Hindawi Publishing Corporation
Boundary Value Problems
Volume 2010, Article ID 856932, 18 pages
doi:10.1155/2010/856932
Research Article
Multiple Positive Solutions for a Class of
Concave-Convex Semilinear Elliptic Equations in
Unbounded Domains with Sign-Changing Weights
Tsing-San Hsu
Center for General Education, Chang Gung University, Kwei-Shan, Tao-Yuan 333, Taiwan
Correspondence should be addressed to Tsing-San Hsu, tshsu@mail.cgu.edu.tw
Received 8 September 2010; Accepted 18 October 2010
Academic Editor: Julio Rossi
Copyright q 2010 Tsing-San Hsu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We study the existence and multiplicity of positive solutions for the following Dirichlet equations:
−Δu u λax|u|q−2 u bx|u|p−2 u in Ω, u 0 on ∂Ω, where λ > 0, 1 < q < 2 < p < 2∗ 2∗ 2N/N − 2 if N ≥ 3; 2∗ ∞ if N 1, 2, Ω is a smooth unbounded domain in ÊN , ax, bx
satisfy suitable conditions, and ax maybe change sign in Ω.
1. Introduction and Main Results
In this paper, we deal with the existence and multiplicity of positive solutions for the
following semilinear elliptic equation:
−Δu u λax|u|q−2 u bx|u|p−2 u
u>0
u0
in Ω,
in Ω,
Eλa,b on ∂Ω,
where λ > 0, 1 < q < 2 < p < 2∗ 2∗ 2N/N − 2 if N ≥ 3, 2∗ ∞ if N 1, 2, Ω ⊂ ÊN is an
unbounded domain, and a, b are measurable functions and satisfy the following conditions:
∗
A1 a ∈ CΩ ∩ Lq Ω q∗ p/p − q with a max{a, 0} /
≡ 0 in Ω.
B1 b ∈ CΩ ∩ L∞ Ω and b max{b, 0} /
≡ 0 in Ω.
2
Boundary Value Problems
Semilinear elliptic equations with concave-convex nonlinearities in bounded domains
are widely studied. For example, Ambrosetti et al. 1 considered the following equation:
−Δu λuq−1 up−1
u>0
u0
in Ω,
Eλ in Ω,
on ∂Ω,
where λ > 0, 1 < q < 2 < p < 2∗ . They proved that there exists λ0 > 0 such that Eλ admits at
least two positive solutions for all λ ∈ 0, λ0 and has one positive solution for λ λ0 and no
positive solution for λ > λ0 . Actually, Adimurthi et al. 2, Damascelli et al. 3, Ouyang and
Shi 4, and Tang 5 proved that there exists λ0 > 0 such that Eλ in the unit ball BN 0; 1
has exactly two positive solutions for λ ∈ 0, λ0 and has exactly one positive solution for
λ λ0 and no positive solution exists for λ > λ0 . For more general results of Eλ involving
sign-changing weights in bounded domains, see Ambrosetti et al. 6, Garcia Azorero et al.
7, Brown and Wu 8, Brown and Zhang 9, Cao and Zhong 10, de Figueiredo et al. 11,
and their references. However, little has been done for this type of problem in unbounded
domains. For Ω ÊN , we are only aware of the works 12–15 which studied the existence
of solutions for some related concave-convex elliptic problems not involving sign-changing
weights.
Wu in 16 has studied the multiplicity of positive solutions for the following equation
involving sign-changing weights:
−Δu u fλ xuq−1 gμ xup−1
u>0
u ∈ H1
in
ÊN ,
ÊN
in
ÊN ,
Efλ ,gμ ,
where 1 < q < 2 < p < 2∗ , the parameters λ, μ ≥ 0. He also assumed that fλ x λf xf− x
is sign-changing and gμ x ax μbx, where a and b satisfy suitable conditions, and
proved Efλ ,gμ has at least four positive solutions.
When Ω Ω × Ê Ω ⊂ ÊN−1 , N ≥ 2 is an infinite strip domain, Wu in 17 considered
≤ a ∈ L2/2−q Ω, 0 ≤ b ∈
Eλa,b not involving sign-changing weights assuming that 0 /
CΩ satisfies lim|xN √
| → ∞ bx , xN 1 in Ω and there exist δ > 0 and 0 < C0 < 1 such that
bx , xN ≥ 1 − C0 e−2 1θ1 δ|xN | for all x x , xN ∈ Ω, where θ1 is the first eigenvalue of the
Dirichlet problem −Δ in Ω . The author proved that there exists a positive constant Λ0 such
that for λ ∈ 0, Λ0 , Eλa,b possesses at least two positive solutions.
