SEMILINEAR ELLIPTIC EQUATIONS HAVING
ASYMPTOTIC LIMITS AT ZERO AND INFINITY
KANISHKA PERERA AND MARTIN SCHECHTER
Received 22 October 1999
We obtain nontrivial solutions for semilinear elliptic boundary value problems having
resonance both at zero and at infinity, when the nonlinear term has asymptotic limits.
1. Introduction
Let be a smooth, bounded domain in Rn , and let A be a selfadjoint operator on
L2 (). We assume that
C0∞ () ⊂ D := D |A|1/2 ⊂ H m,2 ()
(1.1)
holds for some m > 0, and σe (A) = φ with A bounded from below. Let f (x, t) be a
Carathéodory function on × R satisfying
f (x, t) = a0 t + p0 (x, t),
f (x, t) = at + p(x, t),
p0 (x, t) = o(t) as t −→ 0,
p(x, t) = o(t)
as |t| −→ ∞.
(1.2)
The object of this paper is to prove the following theorem.
Theorem 1.1. Assume that there is a λ ∈ σ (A) such that either
a0 ≤ λ ≤ a
(1.3)
a ≤ λ ≤ a0 .
(1.4)
or
If a0 ∈ σ (A), assume also that there is a σ ∈ (2, 2∗ ), 2∗ = 2n/(n − 2), such that
tp0 (x, t)
−→ α±
|t|σ
as t −→ ±0
Copyright © 1999 Hindawi Publishing Corporation
Abstract and Applied Analysis 4:4 (1999) 231–242
1991 Mathematics Subject Classification: 35J65, 58E05
URL: http://aaa.hindawi.com/volume-4/S1085337599000159.html
(1.5)
232
Semilinear elliptic equations having asymptotic limits …
and
y>0
α+ |y|σ +
y<0
α− |y|σ > 0,
y ∈ E a0 \{0},
(1.6)
if λ ≤ a0 and < 0 if a0 ≤ λ, where E(b) = {u ∈ D : (A−b)u = 0}. If a ∈ σ (A), assume
also that there is a τ ∈ (1, 2) such that
tp(x, t)
−→ β±
|t|τ
and
y>0
β+ |y|τ +
y<0
as t −→ ±∞
β− |y|τ > 0,
if λ ≤ a and < 0 if a ≤ λ. Finally assume that
|f (x, t)| ≤ C |t| + 1 ,
y ∈ E(a)\{0}
x ∈ , t ∈ R.
(1.7)
(1.8)
(1.9)
Then
Au = f (x, u)
(1.10)
has a nontrivial solution.
The proof of Theorem 1.1 will be accomplished by means of a series of lemmas
given in the next section.
Many authors have studied special cases of problem (1.10) under hypotheses (1.2)
beginning with the work of Amann-Zehnder [1], who considered the Dirichlet problem
−!u = f (u) in ,
u = 0 on ∂.
(1.11)
They assumed that f (t) ∈ C 1 (R) and that either
f (0) < λ < f (∞)
(1.12)
f (∞) < λ < f (0).
(1.13)
or
They did not allow f (∞) to be in σ (A). Since then many authors have weakened
some of these requirements (see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22], and the references therein). In most cases, f (x, t) is required to be
continuously differentiable with respect to t, and a and a0 are not both allowed to be
in σ (A). In Theorem 1.1, we only require the continuity of f (x, t) with respect to t,
allow either or both a0 and a to be in σ (A) and permit a = a0 = λ.
2. Lemmas
Theorem 1.1 will be established via a series of lemmas. In describing them, we let be a smooth, bounded domain in Rn , and we let A be a selfadjoint operator on L2 ().
K. Perera and M. Schechter
We assume that
C0∞ () ⊂ D := D |A|1/2 ⊂ H m,2 ()
233
(2.1)
holds for some m > 0, and σe (A) ⊂ (0, ∞). We use the notation
a(u, v) = (Au, v),
a(u) = a(u, u),
u, v ∈ D.
