Electronic Journal of Differential Equations, Vol. 2006(2006), No. 131, pp. 1–15.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
A SEMILINEAR ELLIPTIC PROBLEM INVOLVING NONLINEAR
BOUNDARY CONDITION AND SIGN-CHANGING POTENTIAL
TSUNG-FANG WU
Abstract. In this paper, we study the multiplicity of nontrivial nonnegative
solutions for a semilinear elliptic equation involving nonlinear boundary condition and sign-changing potential. With the help of the Nehari manifold, we
prove that the semilinear elliptic equation:
−∆u + u = λf (x)|u|q−2 u
in Ω,
∂u
= g(x)|u|p−2 u on ∂Ω,
∂ν
has at least two nontrivial nonnegative solutions for λ is sufficiently small.
1. Introduction
In this paper, we consider the multiplicity of nontrivial nonnegative solutions for
the following semilinear elliptic equation
−∆u + u = λf (x)|u|q−2 u
∂u
= g(x)|u|p−2 u
∂ν
in Ω,
(1.1)
on ∂Ω,
−1)
N
where 1 < q < 2 < p < 2(N
with smooth
N −2 , λ > 0, Ω is a bounded domain in R
∂
boundary, ∂ν is the outer normal derivative and f, g : Ω → R are continuous
functions which change sign in Ω. Associated with (1.1), we consider the energy
functional Jλ in H 1 (Ω),
Z
Z
1
λ
1
Jλ (u) = kuk2H 1 −
f |u|q dx −
g|u|p ds.
2
q Ω
p ∂Ω
R
where ds is the measure on the boundary and kuk2H 1 = Ω |∇u|2 + u2 dx. It is well
known that Jλ is of C 1 in H 1 (Ω) and the solutions of equation (1.1) are the critical
points of the energy functional Jλ .
The fact that the number of solutions of equation (1.1) is affected by the nonlinear boundary conditions has been the focus of a great deal of research in recent
years. Garcia-Azorero, Peral and Rossi [10] have investigated (1.1) when f ≡ g ≡ 1.
2000 Mathematics Subject Classification. 35J65, 35J50, 35J55.
Key words and phrases. Semilinear elliptic equations; Nehari manifold;
Nonlinear boundary condition.
c
2006
Texas State University - San Marcos.
Submitted July 6, 2006. Published October 17, 2006.
Partially supported by the National Science Council of Taiwan (R.O.C.).
1
2
T.-F. WU
EJDE-2006/131
They found that there exist positive numbers Λ1 , Λ2 with Λ1 ≤ Λ2 such that equation (1.1) admits at least two positive solutions for λ ∈ (0, Λ1 ) and no positive solution exists for λ > Λ2 . Also see Chipot-Chlebik-Fila-Shafrir [4], Chipot-Shafrir-Fila
[5], Flores-del Pino [8], Hu [11], Pierrotti-Terracini [14] and Terraccini [16] where
problems similar to equation (1.1) have been studied.
The purpose of this paper is to consider the multiplicity of nontrivial nonnegative
solutions of equation (1.1) with sign-changing potential. We prove that equation
(1.1) has at least two nontrivial nonnegative solutions for λ is sufficiently small.
Theorem 1.1. There exists λ0 > 0 such that for λ ∈ (0, λ0 ), equation (1.1) has at
least two nontrivial nonnegative solutions.
Among the other interesting problems which are similar of equation (1.1), Ambrosetti-Brezis-Cerami [3] have investigated the equation
−∆u = λ|u|q−2 u + |u|p−2 u
in Ω,
(1.2)
u = 0 on ∂Ω,
where 1 < q < 2 < p ≤ N2N
−2 . They proved that there exists λ0 > 0 such that (1.2)
admits at least two positive solutions for λ ∈ (0, λ0 ), has a positive solution for
λ = λ0 , and no positive solution for λ > λ0 . Actually, Adimurthi-Pacella-Yadava
[1], Damascelli-Grossi-Pacella [6], Ouyang-Shi [13] and Tang [17] proved that there
exists λ0 > 0 such that equation (1.2) in the unit ball B N (0; 1) has exactly two
positive solutions for λ ∈ (0, λ0 ), has exactly one positive solution for λ = λ0 and
no positive solution exists for λ > λ0 . Generalizations of the result of equation
(1.2) were done by Ambrosetti-Azorero-Peral [2], de Figueiredo-Gossez-Ubilla [9]
and Wu [18].
This paper is organized as follows. In section 2, we give some notation and preliminaries. In section 3, we prove that (1.1) has at least two nontrivial nonnegative
solutions for λ is sufficiently small.
2. Notation and Preliminaries
Throughout this section, we denote by Sp , Cp the best Sobolev embedding and
trace constant for the operators H 1 (Ω) ,→ Lp (Ω), H 1 (Ω) ,→ Lp (∂Ω), respectively.
Now, we consider the Nehari minimization problem: For λ > 0,
αλ = inf{Jλ (u) : u ∈ Mλ },
where Mλ = {u ∈ H (Ω)\{0} : hJλ0 (u), ui = 0}. Define
Z
Z
ψλ (u) = hJλ0 (u), ui = kuk2H 1 − λ
f |u|q dx −
1
Ω
g|u|p ds.
∂Ω
Then for u ∈ Mλ ,
hψλ0 (u), ui = 2kuk2H 1 − λq
Z
Ω
f |u|q dx − p
Z
g|u|p ds.