Miotto and Miyagaki in 18 have studied Eλa,b in Ω Ω × Ê, under the assumption
≡ 0 and a− is bounded and has a compact support
that a ∈ Lγ /γ −q Ω q < γ ≤ 2∗ with a /
∞
in Ω and 0 ≤ b ∈ L √Ω satisfies lim|xN | → ∞ bx , xN 1 and there exists C0 > 0 such that
bx , xN ≥ 1 − C0 e−2 1θ1 |xN | for all x x , xN ∈ Ω, where θ1 is the first eigenvalue of the
Dirichlet problem −Δ in Ω . It was obtained there existence of Λ0 > 0 such that for λ ∈ 0, Λ0 ,
Eλa,b possesses at least two positive solutions.
In a recent work 19, Hsu and Lin have studied Eλa,b in ÊN under the assumptions
A1-A2, B1, and Ωb . They proved that there exists a constant Λ0 > 0 such that for
Boundary Value Problems
3
λ ∈ 0, q/2Λ0 , Eλa,b possesses at least two positive solutions. The main aim of this paper
is to study Eλa,b on the general unbounded domains see the condition Ωb and extend
the results of 19 to more general unbounded domains. We will apply arguments similar
to those used in 20 and prove the existence and multiplicity of positive solutions by using
Ekeland’s variational principle 21.
Set
Λ0 2−q
p − q b L∞
2−q/p−2 p−2
p − q a Lq∗
Sp Ωp2−q/2p−2q/2 > 0,
1.1
∗
∗
where b L∞ supx∈Ω b x, a Lq∗ Ω |a |q dx1/q , and Sp Ω is the best Sobolev
constant for the imbedding of H01 Ω into Lp Ω. Now, we state the first main result about
the existence of positive solution of Eλa,b .
Theorem 1.1. Assume that (A1) and (B1) hold. If λ ∈ 0, Λ0 , then Eλa,b admits at least one
positive solution.
Associated with Eλa,b , we consider the energy functional Jλa,b in H01 Ω:
Jλa,b u λ
1
u2H 1 −
2
q
Ω
ax|u|q dx −
1
p
Ω
bx|u|p dx,
1.2
1/2
where uH 1 Ω ∇u|2 u2 dx . By Rabinowitz 22, Proposition B.10, Jλa,b ∈
C1 H01 Ω, Ê. It is well known that the solutions of Eλa,b are the critical points of the energy
functional Jλa,b in H01 Ω.
Under the assumptions A1, B1, and λ > 0, Eλa,b can be regarded as a perturbation
problem of the following semilinear elliptic equation:
−Δu u bxup−1
u>0
u0
in Ω,
Eb in Ω,
on ∂Ω,
where bx ∈ CΩ ∩ L∞ Ω and bx > 0 for all x ∈ Ω. We denote by Sbp Ω the best constant
which is given by
Sbp Ω
inf
u∈H01 Ω\{0}
Ω
u2H 1
bx|u|p dx
2/p .
1.3
4
Boundary Value Problems
A typical approach for solving problem of this kind is to use the following Minimax method:
αbΓ Ω inf max J0b γ t ,
γ ∈ΓΩ t∈0,1
1.4
ΓΩ γ ∈ C 0, 1, H01 Ω : γ 0 0, γ 1 e ,
1.5
where
J0b e 0 and e / 0. By the Mountain Pass Lemma due to Ambrosetti and Rabinowitz 23,
we called the nonzero critical point u ∈ H01 Ω of J0b a ground state solution of Eb in Ω if
J0b u αbΓ Ω. We remark that the ground state solutions of Eb in Ω can also be obtained
by the Nehari minimization problem
αb0 Ω where Mb0 Ω {u ∈ H01 Ω \ {0} : u2H 1 nonzero solution of Eb in Ω,
αbΓ Ω αb0 Ω inf J0b v,
v∈Mb0 Ω
Ω
1.6
bx|u|p dx}. Note that Mb0 Ω contains every
p−2 b
S Ωp/p−2 > 0
2p p
1.7
see Willem 24, and if bx ≡ b∞ > 0 is a constant, then J0b and αb0 Ω replace J0 and α∞
0 Ω,
respectively.
The existence of ground state solutions of Eb is affected by the shape of the domain
Ω and bx that satisfies some suitable conditions and has been the focus of a great deal of
research in recent years. By the Rellich compactness theorem and the Minimax method, it is
easy to obtain a ground state solution for Eb in bounded domains. When Ω is an unbounded
domain and bx ≡ b∞ , the existence of ground state solutions has been established by several
authors under various conditions. We mention, in particular, results by Berestycki and Lions
25, Lien et al. 26, Chen and Wang 27, and Del Pino and Felmer 28, 29. In 25, Ω ÊN .
