Normal Uni¨ ersity, Chongqing 400715, People’s Republic of China
and Xing-Ping Wu Department of Mathematics, Southwest Normal Uni¨ ersity, Chongqing 400715, People’s Republic of China Submitted by Irena Lasiecka Received February 1, 1999
Two nontrivial solutions are obtained by the reduction method for the nonautonomous semilinear elliptic Dirichlet boundary value problem. Some well-known multiplicity results are generalized. 䊚 2000 Academic Press Key Words: semilinear elliptic equation; Dirichlet boundary value problem; nontrivial solution; reduction method; critical point; ŽPS. condition; Sobolev’s embedding theorem.
1. INTRODUCTION AND MAIN RESULTS Consider the semilinear elliptic Dirichlet boundary problem y⌬u s f Ž x, u . in ⍀ , u s 0 on ⭸ ⍀ , Ž 1. where ⍀ ; R Ž N G 1. is a bounded smooth domain and f : ⍀ = R ª R is a subcritical Caratheodory function; that is, there are positive constants ´ N
1
Project 19871067 supported by National Natural Science Foundation of China. 289 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
290
LIU, TANG, AND WU
C1 , C2 such that f Ž x, t . F C1 < t < py 1 q C2
Ž 2.
for all t g R and a.e. x g ⍀, where p gx2, 2 NrŽ N y 2.w for N G 3, p gx2, q⬁w for N s 1, 2. Let 0 - 1 - 2 - 3 - ⭈⭈⭈ be the sequence of the distinct eigenvalues of the eigenvalue problem y⌬u s u in ⍀ ,
u s 0 on ⭸ ⍀
and let k be a fixed positive integer. With the reduction method, two nontrivial solutions are obtained for the nonresonant or resonant elliptic problem Ž1. Žsee w2, 5᎐8x.. Recall the following multiple existence results. THEOREM A. Suppose that f g C 1 Ž R, R ., f Ž0. s 0, and f X is bounded. Assume that there exists 0 - m F k such that
my 1 - f X Ž 0 . - m ,
k - lim f X Ž t . - kq1 < t <ª⬁
Ž 3.
and such that sup f X Ž t . - kq 1 .
Ž 4.
tgR
Then problem Ž1. has at least two nontrivial solutions. THEOREM B.
Suppose that f Ž t . s k t q g Ž t ., g g C 2 Ž R, R ., lim
< t <ª⬁
gŽ t. t
s 0,
and lim g Ž t . t s q⬁
< t <ª⬁
and that g is bounded or lim inf < t < ª⬁ < g Ž t .< ) 0 holds. Assume that there exists 0 - m F k such that
my 1 F k q
inf
tgR _ 0 4
gŽ t. t
F k q gX Ž 0. - m
and that
k q sup g X Ž t . - kq1 . tgR
Then problem Ž1. has at least two nontrivial solutions.
SEMILINEAR ELLIPTIC EQUATIONS
291
Theorem A is due to Castro and Lazer w6x. Their approach is based on the reduction method and finite dimensional critical point theory. That is, under assumptions Ž3. and Ž4., there exist a function on the finite dimensional space V spanned by the eigenfunctions corresponding to the eigenvalues 1 , . . . , k and a function which is from V to the Sobolev space H01 Ž ⍀ . such that ¨ g V is a critical point of if and only if ¨ q Ž ¨ . is a solution of problem Ž1.. Under assumptions Ž3. and Ž4., one of the nontrivial solutions of problem Ž1. is a critical point of at which attains its maximum. The existence of the second nontrivial solution is deduced from the calculation of the Leray᎐Schauder index of critical points. Theorem B is due to Hirano and Nishimura w7x. Their approach is based on the reduction method and an abstract multiplicity result which is based on the minimax method. In this paper, we obtain some multiplicity results which unify and generalize the results mentioned above. Our approach is based on the reduction method and a three-critical-point theorem due to Brezis and Nirenberg w4x. The main results are the following theorems. THEOREM 1. that
Suppose that Ž2. holds and that there exists a - kq 1 such f Ž x, s . y f Ž x, t . syt
Fa
Ž 5.