D becomes a Hilbert space if we use the scalar product
(u, v)D = |A|u, v + P0 u, v , u, v ∈ D,
(2.2)
(2.3)
and its corresponding norm, where P0 is the projection onto N(A). Let f (x, t) be a
Carathéodory function on × R satisfying
|f (x, t)| ≤ V (x)q |t|q−1 + 1 , x ∈ , t ∈ R,
(2.4)
and
f (x, t)
= o |t|q−1 as |t| −→ ∞, uniformly,
q
V (x)
(2.5)
where q > 2 satisfies
q≤
2n
,
n − 2m
2m < n, q < ∞, n ≤ 2m,
(2.6)
and V (x) > 0 is a function in Lq () such that
V uq ≤ CuD ,
u ∈ D.
(The norm on the left in (2.7) is that of Lq ().)
Let
V =
N(A − λ).
(2.7)
(2.8)
λ<0
By assumption, p = dim N (A)+dim V < ∞. Let W = [V
N(A)]⊥ , and let P− , P0 , P+
be the projections onto V , N (A), W , respectively. Let λ(λ) denote the largest (smallest)
point in the negative (positive) spectrum of A. Then
(Av, v) ≤ λv2 ,
v ∈ V,
(Aw, w) ≥ λw2 ,
We let
w ∈ W.
(2.9)
2G(u) = a(u) − 2
where
F (x, t) =
0
t
F (x, u),
f (x, s) ds.
(2.10)
(2.11)
234
Semilinear elliptic equations having asymptotic limits …
As is well known, G is in C 1 in D, and
G (u), h = a(u, h) − f (u), h ,
u, h ∈ D,
(2.12)
where we write f (u) in place of f (x, u(x)). Moreover, u is a solution of
Au = f (x, u)
(2.13)
G (u) = 0.
(2.14)
if and only if it satisfies
In our first result we make use of the following assumption:
(A) there is a constant σ ∈ (2, 2∗ ) such that
f (x, t)t
−→ α± (x) as t −→ ±0, uniformly in x,
|t|σ
where
y>0
α+ |y|σ +
y<0
α− |y|σ > 0,
y ∈ N(A)\{0}.
(2.15)
(2.16)
We have the following lemma.
Lemma 2.1. If 0 is an isolated solution of (2.13), and (A) holds, then
Ck (G, 0) ∼
= δpk Z ∀k,
(2.17)
2J (u) = P+ u2 − P− u2 − P0 u2 ,
(2.18)
where p = dim V + dim N (A).
Proof. We define
and let
Ht (u) = a(u) − 2(1 − t)
Gt (u) = Ht (u) + tJ (u),
We note that there is a ρ > 0 such that
Ht (u), J (u) > 0,
F (x, u),
(2.19)
t ∈ [0, 1].
0 < uD ≤ ρ.
For if (2.20) did not hold, there would be a sequence {uk } ⊂ D such that
Ht uk , J uk ≤ 0,
(2.20)
(2.21)
and ρk = uk D → 0. Let ũk = uk /ρk , and write ũk = ṽk + ỹk + w̃k , ṽk ∈ V , ỹk ∈ N(A),
and w̃k ∈ W . In particular, we have
G uk , h = a u k , h − f u k , h .
(2.22)
K. Perera and M. Schechter
Thus,
Ht uk , J uk
ρk2
2 2 (1 − t) f uk , ûk
= w̃k D + ṽk D −
.
ρk2
(Here we take û = w − v − y.) From this we conclude that (2.21) implies
ṽk + w̃k −→ 0.
D
D
235
(2.23)
(2.24)
Since ũk D = 1, we must have a renamed subsequence such that ỹk → ỹ strongly in
D with ỹD = 1. Consequently,
(1 − t) f uk , ûk
H t uk , J uk
≥−
.
(2.25)
ρkσ
ρkσ
But
f x, uk ỹk
−
ρkσ −1
=−
−→
ỹ>0
uk f x, uk σ −2
ũk ũ
ỹ
k
k
|uk |σ
σ
σ
α+ ỹ +
α− ỹ > 0
(2.26)
ỹ<0
for a subsequence by hypothesis (A), since ỹ = 0. Moreover,
f x, uk ṽk
f x, uk w̃k
−→
0,
−→ 0.