∂Ω
Similarly to the method used in Tarantello [15], we split Mλ into three parts:
0
M+
λ = {u ∈ Mλ : hψλ (u), ui > 0},
M0λ = {u ∈ Mλ : hψλ0 (u), ui = 0},
0
M−
λ = {u ∈ Mλ : hψλ (u), ui < 0}.
Then, we have the following results.
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A SEMILINEAR ELLIPTIC PROBLEM
3
Lemma 2.1. There exists λ1 > 0 such that for each λ ∈ (0, λ1 ) we have M0λ = φ.
Proof. We consider theR following two cases.
Case (I): u ∈ Mλ and ∂Ω g|u|p ds ≤ 0. We have
Z
Z
λ
f |u|q dx = kuk2H 1 −
Ω
g|u|p ds.
∂Ω
Thus,
hψλ0 (u), ui
=
2kuk2H 1
Z
− λq
Z
q
f |u| dx − p
g|u|p ds
∂Ω
Z
+ (q − p)
g|u|p ds > 0
Ω
= (2 − q)kuk2H 1
∂Ω
and so u ∈ M+
λ.
R
Case (II): u ∈ Mλ and ∂Ω g|u|p ds > 0. Suppose that M0λ 6= φ for all λ > 0. If
u ∈ M0λ , then we have
Z
Z
0 = hψλ0 (u), ui = 2kuk2H 1 − λq
f |u|q dx − p
g|u|p ds
∂Ω
Z Ω
2
p
= (2 − q)kukH 1 − (p − q)
g|u| ds.
∂Ω
Thus,
kuk2H 1 =
p−q
2−q
Z
g|u|p ds
(2.1)
∂Ω
and
Z
λ
f |u|q dx = kuk2H 1 −
Ω
Z
g|u|p ds =
∂Ω
p−2
2−q
Z
g|u|p ds.
(2.2)
∂Ω
Moreover,
(
Z
p−2
)kuk2H 1 = kuk2H 1 −
g|u|p ds
p−q
∂Ω
Z
=λ
f |u|q dx
Ω
≤ λkf kLp∗ kukqLp
≤ λkf kLp∗ Spq kukqH 1 ,
where p∗ =
p
p−q .
This implies
h p−q
i1/(2−q)
kukH 1 ≤ λ(
)kf kLp∗ Spq
.
p−2
Let Iλ : Mλ → R be given by
Z
kuk2(p−1) 1/(p−2)
1
−λ
f |u|q dx,
Iλ (u) = K(p, q) R H p
g|u|
ds
Ω
∂Ω
(2.3)
4
T.-F. WU
EJDE-2006/131
2−q (p−1)/(p−2) p−2
where K(p, q) = ( p−q
)
( 2−q ). Then Iλ (u) = 0 for all u ∈ M0λ . Indeed,
from (2.1) and (2.2) it follows that for u ∈ M0λ we have
Z
kuk2(p−1) 1/(p−1)
H1
R
−λ
Iλ (u) = K(p, q)
f |u|q dx
g|u|p ds
Ω
∂Ω
p−1
R
p−q p−1
1
p−2
p
g|u|p ds
p − 2 ( 2−q )
2 − q p−1
∂Ω
(2.4)
R
(
)
=
p ds
p−q
2−q
g|u|
∂Ω
Z
p−2
p
−
g|u| ds = 0.
2 − q ∂Ω
However, by (2.3), the Hölder and Sobolev trace inequality, for u ∈ M0λ
kuk2(p−1) 1/(p−2)
1
Iλ (u) ≥ K(p, q) R H p
− λSpq kf kLp∗ kukqH 1
g|u|
ds
∂Ω
2(p−1)
1/(p−2)
kukH 1
q
≥ kukqH 1 K(p, q)
−
λS
kf
k
p∗
L
p
p+q(p−2)
Cpp kgk∞ kukH 1
h p − q
i 1−q
p
1−q
2−q
kf kLp∗ Spq
− λSpq kf kLp∗ .
≥ kukqH 1 K(p, q)Cp2−p λ 2−q
p−2
This implies that for λ sufficiently small we have Iλ (u) > 0 for all u ∈ M0λ , this
contradicts (2.4). Thus, we can conclude that there exists λ1 > 0 such that for
λ ∈ (0, λ1 ), we have M0λ = φ.
−
By Lemma 2.1, for λ ∈ (0, λ1 ) we write Mλ = M+
λ ∪ Mλ and define
αλ− (Ω) = inf − Jλ (u).
αλ+ = inf + Jλ (u);
u∈Mλ
u∈Mλ
The following lemma shows that the minimizers on Mλ are “usually” critical points
for Jλ .
Lemma 2.2. For λ ∈ (0, λ1 ). If u0 is a local minimizer for Jλ on Mλ , then
Jλ0 (u0 ) = 0 in H ∗ (Ω).
Proof. If u0 is a local minimizer for Jλ on Mλ , then u0 is a solution of the optimization problem
minimize Jλ (u)
subject to ψλ (u) = 0.
Hence, by the theory of Lagrange multipliers, there exists θ ∈ R such that
Jλ0 (u0 ) = θψλ0 (u0 )
in H ∗ (Ω).
Thus,
hJλ0 (u0 ), u0 iH 1 = θhψλ0 (u0 ), u0 iH 1 .
