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Volume 2010, Article ID 856932, 18 pagesdoi:10.1155/2010/856932 Research Article Multiple Positive Solutions for a Class of Concave-Convex Semilinear Elliptic Equations in Unbounded Doma

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

Copyrightq 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−2u  bx|u| p−2u 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 inÊ

N , 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−2u  bx|u| p−2u in Ω,

u > 0 in Ω,

u  0 on ∂Ω,

E λa,b

where λ > 0, 1 < q < 2 < p < 2∗2∗ 2N/N − 2 if N ≥ 3, 2∗  ∞ if N  1, 2, Ω ⊂Ê

N is an

unbounded domain, and a, b are measurable functions and satisfy the following conditions:

A1 a ∈ CΩ ∩ L q∗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  λu q−1 u p−1 inΩ,

u > 0 in Ω,

u  0 on ∂Ω,

E λ

where λ > 0, 1 < q < 2 < p < 2∗ They proved that there exists λ0> 0 such thatE λ admits at

least two positive solutions for all λ ∈ 0, λ0 and has one positive solution for λ  λ0and no

positive solution for λ > λ0 Actually, Adimurthi et al.2 , Damascelli et al 3 , Ouyang and Shi4 , and Tang 5 proved that there exists λ0 > 0 such thatE λ  in the unit ball B N0; 1

has exactly two positive solutions for λ ∈ 0, λ0 and has exactly one positive solution for

λ  λ0and no positive solution exists for λ > λ0 For more general results ofE λ 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 ForΩ  Ê

N, we are only aware of the works12–15 which studied the existence

of solutions for some related concave-convex elliptic problemsnot involving sign-changing weights

Wu in16 has studied the multiplicity of positive solutions for the following equation involving sign-changing weights:

−Δu  u  f λ xu q−1 g μ xu p−1 inÊ

N ,

u > 0 inÊ

N ,

u ∈ H1

Ê

,

E f λ ,g μ

where 1 < q < 2 < p < 2∗, the parameters λ, μ ≥ 0 He also assumed that f λ x  λfxf−x

is sign-changing and g μ x  ax  μbx, where a and b satisfy suitable conditions, and

provedE f λ ,g μ has at least four positive solutions

WhenΩ  Ω ×ÊΩ ⊂Ê

N−1, N≥ 2 is an infinite strip domain, Wu in 17 considered

E λa,b  not involving sign-changing weights assuming that 0 /≤ a ∈ L 2/2−q Ω, 0 ≤ b ∈

CΩ satisfies lim|x N| → ∞b x , xN   1 in Ω and there exist δ > 0 and 0 < C0 < 1 such that

b x , xN  ≥ 1 − C0e−2√

1θ 1δ|x N|for all x  x , xN  ∈ Ω, where θ1is the first eigenvalue of the Dirichlet problem−Δ in Ω The author proved that there exists a positive constantΛ0such

that for λ ∈ 0, Λ0, E λa,b possesses at least two positive solutions

Miotto and Miyagaki in18 have studied E λa,b in Ω  Ω ×Ê, under the assumption

that a ∈ L γ / γ−q Ω q < γ ≤ 2∗ with a/ ≡ 0 and a− is bounded and has a compact support

inΩ and 0 ≤ b ∈ L∞Ω satisfies lim|x N| → ∞b x , x N   1 and there exists C0 > 0 such that

b x , xN  ≥ 1 − C0e−2√

1θ 1|x N|for all x  x , xN  ∈ Ω, where θ1 is the first eigenvalue of the Dirichlet problem−Δ in Ω It was obtained there existence ofΛ0> 0 such that for λ ∈ 0, Λ0,

E λa,b possesses at least two positive solutions

In a recent work19 , Hsu and Lin have studied E λa,b inÊ

N under the assumptions

A1-A2, B1, and Ωb They proved that there exists a constant Λ0 > 0 such that for

λ ∈ 0, q/2Λ0, E λa,b possesses at least two positive solutions The main aim of this paper

is to studyE λa,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 in20 and prove the existence and multiplicity of positive solutions by using Ekeland’s variational principle21

Set

Λ0 



2− q



p − q 

L∞

2−q/p−2

p− 2



p − q 

L q∗



SpΩp2−q/2p−2q/2 > 0, 1.1

where  L∞  supx∈Ωb  L q∗  Ω|a|q∗dx1/q∗, and S pΩ is the best Sobolev

constant for the imbedding of H1

0Ω into L pΩ Now, we state the first main result about the existence of positive solution ofE λa,b

Theorem 1.1 Assume that (A1) and (B1) hold If λ ∈ 0, Λ0, then  E λa,b  admits at least one

positive solution.