ρkσ −1
ρkσ −1
(2.27)
This contradicts (2.21) and shows that (2.20) holds for t < 1. It is obvious for t = 1.
Now
2
Gt (u), J (u) = Ht (u), J (u) + t J (u), J (u) ≥ t J (u) .
(2.28)
If u is a critical point of Gt , then J (u) = 0, from which it follows that u = 0. Thus 0
is an isolated critical point of Gt . Since 2G1 (u) = [a(u) + J (u)],
G1 (0) = A + P+ − P− − P0 .
By hypothesis,
AP+ > 0,
A P− + P0 < 0.
(2.29)
(2.30)
Consequently, the Morse index of G1 (0) is p. By the homotopy invariance of critical
groups, we have
(2.31)
Ck (G, 0) ∼
= C k G1 , 0 ∼
= δpk Z.
This gives the desired conclusion.
In our second result we make use of the following assumption:
236
Semilinear elliptic equations having asymptotic limits …
(B) there is a constant σ ∈ (2, 2∗ ) such that
f (x, t)t
−→ α± (x) as t −→ ±0, uniformly in x,
|t|σ
where
σ
y>0
α+ |y| +
y<0
α− |y|σ < 0,
y ∈ N(A)\{0}.
(2.32)
(2.33)
We have the following lemma.
Lemma 2.2. If 0 is an isolated solution of (2.13), and (B) holds, then
Ck (G, 0) ∼
= δp1 k Z ∀k,
(2.34)
2 2 2
2J (u) = P+ u − P− u + P0 u ,
(2.35)
where p1 = dim V .
Proof. Now we define
and let
Ht (u) = a(u) − 2(1 − t)
F (x, u),
Gt (u) = Ht (u) + tJ (u),
We note that there is a ρ > 0 such that
Ht (u), J (u) > 0,
t ∈ [0, 1].
0 < uD ≤ ρ.
For if (2.37) did not hold, there would be a sequence {uk } ⊂ D such that
Ht uk , J uk ≤ 0,
(2.36)
(2.37)
(2.38)
and ρk = uk D → 0. Let ũk = uk /ρk , and write ũk = ṽk + ỹk + w̃k , ṽk ∈ V , ỹk ∈ N(A),
and w̃k ∈ W . In particular, we have
G uk , h = a u k , h − f u k , h .
(2.39)
Thus,
Ht uk , J uk
ρk2
2 2 (1 − t) f uk , ûk
= w̃k D + ṽk D −
.
ρk2
(Here we take û = w − v + y.) From this we conclude that (2.38) implies
ṽk + w̃k −→ 0.
D
D
(2.40)
(2.41)
Since ũk D = 1, we must have a renamed subsequence such that ỹk → ỹ strongly in
D with ỹD = 1. Consequently,
H t uk , J uk
(1 − t) f uk , ûk
≥−
.
(2.42)
ρkσ
ρkσ
K. Perera and M. Schechter
237
But
f x, uk ỹk
ρkσ −1
uk f x, uk σ −2
ũk =
ũ
ỹ
k
k
|uk |σ
σ
α+ |ỹ| +
α− |ỹ|σ < 0
−→
ỹ>0
(2.43)
ỹ<0
for a subsequence by hypothesis (B), since ỹ = 0. Moreover,
f x, uk ṽk
f x, uk w̃k
−→ 0,
−→ 0.
ρkσ −1
ρkσ −1
(2.44)
This contradicts (2.21) and shows that (2.37) holds. Now
2
Gt (u), J (u) = Ht (u), J (u) + t J (u), J (u) ≥ t J (u) .
(2.45)
If u is a critical point of Gt , then J (u) = 0, from which it follows that u = 0. Thus 0
is an isolated critical point of Gt . Since 2G1 (u) = [a(u) + J (u)],
G1 (0) = A + P+ − P− + P0 .
By hypothesis,
A P+ + P0 > 0,
AP− < 0.
(2.46)
(2.47)
Consequently, the Morse index of G1 (0) is p1 . By the homotopy invariance of critical
groups, we have
(2.48)
Ck (G, 0) ∼
= C k G1 , 0 ∼
= δp1 k Z.