M+
λ
∪ M−
λ,
By Lemma 2.1, u0 ∈
This completes the proof.
we have
hψλ0 (u0 ), u0 iH 1
R
q
Lemma 2.3.
(i) If u ∈ M+
λ , then Ω f |u| dx > 0;
R
−
p
(ii) If u ∈ Mλ , then ∂Ω g|u| ds > 0.
(2.5)
6= 0 and so by (2.5) θ = 0.
EJDE-2006/131
A SEMILINEAR ELLIPTIC PROBLEM
Proof. (i) Case (I):
R
g|u|p ds ≤ 0. We have
Z
Z
q
2
λ
f |u| dx = kukH 1 −
∂Ω
g|u|p ds > 0.
∂Ω
Ω
Case (II):
g|u|p ds > 0. We have
Z
Z
2
q
kukH 1 − λ
f |u| dx −
R
∂Ω
5
Ω
and
kuk2H 1
g|u|p ds = 0
∂Ω
p−q
>
2−q
Z
g|u|p ds.
∂Ω
Thus,
Z
q
f |u| dx =
λ
kuk2H 1
Ω
p−2
−
g|u| ds >
2−q
∂Ω
Z
p
Z
g|u|p ds > 0.
∂Ω
(ii) Since
Z
(2 −
− (p − q)
g|u|p ds = hψλ0 (u), ui < 0.
∂Ω
R
It follows that ∂Ω g|u|p ds > 0. This completes the proof.
q)kuk2H 1
For each u ∈
M−
λ,
we write
1/(p−2)
(2 − q)kuk2
H1
R
< 1.
tmax =
p
(p − q) ∂Ω g|u| ds
Then we have the following lemma.
2−q
p(2−q)
p
2−q p−2
Cp 2−p Sp−q kf k−1
. Then for
Lemma 2.4. Let p∗ = p−q
and λ2 = ( p−2
p−q )( p−q )
Lp∗
−
each u ∈ Mλ and λ ∈ (0, λ2 ), we have
R
(i) if RΩ f |u|q dx ≤ 0, then Jλ (u) = supt≥0 Jλ (tu) > 0;
(ii) if Ω f |u|q dx > 0, then there is a unique 0 < t+ = t+ (u) < tmax such that
t+ u ∈ M+
λ and
Jλ (t+ u) =
Proof. Fix u ∈
M−
λ.
inf
0≤t≤tmax
Jλ (tu), Jλ (u) = sup Jλ (tu).
t≥tmax
Let
h(t) = t2−q kuk2H 1 − tp−q
Z
g|u|p ds
for t ≥ 0.
∂Ω
We have h(0) = 0, h(t) → −∞ as t → ∞, h(t) achieves its maximum at tmax ,
increasing for t ∈ [0, tmax ) and decreasing for t ∈ (tmax , ∞). Moreover,
h(tmax )
Z
2−q
(2 − q)kuk2
p−2
(2 − q)kuk2
p−q
p−2
1
1
2
H
H
R
R
=
kukH 1 −
g|u|p ds
(p − q) ∂Ω g|u|p ds
(p − q) ∂Ω g|u|p ds
∂Ω
2−q
h 2 − q 2−q
i kukp
p−2
2 − q p−q
H1
= kukqH 1 (
) p−2 − (
) p−2 R
p−q
p−q
g|u|p ds
∂Ω
p(2−q)
2−q
p − 2 2 − q p−2
≥ kukqH 1 (
)(
)
Cp 2−p
p−q p−q
or
h(tmax ) ≥ kukqH 1 (
p(2−q)
2−q
p − 2 2 − q p−2
)(
)
Cp 2−p .
p−q p−q
(2.6)
6
T.-F. WU
EJDE-2006/131
R
R
(i): Ω f |u|q dx ≤ 0. There is a unique t− > tmax such that h(t− ) = λ Ω f |u|q dx
and h0 (t− ) < 0. Now,
Z
|t− u|p ds
(2 − q)kt− uk2H 1 − (p − q)
∂Ω
Z
h
i
− 1+q
− 1−q
= (t )
(2 − q)(t ) kuk2H 1 − (p − q)(t− )p−q−1
g|u|p ds
∂Ω
= (t− )1+q h0 (t− ) < 0,
and
hJλ0 (t− u), t− ui
− 2
kuk2H 1
− q
Z
− p
q
Z
− (t ) λ
f |u| dx − (t )
Ω
Z
h
i
= (t− )q h(t− ) − λ
f |u|q dx = 0.
= (t )
g|u|p ds
∂Ω
Ω
−
Thus, t u ∈
M−
λ
−
or t = 1. Since for t > tmax , we have
Z
(2 − q)ktuk2H 1 − (p − q)
g|tu|p ds < 0,
∂Ω
d
Jλ (tu) = tkuk2H 1
dt
d2
J (tu) < 0,
2 λ
Z dt
Z
− λtq−1
f |u|q dx − tp−1
Ω
g|u|p ds = 0
for t = t− .
∂Ω
Thus, Jλ (u) = supt≥0 Jλ (tu). Moreover,
tp
t2
Jλ (u) ≥ Jλ (tu) ≥ kuk2H 1 −
2
p
2
Z
p
By routine computations, g(t) = t2 kuk2H 1 − tp
R
t0 = (kuk2H 1 / ∂Ω g|u|p ds)1/(p−2) . Thus,
Jλ (u) ≥
(ii):
R
Ω
g|u|p ds
for all t ≥ 0.