Actually, Kwong 30 proved that the positive solution of Eb in ÊN is unique. In 26, for Ω
is a periodic domain. In 26, 27, the domain Ω is required to satisfy
Ω1 Ω Ω1 ∪ Ω2 , where Ω1 , Ω2 are domains in ÊN and Ω1 ∩ Ω2 is bounded;
∞
∞
Ω2 α∞
0 Ω < min{α0 Ω1 , α0 Ω2 }.
In 28, 29 for 1 ≤ l ≤ N − 1, ÊN Êl × ÊN−l . For a point x ∈ ÊN , we have x y, z,
where y ∈ Êl and z ∈ ÊN −l . Let y ∈ Êl , we denote by Ωy ⊂ ÊN−l the projection of Ω onto ÊN−l ,
that is,
Ωy z ∈ ÊN−l : y, z ∈ Ω .
1.8
Boundary Value Problems
5
The domain Ω satisfies the following conditions:
Ω3 Ω is a smooth subset of ÊN and the projections Ωy are bounded uniformly in y ∈ Êl ;
Ω4 there exists a nonempty closed set D ⊂ ÊN−l such that D ⊂ Ωy for all y ∈ Êl ;
Ω5 for each δ > 0, there exists R0 > 0 such that
Ωy ⊂ z ∈ ÊN−l : distz, D < δ
1.9
for all |y| ≥ R0 .
When bx /
≡ b∞ and bx ∈ CΩ ∩ L∞ Ω, the existence of ground state solutions
of Eb has been established by the condition bx ≥ b∞ and the existence of ground state
solutions of limit equation
−Δu u b∞ up−1
u>0
u0
in Ω,
in Ω,
Eb∞ on ∂Ω.
In order to get the second positive solution of Eλa,b , we need some additional
assumptions for ax, bx, and Ω. We assume the following conditions on ax, bx, and
Ω:
Ωb bx > 0 for all x ∈ Ω and Eb in Ω has a ground state solution w0 such that
J0b w0 αb0 Ω.
A2 Ω ax|w0 |q dx > 0 where w0 is a positive ground state solution of Eb in Ω.
Theorem 1.2. Assume that (A1)-(A2), (B1), and (Ωb ) hold. If λ ∈ 0, q/2Λ0 , Eλa,b admits at
least two positive solutions.
Throughout this paper, A1 and B1 will be assumed. H01 Ω denotes the standard
Sobolev space, whose norm ·H 1 is induced by the standard inner product. The dual space of
H01 Ω will be denoted by H −1 Ω. ·, · denote the usual scalar product in H01 Ω. We denote
the norm in Ls Ω by · Ls for 1 ≤ s ≤ ∞. on 1 denotes on 1 → 0 as n → ∞. C, Ci will
denote various positive constants, the exact values of which are not important. This paper is
organized as follows. In Section 2, we give some properties of Nehari manifold. In Sections 3
and 4, we complete proofs of Theorems 1.1 and 1.2.
2. Nehari Manifold
In this section, we will give some properties of Nehari manifold. As the energy functional
Jλa,b is not bounded below on H01 Ω, it is useful to consider the functional on the Nehari
manifold
Mλa,b Ω u ∈ H01 Ω \ {0} : Jλa,b u, u 0 .
2.1
6
Boundary Value Problems
Thus, u ∈ Mλa,b Ω if and only if
Jλa,b u, u u2H 1
q
−λ
Ω
ax|u| dx −
Ω
bx|u|p dx 0.
2.2
Note that Mλa,b Ω contains every nonzero solution of Eλa,b . Moreover, we have the
following results.
Lemma 2.1. The energy functional Jλa,b is coercive and bounded below on Mλa,b Ω.
Proof. If u ∈ Mλa,b Ω, then by A1, 2.2, Hölder and Sobolev inequalities
p−q
p−2
2
Jλa,b u ax|u|q dx
uH 1 − λ
2p
pq
Ω
p−q
p−2
q
≥
Sp Ω−q/2 a Lq∗ uH 1 .
u2H 1 − λ
2p
pq
2.3
2.4
Thus, Jλa,b is coercive and bounded below on Mλa,b Ω.
Define
ψλa,b u Jλa,b u, u .