Associated withE λa,b , we consider the energy functional J λa,b in H1

0Ω:

J λa,b u 1

2

2

H1−λ

q

Ωa x|u| q dx− 1

p

Ωb x|u| p dx, 1.2

where H1  Ω∇u|2 u2dx 1/2

By Rabinowitz 22, Proposition B.10 , Jλa,b ∈

C1H1

0Ω,Ê It is well known that the solutions of E λa,b are the critical points of the energy

functional J λa,b in H1

0Ω

Under the assumptionsA1, B1, and λ > 0,  E λa,b can be regarded as a perturbation problem of the following semilinear elliptic equation:

−Δu  u  bxu p−1 inΩ,

u > 0 in Ω,

u  0 on ∂Ω,

E b

where bx ∈ CΩ ∩ L∞Ω and bx > 0 for all x ∈ Ω We denote by S b

pΩ the best constant which is given by

S b pΩ  inf

u ∈H1 Ω\{0}

2

H1



Ωb x|u| p dx2/p 1.3

4 Boundary Value Problems

A typical approach for solving problem of this kind is to use the following Minimax method:

α bΓΩ  inf

γ∈ΓΩmax

t ∈0,1 J0b

where

ΓΩ  γ ∈ C0, 1 , H1

0Ω: γ 0  0, γ1  e, 1.5

J b

0e  0 and e / 0 By the Mountain Pass Lemma due to Ambrosetti and Rabinowitz 23 ,

we called the nonzero critical point u ∈ H1

0Ω of J b

0 a ground state solution ofE b in Ω if

J b

0u  α b

ΓΩ We remark that the ground state solutions of E b in Ω can also be obtained

by the Nehari minimization problem

α b0Ω  inf

v∈MbΩJ0b v, 1.6

whereMb

0Ω  {u ∈ H1

0 2H1Ωb x|u| p dx} Note that Mb

0Ω contains every nonzero solution ofE b in Ω,

α bΓΩ  α b

0Ω  p− 2

2p S

b

pΩp/ p−2 > 0 1.7

see Willem 24 , and if bx ≡ b∞> 0 is a constant, then J b

0 and α b

0Ω replace J0and α∞0 Ω, respectively

The existence of ground state solutions ofE b 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 forE b 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 Ê

N Actually, Kwong30 proved that the positive solution of E b inÊ

N is unique In26 , for Ω

is a periodic domain In26,27 , the domain Ω is required to satisfy

Ω1 Ω  Ω1∪ Ω2, whereΩ1,Ω2are domains inÊ

N andΩ1∩ Ω2is bounded;

Ω2 α∞

0Ω < min{α∞

0 Ω1, α∞

0Ω2}

In28,29 for 1 ≤ l ≤ N − 1,Ê

l ×Ê

N −l For a point x∈Ê

N , we have x  y, z, where y∈Ê

l and z∈Ê

N−l Let y∈Ê

l, we denote byΩy ⊂Ê

N −l the projection ofΩ ontoÊ

N −l, that is,

Ωy z∈ N −l :

y, z

The domainΩ satisfies the following conditions:

Ω3 Ω is a smooth subset ofÊ

N and the projectionsΩy are bounded uniformly in y∈Ê

l;

Ω4 there exists a nonempty closed set D ⊂Ê

N −l such that D⊂ Ωy for all y∈Ê

l;

Ω5 for each δ > 0, there exists R0> 0 such that

Ωy ⊂ z∈Ê

N −l : distz, D < δ 1.9

for all|y| ≥ R0

When bx /≡ b∞ and bx ∈ CΩ ∩ L∞Ω, the existence of ground state solutions

ofE b  has been established by the condition bx ≥ b∞ and the existence of ground state solutions of limit equation

−Δu  u  b∞u p−1 inΩ,

u > 0 in Ω,

u  0 on ∂Ω.

E b∞

In order to get the second positive solution of E λa,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  E b  in Ω has a ground state solution w0 such that

J0b w0  α b

0Ω

A2Ωa x|w0|q dx > 0 where w0is a positive ground state solution ofE b in Ω

Theorem 1.2 Assume that (A1)-(A2), (B1), and (Ω b ) hold If λ ∈ 0, q/2Λ0,  E λa,b  admits at

least two positive solutions.

Throughout this paper,A1 and B1 will be assumed H1

0Ω denotes the standard Sobolev space, whose norm H1is induced by the standard inner product The dual space of

H01Ω will be denoted by H−1Ω ·, · denote the usual scalar product in H1

0Ω We denote

the norm in L s

L s for 1 ≤ s ≤ ∞ o n 1 denotes o n 1 → 0 as n → ∞ C, C iwill denote various positive constants, the exact values of which are not important This paper is organized as follows In Section2, we give some properties of Nehari manifold In Sections 3 and4, we complete proofs of Theorems1.1and1.2

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 H01Ω, it is useful to consider the functional on the Nehari manifold

Mλa,bΩ  u ∈ H1

0Ω \ {0} :J λa,b u, u  0. 2.1

6 Boundary Value Problems

Thus, u∈ Mλa,bΩ if and only if

J λa,b u, u 2

H1− λ

Ωa x|u| q dx−

Ωb x|u| p dx  0. 2.2

Note that Mλa,bΩ contains every nonzero solution of E λa,b Moreover, we have the following results

Lemma 2.1 The energy functional J λa,b is coercive and bounded below onMλa,b Ω.