This gives the desired conclusion.
Lemma 2.3. If N (A) = {0}, 0 is an isolated solution of (2.13), and
f (x, t)
−→ 0
t
as t −→ 0,
(2.49)
then (2.34) holds.
Proof. We follow the proof of Lemma 2.2. In this case P0 = 0, and (2.37) holds because
(2.38) implies (2.41), which is now the same as uk D → 0. This contradicts the fact
that uk D = 1. Thus, (2.45) holds. We can now follow the continuation of the proof
of Lemma 2.2 keeping in mind that P0 = 0.
Our next result assumes
(C) there is a constant σ ∈ (1, 2) such that
f (x, t)t
−→ α± (x) as t −→ ±∞, uniformly in x,
|t|σ
(2.50)
238
Semilinear elliptic equations having asymptotic limits …
where
y>0
α+ |y|σ +
α− |y|σ > 0,
y ∈ N(A)\{0}.
(2.51)
as uD −→ ∞, u ∈ V ⊕ N(A).
(2.52)
y<0
We have the following lemma.
Lemma 2.4. If (C) holds, then
G(u) −→ −∞
Proof. Assume that there is a sequence {uk } ⊆ V ⊕ N(A) such that ρk = uk D → ∞
and G(uk ) is bounded from below. Let ũk = uk /ρk , and write ũk = ṽk + ỹk , ṽk ∈ V ,
ỹk ∈ N(A). Since
2
G uk
F x, uk
= −ṽk D − 2
dx,
(2.53)
ρk2
ρk2
and f (x, t)/t → 0 as t → ∞, we see that ṽk D → 0. Thus, there is a renamed
subsequence such that ũk → ỹ in D. Consequently,
σ
G uk
F x, uk
−vk 2D
=
−2
dx −→ −
α+ ỹ −
α− |ỹ|σ < 0,
ρkσ
ρkσ
ρkσ
ỹ>0
ỹ<0
(2.54)
since ỹ = 0. This contradicts the assumption that G(uk ) is bounded from below.
Similarly, we have the following lemma.
Lemma 2.5. Assume
(D) there is a constant σ ∈ (1, 2) such that
f (x, t)t
−→ α± (x) as t −→ ±∞, uniformly in x,
|t|σ
where
y>0
α+ |y|σ +
α− |y|σ < 0,
(2.55)
y ∈ N(A)\{0}.
(2.56)
as uD −→ ∞, u ∈ W ⊕ N(A).
(2.57)
y<0
Then
G(u) −→ ∞
Lemma 2.6. If
f (x, t)
−→ 0
t
as |t| −→ ∞,
(2.58)
as uD −→ ∞, u ∈ V ,
(2.59)
then
G(u) −→ −∞
G(u) −→ ∞
as uD −→ ∞, u ∈ W.
(2.60)
K. Perera and M. Schechter
239
Proof. Assume {vk } ⊂ V , ρk = vk D → ∞, and G(vk ) → m > −∞. Let ṽk = vk /ρk .
Then ṽk = 1, and there is a renamed subsequence such that ṽk → ṽ in D and a.e. in .
Thus
2 2 F x, vk dx
2G vk
−→ −ṽ D < 0.
(2.61)
= − ṽk D −
2
2
ρk
ρk
This proves (2.59). Similarly, if {wk } ⊂ W , and ρk = wk D → ∞, let w̃k = wk /ρk .
Then w̃k = 1, and there is a renamed subsequence such that w̃k → w̃ weakly in D,
strongly in L2 (), and a.e. in . Then,
2 2 F x, wk dx
2 F x, wk dx
2G wk
= w̃k D −
≥ 1−
−→ 1.
(2.62)
ρk2
ρk2
ρk2
This proves (2.60).
Lemma 2.7. Assume (2.58). If N (A) = {0}, assume also that there is a constant
σ ∈ (1, 2) such that
f (x, t)t
−→ α± (x) as t −→ ±∞, uniformly in x,
|t|σ
where
σ
y>0
α+ |y| +
y<0
α− |y|σ = 0,
y ∈ N(A)\{0}.