2.5
Then for u ∈ Mλa,b Ω,
ψλa,b u, u 2u2H 1 − λq
ax|u|q dx − p
bx|u|p dx
Ω
2−q
u2H 1
λ p−q
Ω
− p−q
Ω
2.6
p
Ω
bx|u| dx
ax|u|q dx − p − 2 u2H 1 .
2.7
Similar to the method used in Tarantello 20, we split Mλa,b Ω into three parts:
Mλa,b Ω u ∈ Mλa,b Ω : ψλa,b u, u > 0 ,
M0λa,b Ω u ∈ Mλa,b Ω : ψλa,b u, u 0 ,
M−λa,b Ω u ∈ Mλa,b Ω : ψλa,b u, u < 0 .
Then, we have the following results.
2.8
Boundary Value Problems
7
∈ M0λa,b Ω. Then
Lemma 2.2. Assume that uλ is a local minimizer for Jλa,b on Mλa,b Ω and uλ /
Jλa,b uλ 0 in H −1 Ω.
Proof. Our proof is almost the same as that in Brown and Zhang 9, Theorem 2.3 or see
Binding et al. 31.
Lemma 2.3. We have the following.
i If u ∈ Mλa,b Ω ∪ M0λa,b Ω, then Ω ax|u|q dx > 0;
ii If u ∈ M−λa,b Ω, then Ω bx|u|p dx > 0.
Proof. The proof is immediate from 2.6 and 2.7.
Moreover, we have the following result.
Lemma 2.4. If λ ∈ 0, Λ0 , then M0λa,b Ω ∅ where Λ0 is the same as in 1.1.
Proof. Suppose the contrary. Then there exists λ ∈ 0, Λ0 such that M0λa,b Ω / ∅. Then for
0
u ∈ Mλa,b Ω by 2.6 and Sobolev inequality, we have
2−q
u2H 1 p−q
bx|u|p dx ≤ b L∞ Sp Ω−p/2 uH 1
p
Ω
2.9
and so
uH 1 ≥
2−q
p − q b L∞
1/p−2
Sp Ωp/2p−2 .
2.10
Similarly, using 2.7 and Hölder and Sobolev inequalities, we have
u2H 1
p−q
λ
p−2
Ω
ax|u|q dx ≤ λ
p−q q
a Lq∗ Sp Ω−q/2 uH 1 ,
p−2
2.11
which implies
uH 1 ≤
λ
p−q a Lq∗
p−2
1/2−q
Sp Ω−q/22−q .
2.12
Sp Ωp2−q/2p−2q/2 Λ0 ,
2.13
Hence, we must have
λ≥
2−q
p − q b L∞
2−q/p−2 p−2
p − q a Lq∗
which is a contradiction. This completes the proof.
8
Boundary Value Problems
For each u ∈ H01 Ω with
Ω
bx|u|p dx > 0, we write
tmax u 1/p−2
2 − q u2H 1
dx
> 0.
p − q Ω bx|u|p
2.14
Then the following lemma holds.
Lemma 2.5. Let λ ∈ 0, Λ0 . For each u ∈ H01 Ω with Ω bx|u|p dx > 0, we have the following.
i If Ω ax|u|q dx ≤ 0, then there is a unique t− t− u > tmax u such that t− u ∈ M−λa,b Ω
and
Jλa,b t− u supJλa,b tu.
2.15
t≥0
ii If
Ω
ax|u|q dx > 0, then there are unique
0 < t t u < tmax u < t− t− u
2.16
such that t u ∈ Mλa,b Ω, t− u ∈ M−λa,b Ω, and
Jλa,b t u inf
Jλa,b t− u supJλa,b tu.
Jλa,b tu,
0≤t≤tmax u
t≥0
2.17
Proof. The proof is almost the same as that in Wu 32, Lemma 5 and is omitted here.
3. Proof of Theorem 1.1
First, we remark that it follows Lemma 2.4 that
Mλa,b Ω Mλa,b Ω ∪ M−λa,b Ω
3.1
for all λ ∈ 0, Λ0 . Furthermore, by Lemma 2.5 it follows that Mλa,b Ω and M−λa,b Ω are
nonempty, and by Lemma 2.1 we may define
αλa,b inf
Jλa,b u; αλa,b u∈Mλa,b Ω
inf
Jλa,b u; α−λa,b u∈Mλa,b Ω
Then we get the following result.
Theorem 3.1. We have the following.
i If λ ∈ 0, Λ0 , then we have αλa,b < 0.
ii If λ ∈ 0, q/2Λ0 , then α−λa,b > d0 for some d0 > 0.
In particular, for each λ ∈ 0, q/2Λ0 , we have αλa,b αλa,b .
inf
u∈M−λa,b Ω
Jλa,b u.