∂Ω
R
∂Ω
g|u|p ds achieves its maximum at
p
2
p − 2 kukH 1 p−2
R
> 0.
2p
g|u|p ds
∂Ω
f |u|q dx > 0. By (2.6) and
Z
h(0) = 0 < λ
f |u|q dx ≤ λkf kLp∗ Spq kukqH 1
Ω
p(2−q)
2−q
− 2 2 − q p−2
)(
)
Cp 2−p
−q p−q
≤ h(tmax ) for λ ∈ (0, λ2 ),
<
p
kukqH 1 (
p
there are unique t+ and t− such that 0 < t+ < tmax < t− ,
Z
+
h(t ) = λ
f |u|q dx = h(t− ),
Ω
h0 (t+ ) > 0 > h0 (t− ).
EJDE-2006/131
A SEMILINEAR ELLIPTIC PROBLEM
7
−
−
−
+
We have t+ u ∈ M+
λ , t u ∈ Mλ , and Jλ (t u) ≥ Jλ (tu) ≥ Jλ (t u) for each
+ −
+
+
−
t ∈ [t , t ] and Jλ (t u) ≤ Jλ (tu) for each t ∈ [0, t ]. Thus, t = 1 and
Jλ (u) = sup Jλ (tu), Jλ (t+ u) =
t≥0
inf
0≤t≤tmax
Jλ (tu).
This completes the proof.
Next, we establish the existence of nontrivial nonnegative solutions for the equation
−∆u + u = λf (x)|u|q−2 u in Ω,
(2.7)
u = 0 on ∂Ω,
Associated with equation (2.7), we consider the energy functional
Z
1
λ
2
f |u|q dx
Kλ (u) = kukH 1 −
2
q Ω
and the minimization problem
βλ = inf{Kλ (u) : u ∈ Nλ },
where Nλ = {u ∈ H01 (Ω)\{0} : hKλ0 (u), ui = 0}. Then we have the following result.
Theorem 2.5. Suppose that λ > 0. Then equation (2.7) has a nontrivial nonnegative solution vλ with Kλ (vλ ) = βλ < 0.
Proof. First, we need to show that Kλ is bounded below on Nλ and βλ < 0. Then
for u ∈ Nλ ,
Z
−q
kuk2H 1 = λ
f |u|q dx ≤ λkf kLq∗ Sp 2 kukqH 1 .
Ω
where p∗ =
p
p−q .
This implies
−q
1
kukH 1 ≤ (λkf kLp∗ Sp 2 ) 2−q .
(2.8)
Hence,
Z
1
λ
kukH 1 −
f |u|q dx
2
q Ω
1 1
− kuk2H 1
=
2 q
1 1
−q 1
≤
−
λkf kLp∗ Sp 2 2−q
2 q
Kλ (u) =
for all u ∈ Nλ and βλ < 0. Let {vn } be a minimizing sequence for Kλ on Nλ . Then
by (2.8) and the compact imbedding theorem, there exist a subsequence {vn } and
vλ in H01 (Ω) such that
vn * vλ weakly in H01 (Ω)
and
vn → vλ
q
R
strongly in Lq (Ω).
f |vλ | dx > 0. If not,
Z
λ
1
1
2
f |vλ |q dx + o(1) ≥ kvλ k2H 1 + o(1),
Kλ (vn ) ≥ kvλ kH 1 −
2
q Ω
2
First, we claim that
Ω
(2.9)
8
T.-F. WU
EJDE-2006/131
R
this contradicts Kλ (vn ) → βλ (Ω) < 0 as n → ∞. Thus, Ω f |vλ |q dx > 0. In
particular, vλ 6≡ 0. Now, we prove that vn → vλ strongly in H01 (Ω). Suppose
otherwise, then kvλ kH 1 < lim inf n→∞ kvn kH 1 and so
Z
Z
2
q
2
kvλ kH 1 − λ
f |vλ | dx < lim inf kvn kH 1 − λ
f |vn |q dx = 0.
n→∞
Ω
Since
Ω
q
R
Ω
f |vλ | dx > 0, there is a unique t0 6= 1 such that t0 vλ ∈ Nλ . Thus,
t0 vn * t0 vλ
weakly in H01 (Ω).
Moreover,
Kλ (t0 vλ ) < Kλ (vλ ) < lim Kλ (vn ) = βλ ,
n→∞
which is a contradiction. Hence vn → vλ strongly in H01 (Ω). This implies vλ ∈ Nλ
and
Kλ (vn ) → Kλ (vλ ) = βλ
as n → ∞.
Since Kλ (vλ ) = Kλ (kvλ k) and kvλ k ∈ Nλ , without loss of generality, we may
assume that vλ is a nontrivial nonnegative solution of equation (2.7).
Then we have the following results.
Lemma 2.6.
(i) αλ ≤ αλ+ ≤ βλ < 0;
(ii) Jλ is coercive and bounded below on Mλ for all λ ∈ (0, p−2
p−q ].
Proof. (i) Let vλ be a positive solution
of equation (2.7) such that K(vλ ) = βλ .
R
Since vλ ∈ C 2 (Ω). Then we have ∂Ω g|vλ |p ds = 0 and vλ ∈ M+
λ . This implies
Z
λ
1
f |vλ |q dx = βλ < 0
Jλ (vλ ) = kvλ k2H 1 −
2
q Ω
and so αλ ≤ αλ+ ≤ βλ < 0.