for all s, t g R, s / t, and a.e. x g ⍀. Assume that F Ž x, t . y 12 k t 2 ª q⬁
Ž 6.
as < t < ª ⬁ uniformly for a.e. x g ⍀ and that there exist ␦ ) 0, b ) 0, and 0 - m F k such that 1 2
my 1 t 2 F F Ž x, t . F 12 Ž m y b . t 2
Ž 7.
for all < t < F ␦ and a.e. x g ⍀, where F Ž x, t . s f Ž x, s . ds. Then problem Ž1. has at least two nontri¨ ial solutions in H01 Ž ⍀ .. H0t
THEOREM 2.
Suppose that f g C 1 Ž R, R . is subcritical and sup f X Ž t . - kq 1 .
Ž 8.
tgR
Assume that f Ž t . t y k t 2 ª q⬁
Ž 9.
at < t < ª ⬁ and that there exist ␦ ) 0 and 0 - m F k such that
my 1 F
inf
0- < t <- ␦
f Ž t. t
,
f X Ž 0. - m .
Ž 10 .
292
LIU, TANG, AND WU
Then the problem u s 0 on ⭸ ⍀
y⌬u s f Ž u . in ⍀ , has at least two nontri¨ ial solutions. COROLLARY 1.
Suppose that g g C 1 Ž R, R . and sup g X Ž t . - kq 1 y k . tgR
Assume that g Ž t . t ª q⬁ as < t < ª ⬁ and that there exists ␦ ) 0 and 0 - m F k such that
my 1 y k F
inf
gŽ t.
0- < t
t
gX Ž 0. - m y k .
,
Then the problem y⌬u s k u q g Ž u . in ⍀ ,
u s 0 on ⭸ ⍀
has at least two nontri¨ ial solutions. Remark 1. Theorem A is a consequence of Theorem 2. In fact, lim < t < ª⬁ f X Ž t . ) k implies that f Ž t . t y k t 2 ª q⬁ as < t < ª ⬁ and f X Ž0. - my 1 implies that
my 1 F
inf
f Ž t.
0- < t <- ␦
t
for some ␦ ) 0. There are functions f g C R, R . satisfying our Theorem 2 and not satisfying Theorem A. For example, 1Ž
¡ Ž
f Ž t. s
~
1 2
kq 1
q k . t y 14 Ž kq1 q k y 2 my1 .
t
< t < G 1,
my 1 t q 14 Ž kq1 q k y 2 my1 . < t < t ,
¢
where f X Ž0. s my 1. Remark 2. Corollary 1 generalizes Theorem B. In fact, Corollary 1 has no need of the conditions that lim
< t <ª⬁
gŽ t. t
s0
SEMILINEAR ELLIPTIC EQUATIONS
293
and that g is bounded or lim inf < t < ª⬁ < g Ž t .< ) 0 holds; besides, the condition that
my 1 F k q
gŽ t.
inf
t
tgR _ 0 4
is replaced by the weaker one that
my 1 y k F
inf
gŽ t. t
0- < t
for some ␦ ) 0. There are functions g g C 1 Ž R, R . satisfying our Corollary 1 and not satisfying Theorem B. For example, g Ž t . s f Ž t . y k t , where f is the same as Remark 1 and lim
< t <ª⬁
gŽ t. t
/ 0.
Remark 3. From Remarks 1 and 2, we know that Theorem 2 unifies and generalizes Theorems A and B. Furthermore, Theorem 1 generalizes Theorems A and B to the nonautonomous case; it needs only the basic regularity.