3.2
Boundary Value Problems
9
Proof. i Let u ∈ Mλa,b Ω. By 2.6,
2−q
u2H 1 >
p−q
Ω
bx|u|p dx
3.3
and so
2−q
1 1
1 1
1 1
1 1
2
p
−
−
−
−
Jλ u bx|u| dx <
uH 1 u2H 1
2 q
q p
2
q
q
p
p
−
q
Ω
p−2 2−q
−
u2H 1 < 0.
2pq
3.4
Therefore, αλa,b < 0.
ii Let u ∈ M−λa,b Ω. By 2.6,
2−q
u2H 1 <
p−q
Ω
bx|u|p dx.
3.5
Moreover, by B1 and Sobolev inequality theorem,
Ω
bx|u|p dx ≤ b L∞ Sp Ω−p/2 uH 1 .
p
3.6
∀ u ∈ M−λa,b Ω.
3.7
This implies
uH 1 >
2−q
p − q b L∞
1/p−2
Sp Ωp/2p−2
By 2.4 and 3.7, we have
Jλa,b u
p−2
p−q
2−q
Sp Ω−q/2 a Lq∗
uH 1 − λ
2p
pq
q/p−2
2−q
> Sp Ωpq/2p−2
p − q b L∞
⎤
⎡
2−q/p−2
2
−
q
p−2
p
−
q
Sp Ωp2−q/2p−2 ×⎣
−λ
Sp Ω−q/2 a Lq∗ ⎦.
2p
pq
p − q b L∞
q
≥ uH 1
3.8
10
Boundary Value Problems
Thus, if λ ∈ 0, q/2Λ0 , then
Jλa,b u > d0
∀ u ∈ M−λa,b Ω,
3.9
for some positive constant d0 . This completes the proof.
We define the Palais-Smale simply by P S sequences, P S-values, and P Sconditions in H01 Ω for Jλa,b as follows.
Definition 3.2. (i) For c ∈ Ê, a sequence {un } is a P Sc -sequence in H01 Ω for Jλa,b if Jλa,b un c on 1 and Jλa,b un on 1 strongly in H −1 Ω as n → ∞.
(ii) c ∈ Ê is a P S-value in H01 Ω for Jλa,b if there exists a P Sc -sequence in H10 Ω for
Jλa,b .
(iii) Jλa,b satisfies the P Sc -condition in H01 Ω if any P Sc -sequence {un } in H01 Ω for
Jλa,b contains a convergent subsequence.
Now, we use the Ekeland variational principle 21 to get the following results.
Proposition 3.3. (i) If λ ∈ 0, Λ0 , then there exists a P Sαλa,b -sequence {un } ⊂ Mλa,b Ω in H01 Ω
for Jλa,b .
(ii) If λ ∈ 0, q/2Λ0 , then there exists a P Sα− -sequence {un } ⊂ M−λa,b Ω in H01 Ω for
λa,b
Jλa,b .
Proof. The proof is almost the same as that in Wu 32, Proposition 9.
Now, we establish the existence of a local minimum for Jλa,b on Mλa,b Ω.
Theorem 3.4. Assume (A1) and (B1) hold. If λ ∈ 0, Λ0 , then Jλa,b has a minimizer uλ in Mλa,b Ω
and it satisfies the following.
i Jλa,b uλ αλa,b αλa,b .
ii uλ is a positive solution of Eλa,b in Ω.
iii uλ H 1 → 0 as λ → 0 .
Proof. By Proposition 3.3i, there is a minimizing sequence {un } for Jλa,b on Mλa,b Ω such
that
Jλa,b un αλa,b on 1,
Jλa,b un on 1 in H −1 Ω.
3.10
Since Jλ is coercive on Mλa,b Ω see Lemma 2.1, we get that {un } is bounded in H01 Ω.
Going if necessary to a subsequence, we can assume that there exists uλ ∈ H01 Ω such that
un uλ
weakly in H01 Ω,
un −→ uλ
almost every where in Ω,
un −→ uλ
strongly in Lsloc Ω ∀ 1 ≤ s < 2∗ .
3.11
By A1, Egorov theorem, and Hölder inequality, we have
λ
q
Ω
ax|un | dx λ
Ω
ax|uλ |q dx on 1 as n −→ ∞.