(2.63)
(2.64)
Then G satisfies the PS condition.
Proof. If
G uk −→ c,
G uk −→ 0,
(2.65)
assume that ρk = uk D → ∞. Let ũk = uk /ρk , and write ũk = ṽk + ỹk + w̃k , ṽk ∈ V ,
ỹk ∈ N(A), and w̃k ∈ W . In particular, we have
G uk , h = a uk , h − f uk , h = o hD .
(2.66)
Setting h = w̃k , −ṽk , respectively, and dividing by ρk , we conclude that
ṽk + w̃k −→ 0.
D
D
(2.67)
Since ũk D = 1, we must have a renamed subsequence such that ỹk → ỹ strongly in
D with ỹD = 1. Consequently,
f uk
G uk
(2.68)
, ỹk = −
, ỹk −→ 0.
σ −1
ρ σ −1
ρk
But
f x, uk ỹk
ρkσ −1
=
σ ũk σ
σ
uk f x, uk
−→
α
+
α− ỹ ỹ
ỹ
k
+
σ
|uk |
ũk
ỹ>0
ỹ<0
(2.69)
240
Semilinear elliptic equations having asymptotic limits …
for any subsequence. By hypothesis, this cannot vanish, since ỹ = 0. This contradiction
shows that ρk ≤ C, and the usual methods obtain a convergent subsequence of {uk }
(cf. [20]).
The following lemma is well known (cf. [2]).
Lemma 2.8. If E = V ⊕ W , k = dim V < ∞, G ∈ C 1 (E, R) satisfies the PS condition,
u0 is the only critical point of G, (2.59) holds, and
inf G > −∞,
(2.70)
Ck G, u0 = 0.
(2.71)
W
then
3. The final proof
We can now give the proof of Theorem 1.1.
Proof. Assume that 0 is the only solution of (1.10) and that (1.3) holds. Let
N(A − µ),
W0 = V0⊥ .
V0 =
(3.1)
µ<a0
If a0 ∈
/ σ (A), then
Ck (G, 0) ∼
= δp0 k Z ∀k,
(3.2)
where p0 = dim V0 by Lemma 2.3. If a0 ∈ σ (A), then (3.2) holds by Lemma 2.2. On
the other hand, if a ∈
/ σ (A), then (2.59) and (2.60) hold by Lemma 2.6, where
N(A − µ),
W = V ⊥.
(3.3)
V =
µ≤a
This implies (2.70). For if
G wk −→ −∞,
(3.4)
then we must have wk D ≤ C by (2.60). Then there is a renamed subsequence such
that wk → w weakly in D, strongly in L2 () and a.e. in . It then follows that
(3.5)
G wk ≥ − F x, wk dx −→ − F (x, w) dx > −∞
(cf. [20]). Therefore,
Cp (G, 0) = 0,
p = dim V .
(3.6)
If a ∈ σ (A), then (2.59) holds by Lemma 2.4, while (2.60) holds as before. Thus, (3.6)
holds in this case as well. Now we note that p0 < p, since E(λ) ⊂ V , while E(λ) ⊂ V0 .
K. Perera and M. Schechter
241
This contradiction proves the theorem when (1.3) holds. Assume next that (1.4) holds.
Let
N(A − µ),
W = V ⊥,
V =
µ<a
V0 =
µ≤a0
N(A − µ),
W0 = V0⊥ .
(3.7)
/ σ (A), then (3.2) holds by Lemma 2.3. If a0 ∈ σ (A), then (3.2) holds by
If a0 ∈
Lemma 2.4. However, if a ∈
/ σ (A), then (2.59) and (2.60) hold by Lemma 2.6. Hence,
(2.70) holds as before. This implies (3.6). If a ∈ σ (A), then (2.59) and (2.60) hold
again by Lemma 2.6, implying (3.6) in this case as well. Since E(λ) ⊂ V0 , E(λ) ⊂ V ,
we have p < p0 , providing the necessary contradiction. This completes the proof. References
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Kanishka Perera: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6997, USA
Martin Schechter: Department of Mathematics, University of California Irvine,
Irvine, CA 92697-3875, USA