R
R
(ii) For u ∈ Mλ , we have kuk2H 1 = λ Ω f |u|q dx + ∂Ω g|u|p ds. Then by the Hölder
and Young inequalities,
Z
p−2
p − q
Jλ (u) =
kuk2H 1 − λ
f |u|q dx
2p
pq
Ω
p−2
p − q
2
≥
kukH 1 − λ
kf kLp∗ Spq kukqH 1
2p
pq
2
p − 2
p − q (p − q)(2 − q) ≥
−λ
kuk2H 1 − λ
kf kLp∗ Spq 2−q
2p
2p
2pq
2
1
(p − q)(2 − q) (p − 2) − λ(p − q) kuk2H 1 − λ
kf kLp∗ Spq 2−q .
=
2p
2pq
Thus, Jλ is coercive on Mλ and
Jλ (u) ≥ −λ
for all λ ∈ (0, p−2
p−q ].
2
(p − q)(2 − q) kf kLp∗ Spq 2−q
2pq
EJDE-2006/131
A SEMILINEAR ELLIPTIC PROBLEM
9
3. Proof of Theorem 1.1
First, we will use the idea of Ni-Takagi [12] to get the following results.
Lemma 3.1. For each u ∈ Mλ , there exist > 0 and a differentiable function
ξ : B(0; ) ⊂ H 1 (Ω) → R+ such that ξ(0) = 1, the function ξ(v)(u − v) ∈ Mλ and
R
R
R
2 Ω ∇u∇vdx − λq Ω f |u|q−2 uvdx − p ∂Ω g|u|p−2 uvds
R
hξ 0 (0), vi =
(3.1)
(2 − q)kuk2H 1 − (p − q) ∂Ω g|u|p ds
for all v ∈ H 1 (Ω).
Proof. For u ∈ Mλ , define a function F : R × H 1 (Ω) → R by
Fu (ξ, w) = hJλ0 (ξ(u − w)), ξ(u − w)i
Z
Z
= ξ2
|∇(u − w)|2 + (u − w)2 dx − ξ q λ
f |u − w|q dx
Ω
Ω
Z
p
p
−ξ
g|u − w| ds.
∂Ω
Then Fu (1, 0) =
hJλ0 (u), ui
= 0 and
d
Fu (1, 0) = 2kuk2H 1 − λq
dξ
= (2 − q)kuk2H 1
Z
Z
q
f |u| dx − p
g|u|p ds
∂Ω
Z
− (p − q)
g|u|p ds 6= 0.
∂Ω
∂Ω
According to the implicit function theorem, there exist > 0 and a differentiable
function ξ : B(0; ) ⊂ H 1 (Ω) → R such that ξ(0) = 1,
R
R
R
2 Ω ∇u∇vdx − λq Ω f |u|q−2 uvdx − p ∂Ω g|u|p−2 uvds
R
hξ 0 (0), vi =
(2 − q)kuk2H 1 − (p − q) ∂Ω g|u|p ds
and
Fu (ξ(v), v) = 0
for all v ∈ B(0; )
which is equivalent to
hJλ0 (ξ(v)(u − v)), ξ(v)(u − v)i = 0
for all v ∈ B(0; ),
that is ξ(v)(u − v) ∈ Mλ .
Lemma 3.2. For each u ∈ M−
λ , there exist > 0 and a differentiable function
ξ − : B(0; ) ⊂ H 1 (Ω) → R+ such that ξ − (0) = 1, the function ξ − (v)(u − v) ∈ M−
λ
and
R
R
R
2 Ω ∇u∇vdx − λq Ω f |u|q−2 uvdx − p ∂Ω g|u|p−2 uvds
− 0
R
h(ξ ) (0), vi =
(3.2)
(2 − q)kuk2H 1 − (p − q) ∂Ω g|u|p ds
for all v ∈ H 1 (Ω).
Proof. Similar to the argument in Lemma 3.1, there exist > 0 and a differentiable
function ξ − : B(0; ) ⊂ H 1 (Ω) → R such that ξ − (0) = 1 and ξ − (v)(u − v) ∈ Mλ
for all v ∈ B(0; ). Since
Z
0
2
hψλ (u), ui = (2 − q)kukH 1 − (p − q)
g|u|p ds < 0.
∂Ω
10
T.-F. WU
EJDE-2006/131
Thus, by the continuity of the function ξ − , we have
hψλ0 (ξ − (v)(u − v)), ξ − (v)(u − v)i
−
= (2 − q)kξ (v)(u −
v)k2H 1
Z
g|ξ − (v)(u − v)|p ds < 0
− (p − q)
∂Ω
if sufficiently small, this implies that ξ − (v)(u − v) ∈ M−
λ.
Proposition 3.3. Let λ0 = min{λ1 , λ2 , p−1
p−q }, Then for λ ∈ (0, λ0 ):
(i) There exists a minimizing sequence {un } ⊂ Mλ such that
Jλ (un ) = αλ + o(1),
Jλ0 (un ) = o(1)
in H ∗ (Ω);
(ii) there exists a minimizing sequence {un } ⊂ M−
λ such that
Jλ (un ) = αλ− + o(1),
Jλ0 (un ) = o(1)
in H ∗ (Ω).