3.12
Boundary Value Problems
11
First, we claim that uλ is a nonzero solution of Eλa,b . By 3.10 and 3.11, it is easy to see
that uλ is a solution of Eλa,b . From un ∈ Mλa,b Ω and 2.3, we deduce that
q p−2
pq
Jλa,b un .
ax|un |q dx un 2H 1 −
p
−q
2
p
−
q
Ω
λ
3.13
Let n → ∞ in 3.13; by 3.10, 3.12, and αλa,b < 0, we get
λ
Ω
ax|uλ |q dx ≥ −
pq
αλa,b > 0.
p−q
3.14
Thus, uλ ∈ Mλa,b Ω is a nonzero solution of Eλa,b . Now we prove that un → uλ strongly in
H01 Ω and Jλa,b uλ αλa,b . By 3.13, if u ∈ Mλa,b Ω, then
Jλa,b u p−q
p−2
λ
u2H 1 −
2p
pq
Ω
ax|u|q dx.
3.15
In order to prove that Jλa,b uλ αλa,b , it suffices to recall that un , uλ ∈ Mλa,b Ω, by 3.15
and by applying Fatou’s lemma to get
p−2
p−q
λ
ax|uλ |q dx
uλ 2H 1 −
2p
pq
Ω
p−2
p−q
2
q
≤ lim inf
λ
−
ax|u
dx
un H 1
n|
n→∞
2p
pq
Ω
αλa,b ≤ Jλa,b uλ 3.16
≤ lim inf Jλa,b un αλa,b .
n→∞
This implies that Jλa,b uλ αλa,b and limn → ∞ un 2H 1 uλ 2H 1 . Let vn un − uλ ; then by
Brézis and Lieb, lemma 33 implies that
un 2H 1 un 2H 1 − uλ 2H 1 on 1.
3.17
Therefore, un → uλ strongly in H01 Ω. Moreover, we have uλ ∈ Mλa,b Ω. On the contrary, if
uλ ∈ M−λa,b Ω, then by Lemma 2.5, there are unique t0 and t−0 such that t0 uλ ∈ Mλa,b Ω and
t−0 uλ ∈ M−λa,b Ω. In particular, we have t0 < t−0 1. Since
d
Jλa,b t0 uλ 0,
dt
d2
Jλa,b t0 uλ > 0,
2
dt
3.18
12
Boundary Value Problems
there exists t0 < t ≤ t−0 such that Jλa,b t0 uλ < Jλa,b tuλ . By Lemma 2.5,
Jλa,b t0 uλ < Jλa,b tuλ ≤ Jλa,b t−0 uλ Jλa,b uλ ,
3.19
which is a contradiction. Since Jλa,b uλ Jλa,b |uλ | and |uλ | ∈ Mλa,b Ω, by Lemma 2.2 we
may assume that uλ is a nonzero nonnegative solution of Eλa,b . By Harnack inequality 34,
we deduce that uλ > 0 in Ω. Finally, by 2.3 and Hölder and Sobolev inequalities,
2−q
uλ H 1 < λ
p−q a Lq∗ Sp Ω−q/2
p−2
3.20
and so uλ H 1 → 0 as λ → 0 .
Now, we begin the proof of Theorem 1.1. By Theorem 3.4, we obtain that Eλa,b has a
positive solution uλ in H01 Ω.
4. Proof of Theorem 1.2
In this section, we will establish the existence of the second positive solution of Eλa,b by
proving that Jλa,b satisfies the P Sα− -condition.
λa,b
Lemma 4.1. Assume that (A1) and (B1) hold. If {un } ⊂ H01 Ω is a P Sc -sequence for Jλa,b , then
{un } is bounded in H01 Ω.
n un /un H 1 . We may
Proof. We argue by contradiction. Assume that un H 1 → ∞. Let u
weakly in H01 Ω. This implies that u
n → u
strongly in Lsloc Ω for all
assume that u
n u
1 ≤ s < 2∗ . By A1, Egorov theorem, and Hölder inequality, we have
λ
q
Ω
ax|
un |q dx λ
q
Ω
ax|
u|q dx on 1.
4.1
Since {un } is a P Sc -sequence for Jλa,b and un H 1 → ∞, there hold
q−2
λun H 1
1
un 2H 1 −
2
q
q−2
un 2H 1 − λun H 1
p−2
Ω
ax|
un |q dx −
un H 1
p
p−2
Ω
ax|
un |q dx − un H 1
Ω
Ω
bx|
un |p dx on 1,
bx|
un |p dx on 1.
4.2
4.3
From 4.1–4.3, we can deduce that
un 2H 1
2 p−q
ax|
u|q dx on 1.
un q−2 λ
q p−2
Ω
4.4
Boundary Value Problems
13
Since 1 < q < 2 and un H 1 → ∞, 4.4 implies
un 2H 1 −→ 0
4.5
as n −→ ∞,
which contradicts with the fact that un H 1 1 for all n.