2. PROOFS OF THEOREMS Define the functional on the Sobolev space H01 Ž ⍀ . by
Ž u . s y 12 5 u 5 2 q
u g H01 Ž ⍀ . ,
H⍀ F Ž x, u . dx,
where F Ž x, t . s H0t f Ž x, s . ds, 5 u 5 s Ž H⍀ < ⵜu < 2 dx .1r2 is the usual norm in H01 Ž ⍀ .. Then is continuously differentiable and
² X Ž u . , ¨: s yH ⵜu ⵜ¨ dx q H f Ž x, u . ¨ dx ⍀
⍀
for u, ¨ g H01 Ž ⍀ .. It is well known that u g H01 Ž ⍀ . is a solution of problem Ž1. if and only if u is a critical point of . Let V s E Ž 1 . q ⭈⭈⭈ qE Ž k .
294
LIU, TANG, AND WU
and W s V H , where EŽ i . stands for the eigenspace corresponding to i , i.e., the finite dimensional space spanned by the eigenfunctions corresponding to i . Define the functional
Ž ¨ . s sup Ž ¨ q w . ,
¨ gV
wgW
It follows easily from Theorem 2.3 of Amann w1x that LEMMA 1. Suppose that Ž2. and Ž5. hold. Then : V ª R is continuously differentiable and
X Ž ¨ . s PV X Ž ¨ q Ž ¨ . . ,
¨ g V,
where PV : H01 Ž ⍀ . ª V is the corresponding projection onto V along W, : V ª W and is a continuous mapping satisfying that
Ž¨. s Ž¨ q Ž¨.. for e¨ ery ¨ g V. It is a simple corollary that ¨ q Ž ¨ . is a critical point of if ¨ is a critical point of . LEMMA 2.
Suppose that Ž2. and Ž6. hold. Then is coerci¨ e on V, i.e., as 5 ¨ 5 ª ⬁ in V .
Ž¨. q ⬁ Thus is coerci¨ e.
Proof. By Ž6. there exists M ) 0 such that F Ž x, t . y 12 k t 2 G 0 for all < t < G M and a.e. x g ⍀. It follows from Ž2. that F Ž x, t . F
C1 p
M py1 q C2 M
for all < t < F M and a.e. x g ⍀. Hence we have F Ž x, t . G
1 2
k t 2 y
1 2
k M 2 y
C1 p
M p y C2 M
for all t g R and a.e. x g ⍀. If is not coercive, there exist M0 ) 0 and a sequence Ž ¨ n . in V such that 5 ¨ n 5 ª ⬁ and Ž ¨ n . F M0 . Let ¨ n s a n q bn , a n g EŽ 1 . q ⭈⭈⭈ q EŽ ky 1 ., and bn g EŽ k .. Then 5 ¨ n 5 2 s 5 a n 5 2 q 5 bn 5 2 .
SEMILINEAR ELLIPTIC EQUATIONS
295
In the case that Ž a n . has a subsequence Ž a n i . such that 5 an 5 ª ⬁ i
as i ª ⬁
one has 1 1 Ž ¨ n i . G y 5 a n i 5 2 q k 5 a n i 5 2L2 y C3 2 2 G
1 2
ž
k ky 1
y 1 5 a n i 5 2 y C3
/
ª q⬁ as i ª q⬁, where C3 s Ž 12 k M 2 q Ž C1rp . M p q C2 M . meas ⍀. This is a contradiction. In the case that Ž5 a n 5. is bounded by C4 , one has 5 bn 5 ª ⬁ as n ª ⬁. It follows from the first part of the proof of Lemma 3.2 in w3x that for every ␣ ) 0 there exists m␣ ) 0 such that meas x g ⍀ ¬ ¨ Ž x . - m␣ 5 ¨ 5 4 - ␣ for all ¨ g EŽ k .. Let A n s x g ⍀ ¬ bn Ž x . G m␣ 5 bn 5 4 . Then one has measŽ ⍀ _ A n . - ␣ . By the finite dimensionality of V, there exists C5 ) 0 such that sup a n Ž x . ¬ x g ⍀ 4 F C5 for all n. For every  ) 0, there exists M ) 0 such that F Ž x, t . y 12 k t 2 G  for all < t < G M and a.e. x g ⍀ by Ž6.. Let Bn s x g ⍀ ¬ ¨ n Ž x . G M 4 . For x g A n , one has ¨ n Ž x . G bn Ž x . y a n Ž x . G m␣ 5 bn 5 y C5 ,
296
LIU, TANG, AND WU
which implies A n ; Bn for large n. Now we have
H⍀
ž
F Ž x, ¨ n . y
1 2
k < ¨ n < 2 dx
/
G  meas Bn y G  meas A n y
ž ž
1
C1
k M 2 q
2 1 2
p C1
k M 2 q
G  Ž meas ⍀ y ␣ . y
1
ž
2
p
M p q C2 M meas Ž ⍀ _ Bn .