Proof If u∈ Mλa,bΩ, then by A1, 2.2, H¨older and Sobolev inequalities

Jλa,b u  p− 2

2p

2

H1− λ



p − q

pq



Ωa x|u| q dx 2.3

≥ p− 2

2p

2

H1− λp − q

pq



SpΩ−q/2  L q∗

q

H1. 2.4

Thus, J λa,bis coercive and bounded below onMλa,bΩ

Define

ψ λa,b u J λa,b u, u 2.5

Then for u∈ Mλa,bΩ,



ψ λa,b

u, u 2

H1− λq

Ωa x|u| q dx − p

Ωb x|u| p dx

2− q 2

H1−p − q

Ωb x|u| p dx

2.6

 λp − q

Ωa x|u| q dx−p− 2 2

H1. 2.7

Similar to the method used in Tarantello20 , we split Mλa,bΩ into three parts:

M

λa,bΩ  u∈ Mλa,bΩ :

ψ λa,b

u, u> 0

,

M0

λa,bΩ  u∈ Mλa,bΩ :

ψ λa,b

u, u 0,

M−

λa,bΩ  u∈ Mλa,bΩ :

ψ λa,b

u, u< 0

.

2.8

Then, we have the following results

Lemma 2.2 Assume that u λ is a local minimizer for Jλa,b onMλa,b Ω and u λ /∈ M0

λa,b Ω Then

J λa,b u λ   0 in H−1Ω.

Binding et al.31 

Lemma 2.3 We have the following.

i If u ∈ M

λa,bΩ ∪ M0

λa,b Ω, thenΩa x|u| q dx > 0;

ii If u ∈ M−

λa,b Ω, thenΩb x|u| p dx > 0.

Proof The proof is immediate from2.6 and 2.7

Moreover, we have the following result

Lemma 2.4 If λ ∈ 0, Λ0, then M0

λa,b Ω  ∅ where Λ0is the same as in1.1.

Proof Suppose the contrary Then there exists λ ∈ 0, Λ0 such that M0

λa,b Ω / ∅ Then for

u∈ M0

λa,bΩ by 2.6 and Sobolev inequality, we have

2− q

p − q 2H1

Ωb x|u| p dx  L∞S pΩ−p/2 p

and so

H1≥



2− q



p − q 

L∞

1/p−2

S pΩp/2 p−2 2.10

Similarly, using2.7 and H¨older and Sobolev inequalities, we have

2

H1 λ p − q

p− 2

Ωa x|u| q dx ≤ λ p − q

p− 2  L q∗ SpΩ−q/2 q

H1, 2.11

which implies

H1≤



λ p − q

p− 2  L q∗

1/2−q

SpΩ−q/22−q 2.12

Hence, we must have

λ≥



2− q



p − q 

L∞

2−q/p−2

p− 2



p − q 

L q∗



SpΩp2−q/2p−2q/2  Λ0, 2.13

which is a contradiction This completes the proof

8 Boundary Value Problems

For each u ∈ H1

0Ω withΩb x|u| p dx > 0, we write

tmaxu 

 

2− q 2

H1



p − q Ωb x|u| p dx

1/p−2

Then the following lemma holds

Lemma 2.5 Let λ ∈ 0, Λ0 For each u ∈ H1

0Ω withΩb x|u| p dx > 0, we have the following.

i IfΩa x|u| q dx ≤ 0, then there is a unique t−  t−u > tmaxu such that t−u∈ M−

λa,bΩ

and

Jλa,b

t−u

 sup

ii IfΩa x|u| q dx > 0, then there are unique

0 < t tu < tmaxu < t− t−u 2.16

such that tu∈ M

λa,b Ω, t−u∈ M−

λa,b Ω, and

J λa,b tu  inf

0≤t≤t maxu J λa,b tu, J λa,b

t−u

 sup

t≥0J λa,b tu. 2.17

Proof The proof is almost the same as that in Wu32, Lemma 5 and is omitted here

First, we remark that it follows Lemma2.4that

Mλa,bΩ  M

λa,bΩ ∪ M−

for all λ ∈ 0, Λ0 Furthermore, by Lemma2.5 it follows thatM

λa,bΩ and M−

λa,bΩ are nonempty, and by Lemma2.1we may define

u∈Mλa,bΩJ λa,b u; α

u∈M 

λa,bΩJ λa,b u; α−

u∈M −

λa,bΩJ λa,b u. 3.2

Then we get the following result

Theorem 3.1 We have the following.

i If λ ∈ 0, Λ0, then we have α

λa,b < 0.

ii If λ ∈ 0, q/2Λ0, then α−

λa,b > d0for some d0 > 0.