We assume the condition Ωb holds and recall
Sbp Ω inf
u∈H01 Ω\{0}
u2H 1
Ω
bx|u|p dx
4.6
2/p .
Lemma 4.2. Assume that (A1), (B1), and (Ωb ) hold. If {un } ⊂ H01 Ω is a P Sc -sequence for Jλa,b
with c ∈ 0, αb0 Ω, then there exists a subsequence of {un } converging weakly to a nonzero solution
of Eλa,b .
Proof. Let {un } ⊂ H01 Ω be a P Sc -sequence for Jλa,b with c ∈ 0, αb0 Ω. We know from
Lemma 4.1 that {un } is bounded in H01 Ω, and then there exist a subsequence of {un } still
denoted by {un } and u0 ∈ H01 Ω such that
un u0
weakly in H01 Ω,
4.7
un −→ u0
almost every where in Ω,
un −→ u0
strongly in Lsloc Ω ∀ 1 ≤ s < 2∗ .
It is easy to see that Jλa,b u0 0, and by A1, Egorov theorem, and Hölder inequality, we
have
λ
Ω
ax|un |q dx λ
Ω
ax|u0 |q dx on 1.
4.8
Next we verify that u0 ≡
/ 0. Arguing by contradiction, we assume u0 ≡ 0. Setting
l lim
n→∞
Ω
bx|un |p dx.
4.9
Since Jλa,b un on 1 and {un } is bounded, then by 4.8, we can deduce that
0 lim Jλa,b un , un
n→∞
2
p
lim un H 1 −
bx|un | dx lim un 2H 1 − l,
n→∞
Ω
n→∞
4.10
14
Boundary Value Problems
that is,
lim un 2H 1 l.
4.11
n→∞
If l 0, then we get c limn → ∞ Jλa,b un 0, which contradicts with c > 0. Thus we
conclude that l > 0. Furthermore, by the definition of Sbp Ω we obtain
un 2H 1
≥
Sbp Ω
Ω
bx|un |p dx
2/p
.
4.12
Then as n → ∞ we have
l lim un 2H 1 ≥ Sbp Ωl2/p ,
4.13
l ≥ Sbp Ωp/p−2 .
4.14
n→∞
which implies that
Hence, from 1.7 and 4.8–4.14 we get,
λ
1
1
2
q
ax|un | dx − lim
bx|un |p dx
c lim Jλa,b un lim un H 1 − lim
n→∞
2n→∞
qn→∞ Ω
pn→∞ Ω
p−2 b
1 1
−
S Ωp/p−2 αb0 Ω.
l≥
2 p
2p p
4.15
This is a contradiction to c < αb0 Ω. Therefore u0 is a nonzero solution of Eλa,b .
Lemma 4.3. Assume that (A1)-(A2), (B1), and (Ωb ) hold. Let w0 be a positive ground state solution
of Eb ; then
i supt≥0 Jλa,b tw0 < αb0 Ω for all λ > 0;
ii α−λa,b < αb0 Ω for all λ ∈ 0, Λ0 .
Proof. i First, we consider the functional Q : H01 Ω →
1
1
Qu u2H 1 −
2
p
Ê defined by
Ω
bx|u|p dx
∀ u ∈ H01 Ω.
4.16
Boundary Value Problems
15
Then, from 1.3 and 1.7, we conclude that
⎞p/p−2
⎛
w0 2H 1
p−2 b
p − 2⎝
Sp Ωp/p−2 αb0 Ω,
supQtw0 2/p ⎠
p
2p
2p
t≥0
bx|w
dx
|
0
Ω
4.17
where the following fact has been used:
sup
t≥0
tp
t2
A− B
2
p
p−2
A p/p−2
2p
B2/p
where A, B > 0.
4.18
Using the definitions of Jλa,b , w0 and bx > 0 for all x ∈ Ω, for any λ > 0, we have
Jλa,b tw0 −→ −∞
as t −→ ∞.
4.19
From this we know that there exists t0 > 0 such that
supJλa,b tw0 sup Jλa,b tw0 .
t≥0
0≤t≤t0
4.20
By the continuity of Jλa,b tw0 as a function of t ≥ 0 and Jλa,b 0 0, we can find some
t1 ∈ 0, t0 such that
sup Jλa,b tw0 < αb0 Ω.
4.21
sup Jλa,b tw0 < αb0 Ω.