/ /
M p q C2 M meas Ž ⍀ _ A n .
k M 2 q
C1 p
M p q C2 M ␣
/
for large n. Hence one has lim inf nª⬁
H⍀
ž
1
F Ž x, ¨ n . y
2
k < ¨ n < 2 dx
/
G  Ž meas ⍀ y ␣ . y
ž
1 2
k M 2 q
C1 p
M p q C2 M ␣ .
/
Letting ␣ ª 0, we obtain lim inf nª⬁
H⍀ Ž F Ž x, ¨
n
. y 12 k < ¨ n < 2 . dx G  meas ⍀ .
By the arbitrariness of  we have lim inf nª⬁
H⍀ Ž F Ž x, ¨
n
. y 12 k < ¨ n < 2 . dx s q⬁.
Hence one has
Ž ¨n . G
H⍀ Ž F Ž x, ¨
n
. y 12 k < ¨ n < 2 . dx ª q⬁
as n ª ⬁, which is a contradiction. Therefore is coercive on V. Thus is coercive. LEMMA 3. Let V1 s EŽ 1 . q ⭈⭈⭈ qEŽ my1 ., V2 s EŽ m . q ⭈⭈⭈ qEŽ k .. Suppose that Ž7. holds. Then there exists ␦ 0 ) 0 such that
Ž¨. G 0
for ¨ g V1
and
5¨ 5 F ␦0
Ž¨. F 0
for ¨ g V2
and
5¨ 5 F ␦0 .
and
297
SEMILINEAR ELLIPTIC EQUATIONS
Proof. By the finite dimensionality of V1 , there exists C6 ) 0 such that sup ¨ Ž x . ¬ x g ⍀ 4 F C6 5 ¨ 5 for all ¨ g V1. Let ␦ 1 s Ž1rC6 . ␦ . Then for ¨ g V1 , 5 ¨ 5 F ␦ 1 , one has
Ž ¨ . G Ž ¨ . G y 12 5 ¨ 5 2 q 12 my1 5 ¨ 5 2L2 G 0. On the other hand, it follows from Ž7. that f Ž x, 0. s 0 for a.e. x g ⍀. Associating with Ž5. we have f Ž x, t . t F at 2 for all t g R and a.e. x g ⍀, which implies that F Ž x, t . F 12 at 2 for all t g R and a.e. x g ⍀ by the fact that F Ž x, t . s H01 f Ž x, ts . t ds. Hence one has F Ž x, t . F 12 Ž m y b . t 2 q 12 Ž a q b y m . t 2 F 12 Ž m y b . t 2 q 12 < a q b y m < ␦ 2yp < t < p for all < t < G ␦ and a.e. x g ⍀. It follows from Ž7. that F Ž x, t . F 12 Ž m y b . t 2 q C 7 < t < p
Ž 11 .
for all t g T, a.e. x g ⍀, and some C 7 ) 21 < a q b y m < ␦ 2y p. By Sobolev’s embedding theorem, there exists C8 ) 0 such that 5 u 5 L p F C8 5 u 5 for all u g H01 Ž ⍀ .. It follows from the continuity of that there exists ␦ 0 gx0, ␦ 1w such that ¨ q Ž¨. F
b
ž
2 m C 7 C8p
1r Ž py2 .