Proof. (i) By Lemma 2.6 (ii) and the Ekeland variational principle [7], there exists
a minimizing sequence {un } ⊂ Mλ such that
Jλ (un ) < αλ +
1
,
n
1
kw − un kH 1 for each w ∈ Mλ .
n
By taking n large, from Lemma 2.6 (i), we have
Z
1 1
1 1
2
Jλ (un ) = ( − )kun kH 1 − ( − )λ
f |un |q dx
2 p
q
p
Ω
1
βλ
< αλ + <
.
n
2
This implies
Z
−pq
q
q
∗
kf kLp Sp kun kH 1 ≥
βλ > 0.
f |un |q dx >
2λ(p
− q)
Ω
Jλ (un ) < Jλ (w) +
(3.3)
(3.4)
(3.5)
(3.6)
Consequently, un 6= 0 and putting together (3.5), (3.6) and the Hölder inequality,
we obtain
h −pq
i1/q
kun kH 1 >
βλ Sp−q kf k−1
(3.7)
∗
p
L
2λ(p − q)
h 2(p − q)
i1/(2−q)
kun kH 1 <
kf kLp∗ Spq
(3.8)
(p − 2)q
Now, we show that
kJλ0 (un )kH −1 → 0
as n → ∞.
Applying Lemma 3.1 with un to obtain the functions ξn : B(0; n ) → R+ for some
n > 0, such that ξn (w)(un − w) ∈ Mλ . Choose 0 < ρ < n . Let u ∈ H 1 (Ω) with
ρu
u 6≡ 0 and let wρ = kuk
. We set ηρ = ξn (wρ )(un − wρ ). Since ηρ ∈ Mλ , we
H1
deduce from (3.4) that
1
Jλ (ηρ ) − Jλ (un ) ≥ − kηρ − un kH 1
n
EJDE-2006/131
A SEMILINEAR ELLIPTIC PROBLEM
11
and by the mean value theorem, we have
1
hJλ0 (un ), ηρ − un i + o(kηρ − un kH 1 ) ≥ − kηρ − un kH 1 .
n
Thus,
hJλ0 (un ), −wρ i + (ξn (wρ ) − 1)hJλ0 (un ), (un − wρ )i
1
≥ − kηρ − un kH 1 + o(kηρ − un kH 1 ).
n
Since ξn (wρ )(un − wρ ) ∈ Mλ and (3.9) it follows that
u
− ρhJλ0 (un ),
i + (ξn (wρ ) − 1)hJλ0 (un ) − Jλ0 (ηρ ), (un − wρ )i
kukH 1
1
≥ − kηρ − un kH 1 + o(kηρ − un kH 1 ).
n
Thus,
hJλ0 (un ),
u
kηρ − un kH 1
o(kηρ − un kH 1 )
i≤
+
kukH 1
nρ
ρ
(ξn (wρ ) − 1) 0
+
hJλ (un ) − Jλ0 (ηρ ), (un − wρ )i.
ρ
(3.9)
(3.10)
Since kηρ − un kH 1 ≤ ρkξn (wρ )k + kξn (wρ ) − 1kkun kH 1 and
lim
ρ→0
kξn (wρ ) − 1k
≤ kξn0 (0)k,
ρ
if we let ρ → 0 in (3.10) for a fixed n, then by (3.8) we can find a constant C > 0,
independent of ρ, such that
hJλ0 (un ),
u
C
i ≤ (1 + kξn0 (0)k).
kukH 1
n
The proof will be complete once we show that kξn0 (0)k is uniformly bounded in n.
By (3.1), (3.8) and the Hölder inequality, we have
hξn0 (0), vi ≤
bkvkH 1
R
|(2 − q)kun kH 1 − (p − q) ∂Ω g|un |p ds|
for some b > 0.
We only need to show that
Z
|(2 − q)kun kH 1 − (p − q)
g|un |p ds| > c
(3.11)
∂Ω
for some c > 0 and n large enough. We argue by contradiction. Assume that there
exists a subsequence {un }, we have
Z
(2 − q)kun kH 1 − (p − q)
g|un |p ds = o(1).
(3.12)
∂Ω
Combining (3.12) with (3.7), we can find a suitable constant d > 0 such that
Z
g|un |p ds ≥ d for n sufficiently large.
(3.13)
∂Ω
In addition (3.12), and the fact that un ∈ Mλ also give
Z
Z
Z
p−2
q
2
p
λ
f |un | dx = kun kH 1 −
g|un | ds =
g|un |p ds + o(1)
2 − q ∂Ω
Ω
∂Ω
12
T.-F. WU
EJDE-2006/131
and
1
h p−q
i 2−q
kun kH 1 ≤ λ(
+ o(1).
)kf kLp∗ Spq
p−2
(3.14)
This implies
Z
ku k2(p−1) 1/(p−2)
n H1
Iλ (un ) = K(p, q) R
−
λ
f |un |q dx = o(1).
p ds
g|u
|
n
Ω
∂Ω
(3.15)
However, by (3.13), (3.14) and λ ∈ (0, λ0 ),
ku k2(p−1) 1/(p−2)
n H1
Iλ (un ) ≥ K(p, q) R
− λSpq kf kLp∗ kun kqH 1
p ds
g|u
|
n
∂Ω
ku k2(p−1) 1/(p−2)
n H1
q
q
−
λS
kf
k
≥ kun kH 1 K(p, q)
p∗
L
p
p+q(p−2)
Cpp kun kH 1
n
o
p
1−q
1−q p − q
)kf kLp∗ Spq 2−q − λkf kLp∗ .