4.22
0≤t≤t1
Thus, we only need to show that
t1 ≤t≤t0
To this end, by A2 and 4.17, we have
q
t
sup Jλa,b tw0 ≤ supQtw0 − 1
q
t1 ≤t≤t0
t≥0
Ω
ax|w0 |q dx < αb0 Ω.
4.23
Hence i holds.
ii By A1, A2, and the definition of w0 , we have
p
Ω
bx|w0 | dx > 0,
Ω
ax|w0 |q dx > 0.
4.24
16
Boundary Value Problems
Combining this with lemma 2.5ii, from the definition of α−λa,b and part i, for all λ ∈ 0, Λ0 ,
we obtain that there exists t0 > 0 such that t0 w0 ∈ M−λa,b Ω and
α−λa,b ≤ Jλa,b t0 w0 ≤ supJλa,b tw0 < αb0 Ω.
t≥0
4.25
Therefore, ii holds.
Now, we establish the existence of a local minimum of Jλ on M−λa,b Ω.
Theorem 4.4. Assume that (A1)-(A2), (B1), and (Ωb ) hold. If λ ∈ 0, q/2Λ0 , then Jλa,b has a
minimizer Uλ in M−λa,b Ω and it satisfies the following.
i Jλa,b Uλ α−λa,b .
ii Uλ is a positive solution of Eλa,b in Ω.
Proof. If λ ∈ 0, q/2Λ0 , then by Theorem 3.1ii, Proposition 3.3ii, and Lemma 4.3ii,
there exists a P Sα− -sequence {un } ⊂ M−λa,b Ω in H01 Ω for Jλa,b with α−λa,b ∈ 0, αb0 Ω.
λa,b
From Lemma 4.2, there exist a subsequence still denoted by {un } and a nonzero solution
Uλ ∈ H01 Ω of Eλa,b such that un Uλ weakly in H01 Ω. Now we prove that un → Uλ
strongly in H01 Ω and Jλa,b Uλ α−λa,b . By 3.15, if u ∈ Mλa,b Ω, then
p−q
p−2
λ
Jλa,b u u2H 1 −
2p
pq
Ω
ax|u|q dx.
4.26
First, we prove that Uλ ∈ M−λa,b Ω. On the contrary, if Uλ ∈ Mλa,b Ω, then by M−λa,b Ω being
closed in H01 Ω, we have Uλ 2H 1 < lim infn → ∞ un 2H 1 . From Lemma 2.3i and bx > 0 for
all x ∈ Ω, we get
q
Ω
ax|Uλ | dx > 0,
Ω
bx|Uλ |p dx > 0.
4.27
By Lemma 2.5ii, there exists a unique t−λ such that t−λ Uλ ∈ M−λa,b Ω. Since un ∈ M−λa,b Ω,
Jλa,b un ≥ Jλa,b tun for all t ≥ 0 and by 4.26, we have
α−λa,b ≤ Jλa,b t−λ Uλ < lim Jλa,b t−λ un ≤ lim Jλa,b un α−λa,b
n→∞
n→∞
4.28
and this is a contradiction. In order to prove that Jλa,b Uλ α−λa,b , it suffices to recall that un ,
Uλ ∈ M−λa,b for all n, by 4.26 and applying Fatou’s lemma to get
p−q
p−2
λ
ax|Uλ |q dx
Uλ 2H 1 −
2p
pq
Ω
p−q
p−2
≤ lim inf
λ
ax|un |q dx
un 2H 1 −
n→∞
2p
pq
Ω
α−λa,b ≤ Jλa,b Uλ ≤ lim inf Jλa,b un α−λa,b .
n→∞
4.29
Boundary Value Problems
17
This implies that Jλa,b Uλ α−λa,b and limn → ∞ un 2H 1 Uλ 2H 1 . Let vn un − Uλ ; then by
Brézis and Lieb, lemma 33 implies that
vn 2H 1 un 2H 1 − Uλ 2H 1 on 1.
4.30
Therefore, un → Uλ strongly in H01 Ω.
Since Jλa,b Uλ Jλa,b |Uλ | and |Uλ | ∈ M−λa,b Ω, by Lemma 2.2 we may assume that
Uλ is a nonzero nonnegative solution of Eλa,b . Finally, By the Harnack inequality 34 we
deduce that Uλ > 0 in Ω.
Now, we complete the proof of Theorem 1.2: by Theorems 3.4, 4.4, we obtain that
Eλa,b has two positive solutions uλ and Uλ such that uλ ∈ Mλa,b Ω, Uλ ∈ M−λa,b Ω. Since
Mλa,b Ω ∩ M−λa,b Ω ∅, this implies that uλ and Uλ are distinct.
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