/
for all ¨ g V2 with 5 ¨ 5 F ␦ 0 . Thus Ž11. implies that
Ž¨. s Ž¨ q Ž¨.. Fy
1
¨ q Ž¨.
2
q C7 ¨ q Ž ¨ . Fy
b 2 m
2
q
for all ¨ g V2 with 5 ¨ 5 F ␦ 0 .
2
Ž m y b . ¨ q Ž ¨ .
p Lp
¨ q Ž¨.
F0
1
2
q C 7 C8p ¨ q Ž ¨ .
p
2 L2
298
LIU, TANG, AND WU
Proof of Theorem 1. By the finite dimensionality of V and Lemma 1 we know that satisfies the ŽPS. condition. In the case that inf ¨ g V Ž ¨ . s 0, all ¨ g V2 with 5 ¨ 5 F ␦ 0 are minima of by Lemma 3, which implies that has infinite critical points. In the case that inf ¨ g V Ž ¨ . - 0, from the proof of Lemma 2 one obtains inf Ž ¨ . ) y⬁.
¨ gV
It follows from Theorem 4 in w4x that has at least two nonzero critical points. Hence has at least two nonzero critical points. Thus problem Ž1. has at least two nontrivial solutions in H01 Ž ⍀ .. Proof of Theorem 2. Let a s sup t g R f X Ž t .. It follows from the mean value theorem and Ž8. that f Ž s. y f Ž t . syt
s fXŽ . F a
for all s, t g R, s / t. Hence Ž5. holds. For every  ) 0, there exists M ) 0 such that f Ž t . t y k t 2 G  for every < t < G M by Ž9.. Hence one has f Ž t . y k t G
 t
for all t G M, which implies that F Ž t . y F Ž M . y 12 k t 2 q 12 k M 2 G  ln t y  ln M. Thus we have F Ž t . y 12 k t 2 ª q⬁ as t ª q⬁. In a similar way we obtain F Ž t . y 12 k t 2 ª q⬁ as t ª y⬁. Hence Ž6. holds. Let b s 12 Ž m y f X Ž0... From the first part of Ž10. and the continuity of f, one obtains that f Ž0. s 0. By Ž10., there exists ␦ 0 gx0, ␦ w such that f Ž t. t
F m y b
SEMILINEAR ELLIPTIC EQUATIONS
299
for all 0 - < t < - ␦ 0 . Hence we have
my 1 t 2 F f Ž t . t F Ž m y b . t 2 for all < t < - ␦ 0 . Thus one has 1 2
my 1 t 2 F F Ž t . F 12 Ž m y b . t 2 .
for all < t < - ␦ 0 . Therefore Ž7. is proved. Now Theorem 2 follows from Theorem 1.
REFERENCES 1. H. Amann, Saddle points and multiple solutions of differential equations, Math. Z. 169 Ž1979., 127᎐166. 2. D. Arcoya and D. G. Costa, Nontrivial solutions for a strongly resonant problem, Differential Integral Equations 8, No. 1 Ž1995., 151᎐159. 3. P. Bartolo, V. Benci, and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal. 7, No. 7 Ž1983., 981᎐1012. 4. H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 Ž1991., 939᎐963. 5. N. P. Cac, On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl. 132 Ž1988., 473᎐483. 6. A. Castro and A. C. Lazer, Critical point theory and the number of solutions of a nonlinear Dirichlet problem, Ann. Math. 18 Ž1977., 113᎐137. 7. N. Hirano and T. Nishimura, Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities, J. Math. Anal. Appl. 180 Ž1993., 566᎐586. 8. K. Thews, A reduction method for some nonlinear Dirichlet problems, Nonlinear Anal. 3 Ž1979., 795᎐813.