Multiple Positive Solutions for a Quasilinear Elliptic System Involving Concave Convex Nonlinearities and Sign Changing Weight Functions

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Volume 2012, Article ID 109214,21pages doi:10.1155/2012/109214

Research Article

Multiple Positive Solutions for

a Quasilinear Elliptic System Involving

Concave-Convex Nonlinearities and

Sign-Changing Weight Functions

Tsing-San Hsu

Center for General Education, Chang Gung University, Kwei-Shan, Tao-Yuan 333, Taiwan

Correspondence should be addressed to Tsing-San Hsu,[email protected]

Received 20 April 2011; Accepted 20 December 2011

Academic Editor: Marco Squassina

Copyrightq2012 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.

LetΩ0 be an-open bounded domain inRN_N _≥₃_and_p∗ _pN/_N₋_p_{. We consider the}

following quasilinear elliptic system of two equations inW₀1,pΩ×W₀1,pΩ:−Δpuλfx|u|q−2_u

α/αβhx|u|α−2_u_|_v_|β_,₋_Δp_v _μg_x_|_v_|q−2_v _β/_α_β_h_x_|_u_|α_|_v_|β−2_v_{, where}_{λ, μ >} _0,_Δp

denotes thep-Laplacian operator, 1 ≤ q < p < N,α, β >1 satisfyp < αβ ≤ p∗, andf, g, h

are continuous functions onΩwhich are somewhere positive but which may change sign onΩ. We establish the existence and multiplicity results of positive solutions tothe above mentioned quasilinear elliptic system equationsby variational methods.

1. Introduction and Main Results

Let Ω 0 be a smooth-bounded domain in RN with N ≥ 3. In this paper, we study the existence and multiplicity of positive solutions for the following quasilinear elliptic system:

−Δpuλfx|u|q−2u α

αβhx|u|

α−2_u_|_v_|β _in_Ω_,

−Δpvμgx|v|q−2v β

αβhx|u|

α_|_v_|β−2_v _in_Ω_,

uv0 on∂Ω,

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whereλ, μ >0,1 ≤q < p < N,Δpudiv|∇u|p−2∇uis thep-Laplacian,α >1, β >1 satisfy

p < αβ ≤ p∗, p∗ pN/N−pdenotes the critical Sobolev exponent, and the weight

functionsf, g, hare satisfying the following assumptions:

A1f, g∈CΩ,f max{f,0}/≡0, gmax{g,0}/≡0 and|f|∞|g|∞1;

A2h∈CΩ, hmax{h,0}/≡0 and|h|_∞1;

A3there exist a0, b0, r0 > 0 andx0 ∈ Ω such that Bx0,2r0 ⊂ Ω, fx ≥ a0 and gx ≥ b0 for allx ∈ Bx0,2r0, without loss of generality, we assume below that

x00;

A4hx>0 for allx∈Ω, |h|_∞h0and there existsδ0> N/p−1such that

hx h0 o|x|δ0 asx−→0. 1.1

SystemSλf,μg,his posed in the framework of the Sobolev spaceW W₀1,pΩ×W₀1,pΩwith the standard norm

u, v

Ω|∇u|

p_dx

Ω|∇v|

p_dx

1/p

. 1.2

Moreover, a pair of functionsu, v∈Wis said to be a weak solution of SystemSλf,μg,hif

Ω|∇u|

p−2_∇_u_∇_ϕ

1dx

Ω|∇v|

p−2_∇_v_∇_ϕ

2dx

−λ

Ωfx|u|

q−2_uϕ

1dx−μ

Ωgx|v|

q−2_vϕ

2dx

− α

αβ

Ωhx|u|

α−2_u_|_v_|β_ϕ1dx₋ β

αβ

Ωhx|u|

α_|_v_|β−2_vϕ2dx₀

1.3

for allϕ1, ϕ2∈W. Thus, the corresponding energy functional of SystemSλf,μg,his defined

by

Iλ,μu, v

1

pu, v

p₋ 1

q

Ω

λfx|u|qμgx|v|qdx− 1

αβ

Ωhx|u|

α_|_v_|β_dx. _1.4

Semilinear and quasilinear scalar elliptic equations with concave-convex nonlineari-ties are widely studied: we refer the reader to Ambrosetti et al.1 , de Figueiredo et al.2 , Azorero and Peral3 , Azorero et al.4 , EL Hamidi5 , Hirano et al.6 , Hsu7 , and Wu