≥ kun kqH 1 K(p, q)Cp2−p λ 2−q (
p−2
this contradicts (3.15). We get
hJλ0 (un ),
u
C
i≤ .
kukH 1
n
This completes the proof of (i).
(ii) Similarly, by using Lemma 3.2, we can prove (ii). We will omit detailed proof
here.
Now, we establish the existence of a local minimum for Jλ on M+
λ.
Theorem 3.4. Let λ0 > 0 as in Proposition 3.3, then for λ ∈ (0, λ0 ) the functional
+
Jλ has a minimizer u+
0 in Mλ and it satisfies
+
(i) Jλ (u+
0 ) = αλ = αλ ;
+
(ii) u0 is a nontrivial nonnegative solution of equation (1.1);
(iii) Jλ (u+
0 ) → 0 as λ → 0.
Proof. Let {un } ⊂ Mλ be a minimizing sequence for Jλ on Mλ such that
and Jλ0 (un ) = o(1)
Jλ (un ) = αλ + o(1)
in H ∗ (Ω).
Then by Lemma 2.6 and the compact imbedding theorem, there exist a subsequence
1
{un } and u+
0 ∈ H (Ω) such that
u n * u+
0
weakly in H 1 (Ω), un → u+
0
strongly in Lp (∂Ω)
and
un → u+
0
First, we claim that
conclude that
Z
R
strongly in Lq (Ω).
q
f (x)ku+
0 k dx
Ω
f |un |q dx →
Ω
Z
6= 0. Suppose otherwise, by (3.16) we can
q
f |u+
0 | dx = 0
as n → ∞
Ω
and so
kun k2H 1
(3.16)
Z
=
∂Ω
g|un |p ds + o(1).
EJDE-2006/131
A SEMILINEAR ELLIPTIC PROBLEM
13
Thus,
Z
Z
1
λ
1
2
q
Jλ (un ) = kun kH 1 −
f |un | dx −
g|un |p ds
2
q Ω
p ∂Ω
Z
1 1
g|un |p ds + o(1)
=( − )
2 p ∂Ω
Z
1 1
p
=( − )
g|u+
0 | ds as n → ∞,
2 p ∂Ω
this contradicts Jλ (un ) → αλ < 0 as n → ∞. Moreover,
o(1) = hJλ0 (un ), φi = hJλ0 (u0 ), φi + o(1)
for all φ ∈ H 1 (Ω).
+
Thus, u+
0 ∈ Mλ is a nonzero solution of equation (1.1) and Jλ (u0 ) ≥ αλ . Now
+
we prove that un → u0 strongly in H 1 (Ω). Suppose otherwise, then ku+
0 kH 1 <
lim inf n→∞ kun kH 1 and so
Z
Z
+ q
2
p
ku+
k
−
λ
f
|u
|
dx
−
g|u+
1
0 H
0
0 | ds
Ω
∂Ω
Z
Z
2
q
< lim inf kun kH 1 − λ
f |un | dx −
g|un |p ds = 0,
n→∞
this contradicts
u+
0
Ω
∈ Mλ . Hence un →
∂Ω
u+
0
strongly in H 1 (Ω) and
Jλ (un ) → Jλ (u+
0 ) = αλ
as n → ∞.
+
+
−
Moreover, we have u+
0 ∈ Mλ . If not, then u0 ∈ Mλ and by Lemma 2.4, there are
+
−
+ +
+
− +
unique t0 and t0 such that t0 u0 ∈ Mλ and t0 u0 ∈ M−
λ . In particular, we have
−
t+
<
t
=
1.
Since
0
0
d
+
Jλ (t+
0 u0 ) = 0
dt
and
d2
+
Jλ (t+
0 u0 ) > 0,
dt2
−
+ +
+
there exists t+
0 < t̄ ≤ t0 such that Jλ (t0 u0 ) < Jλ (t̄u0 ). By Lemma 2.4,
+
+
− +
+
Jλ (t+
0 u0 ) < Jλ (t̄u0 ) ≤ Jλ (t0 u0 ) = Jλ (u0 ),
+
+
+
which is a contradiction. Since Jλ (u+
0 ) = Jλ (|u0 |) and |u0 | ∈ Mλ , by Lemma
+
2.2 we may assume that u0 is a nontrivial nonnegative solution of equation (1.1).
From Lemma 2.6 it follows that
(p − q)(2 − q) 2
(kf kLp∗ Spq ) 2−q
0 > Jλ (u+
)
≥
−λ
0
2pq
and so Jλ (u+
0 ) → 0 as λ → 0.
Next, we establish the existence of a local minimum for Jλ on M−
λ.
Theorem 3.5. Let λ0 > 0 as in Proposition 3.3. Then for λ ∈ (0, λ0 ) the functional
−
Jλ has a minimizer u−
0 in Mλ and satisfies
−
(i) Jλ (u−
0 ) = αλ ;
−
(ii) u0 is a nontrivial nonnegative solution of equation (1.1).
Proof. By Proposition 3.3 (ii), there exists a minimizing sequence {un } for Jλ on
M−
λ such that
Jλ (un ) = αλ− + o(1)
and Jλ0 (un ) = o(1)
in H ∗ (Ω).