In particular, for each λ ∈ 0, q/2Λ0, we have α

λa,b  α λa,b.

Proof i Let u ∈ M

λa,bΩ By 2.6,

2− q

p − q 2H1>

Ωb x|u| p dx 3.3 and so

J λ u 



1

2− 1

q



2

H1

 1

q−1

p



Ωb x|u| p dx <



1

2− 1

q





 1

q−1

p

2− q

p − q



2

H1

 −



p− 22− q

2pq

2

H1 < 0.

3.4

Therefore, αλa,b < 0.

ii Let u ∈ M−

λa,bΩ By 2.6,

2− q

p − q 2H1<

Ωb x|u| p dx. 3.5 Moreover, byB1 and Sobolev inequality theorem,

Ωb x|u| p dx  L∞S pΩ−p/2 p

This implies

H1>



2− q



p − q 

L∞

1/p−2

SpΩp/2 p−2 ∀ u ∈ M−

λa,b Ω. 3.7

By2.4 and 3.7, we have

J λa,b u

q

H1



p− 2

2p

2−q

H1 − λ



p − q

pq



S pΩ−q/2 

L q∗



>



2− q



p − q 

L∞

q/ p−2

SpΩpq/2 p−2

×

⎡

⎣p − 2

2p S pΩp 2−q/2p−2



2− q



p − q 

L∞

2−q/p−2

− λ



p − q

pq



S pΩ−q/2 

L q∗

⎤

⎦.

3.8

10 Boundary Value Problems

Thus, if λ ∈ 0, q/2Λ0, then

Jλa,b u > d0 ∀ u ∈ M−

for some positive constant d0 This completes the proof

We define the Palais-Smale simply by PS sequences, values, and PS-conditions in H01Ω for J λa,bas follows

Definition 3.2 (i) For c ∈Ê, a sequence {u n } is a PS c -sequence in H01Ω for J λa,b if Jλa,b u n 

c  o n 1 and J λa,b u n   o n 1 strongly in H−1Ω as n → ∞.

(ii) c ∈ Êis a PS-value in H1

0Ω for J λa,b if there exists a PS c -sequence in H10Ω for

Jλa,b.

(iii) J λa,b satisfies the PS c -condition in H1

0Ω if any PS c -sequence {u n } in H1

0Ω for

J λa,b contains a convergent subsequence.

Now, we use the Ekeland variational principle21 to get the following results

Proposition 3.3 (i) If λ ∈ 0, Λ0, then there exists a PS α λa,b -sequence {u n} ⊂ Mλa,b Ω in H1

0Ω

for J λa,b

(ii) If λ ∈ 0, q/2Λ0, then there exists a PS α−

λa,b -sequence {u n} ⊂ M−

λa,b Ω in H1

0Ω for

Jλa,b

Proof The proof is almost the same as that in Wu32, Proposition 9

Now, we establish the existence of a local minimum for J λa,bonM

λa,bΩ

Theorem 3.4 Assume (A1) and (B1) hold If λ ∈ 0, Λ0, then J λa,b has a minimizer uλ inM

λa,bΩ

and it satisfies the following.

i J λa,b u λ   α λa,b  α

λa,b

ii u λ is a positive solution of E λa,b  in Ω.

λ H1 → 0 as λ → 0.

Proof By Proposition3.3i, there is a minimizing sequence {u n } for J λa,bonMλa,bΩ such that

Jλa,b u n   α λa,b  o n 1, J λa,b u n   o n 1 in H−1Ω. 3.10

Since J λ is coercive onMλa,bΩ see Lemma2.1, we get that {u n } is bounded in H1

0Ω

Going if necessary to a subsequence, we can assume that there exists u λ ∈ H1

0Ω such that

u n u λ weakly in H01Ω,

un −→ u λ almost every where inΩ,

u n −→ u λ strongly in L slocΩ ∀ 1 ≤ s < 2∗.

3.11

ByA1, Egorov theorem, and H¨older inequality, we have

λ

Ωa x|u n|q dx  λ

Ωa x|u λ|q dx  o n 1 as n −→ ∞. 3.12

Lemma 2.2 Assume that u λ is a local minimizer... q/2Λ0, we have α

? ?a, b  α ? ?a, b.

8 Boundary Value Problems

For each u ∈ H1

0Ω with< sup>Ωb