14
T.-F. WU
EJDE-2006/131
By Lemma 2.6 and the compact imbedding theorem, there exist a subsequence {un }
1
and u−
0 ∈ H (Ω) such that
u n * u−
0
un → u−
0
weakly in H 1 (Ω),
strongly in Lp (∂Ω),
un → u−
strongly in Lq (Ω).
0
R
p
Since (2 − q)kun k2H 1 − (p − q)
R ∂Ω g|upn | ds < 0, by the Sobolev trace inequality
there exists C > 0 such that ∂Ω g|un | ds > C. Moreover,
o(1) = hJλ0 (un ), φi = hJλ0 (u0 ), φi + o(1)
for all φ ∈ H 1 (Ω)
and
(2 − q)ku0 k2H 1 − (p − q)
Z
g|u0 |p ds
Z
− (p − q)
∂Ω
≤ lim inf (2 − q)kun k2H 1
n→∞
u−
0
g|un |p ds ≤ 0.
∂Ω
M−
λ
1
Thus,
∈
is a nonzero solution of equation (1.1). Now we prove that un → u−
0
strongly in H (Ω). Suppose otherwise, then ku−
0 kH 1 < lim inf n→∞ kun kH 1 and so
Z
Z
− q
2
p
ku−
k
−
λ
f
|u
|
dx
−
g|u−
1
0 H
0
0 | ds
Ω
∂Ω
Z
Z
2
< lim inf kun kH 1 − λ
f |un |q dx −
g|un |p ds = 0,
n→∞
this contradicts
u−
0
∈
Ω
M−
λ.
Hence un →
∂Ω
u−
0
strongly in H 1 (Ω). This implies
−
Jλ (un ) → Jλ (u−
0 ) = αλ
as n → ∞.
−
−
−
−
Since Jλ (u−
0 ) = Jλ (|u0 |) and |u0 | ∈ Mλ , by Lemma 2.2 we may assume that u0
is a nontrivial nonnegative solution of equation (1.1).
Now, we complete the proof of Theorem 1.1. By Theorems 3.4, 3.5, we obtain
−
+
equation (1.1) has two nontrivial nonnegative solutions u+
0 and u0 such that u0 ∈
+
−
−
+
−
+
−
Mλ and u0 ∈ Mλ . Since Mλ ∩Mλ = φ, this implies that u0 and u0 are different.
References
[1] Adimurthi, F. Pacella, and L. Yadava; On the number of positive solutions of some semilinear
Dirichlet problems in a ball, Diff. Int. Equations 10 (6) (1997) 1157–1170.
[2] A. Ambrosetti, J. Garcia-Azorero and I. Peral; Multiplicity results for some nonlinear elliptic
equations, J. Funct. Anal. 137 (1996) 219–242.
[3] A. Ambrosetti, H. Brezis and G. Cerami; Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519–543.
[4] M. Chipot, M. Chlebik, M. Fila and I. Shafrir; Existence of positive solutions of a semilinear
elliptic equation in RN
+ with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998)
429–471.
[5] M. Chipot, I. Shafrir and M. Fila; On the solutions to some elliptic equations with nonlinear
boundary conditions, Adv. Diff. Eqns 1 (1) (1996) 91–100.
[6] L. Damascelli, M. Grossi and F. Pacella; Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Annls Inst. H.
Poincaré Analyse Non linéaire 16 (1999) 631–652.
[7] I. Ekeland; On the variational principle, J. Math. Anal. Appl. 17 (1974) 324–353.
[8] C. Flores, M. del Pino; Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains, Comm. Partial Diff. Eqns. 26 (11–12) (2001) 2189–2210.
EJDE-2006/131
A SEMILINEAR ELLIPTIC PROBLEM
15
[9] D. G. de Figueiredo, J. P. Gossez and P. Ubilla; Local superlinearity and sublinearity for
indefinite semilinear elliptic problems, J. Funct. Anal. 199 (2003) 452–467.
[10] J. Garcia-Azorero, I. Peral and J. D. Rossi; A convex-concave problem with a nonlinear
boundary condition, J. Diff. Eqns. 198 (2004) 91–128.
[11] B. Hu; Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary
condition, Diff. Integral Eqns. 7 (2) (1994), 301–313.
[12] W. M. Ni and I. Takagi; On the shape of least energy solution to a Neumann problem, Comm.
Pure Appl. Math. 44 (1991) 819–851.
[13] T. Ouyang and J. Shi; Exact multiplicity of positive solutions for a class of semilinear problem
II, J. Diff. Eqns. 158 (1999) 94–151.
[14] D. Pierrotti and S. Terracini; On a Neumann problem with critical exponent and critical
nonlinearity on the boundary, Comm. Partial Diff. Eqns. 20 (7–8) (1995) 1155–1187.
[15] G. Tarantello; On nonhomogeneous elliptic involving critical Sobolev exponent, Ann. Inst. H.
Poincaré Anal. Non Linéaire 9 (1992) 281–304.
[16] S. Terraccini; Symmetry properties of positive solutions to some elliptic equations with nonlinear boundary conditions, Diff. Integral Eqns. 8 (1995) 1911–1922.
[17] M. Tang; Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and
convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 705–717.
[18] T. F. Wu; On semilinear elliptic equations involving concave-convex nonlinearities and signchanging weight function, J. Math. Anal. Appl. 318 (2006) 253–270.
Tsung-Fang Wu
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811,
Taiwan
E-mail address: tfwu@nuk.edu.tw