8 , and so forth. For the nonlinear elliptic systems, we refer to Adriouch and EL Hamidi

9 , Ahammou 10 , Alves et al. 11 , Bozhkov and Mitidieri12 , Cl´ement et al. 13 , de Figueiredo and Felmer 14 , EL Hamidi 15 , Hsu and Lin 16,17 , Squassina 18 , V´elin

19 , and Wu20 , and so forth.

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in the Nehair manifold, and for the definition of Nehair manifold we refer the reader to see Nehari21 or Willem22 .

More recently, in20 the author extends the method of8 to systemSλf,μg,hin the

semilinear casep 2 with the subcritical case 2 < αβ < 2∗and the sign-changing weight functionsf, g. In16 the author also extends the method of 8 to system Sλf,μg,hin the quasilinear case 1 < p < N with critical caseαβ p∗ and the constant weight functions

f≡g≡h≡1. In the present paper, motivated by16,20 we extend and improve the papers by Hsu16 and Wu20 . First, we deal with more general weight functionsf, g, h which may be changing sign, and second, we also deal with quasilinear elliptic systems involving subcritical or critical Sobolev exponents.

LetSbe the best Sobolev constant for the embedding ofW₀1,pΩinLαβΩdefined by

S inf

u∈W1,p

0 Ω\{0}

Ω|∇u|pdx

Ω|u|αβdx

p/αβ, _1.5

and set

Λ1

p−q

αβ−q

_p/αβ−p

αβ−q

αβ−p|Ω|

αβ−q/αβ−p/p−q_Spαβ−q/p−qαβ−p_>₀_, _1.6

where|Ω|is the Lebesgue measure ofΩ. Our main results are as follows.

Theorem 1.1existence of one positive solution. Assume thatA1-A2hold. If 1≤ q < p <

N, p < αβ≤ p∗, andλ, μ >0 satisfy 0< λp/p−qμp/p−q <Λ1, then systemSλf,μg,hhas at

least one positive solution inW₀1,pΩ×W₀1,pΩ.

Theorem 1.2second positive solution in the subcritical case. Assume thatA1-A2hold. If

1≤q < p < N, p < αβ < p∗, andλ, μ >0 satisfy 0< λp/p−qμp/p−q<q/pp/p−qΛ1, then

SystemSλf,μg,hhas at least two positive solutions inW01,pΩ×W 1,p

0 Ω.

Theorem 1.3second positive solution in the critical case. Assume that A1–A4hold. If

1≤q < p < N, αβp∗, andλ, μ >0 satisfy 0< λp/p−qμp/p−q<q/pp/p−qΛ1, then System

Sλf,μg,hhas at least two positive solutions inW₀1,pΩ×W₀1,pΩ.

This paper is organized as follows. In Section 2, we give some notations and pre-liminaries. The proofs of Theorems 1.1 and 1.2 are in Section3. In Section 4, we manage to give the proof of Theorem1.3. Throughout this paper,A1andA2will be assumed.

2. Notations and Preliminaries

In this section, we give some notations and necessary preliminary results.

Notations. We make use of the following notation.

Ls_Ω_, ₁ _≤ _{s <} _∞_{, denote Lebesgue spaces; the norm} _Ls _{is denoted by}_{| · |}

s for

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W W₀1,pΩ 2, endowed with normzp u, vp|∇u|pp|∇v|pp;

The dual space of a Banach spaceWwill be denoted byW−1;

tztu, v tu, tvfor allz u, v∈Wandt∈R;

|z| |u|,|v|for allz u, v∈W;

z u, vis said to be nonnegative inΩifu≥0 andv≥0 inΩ;

z u, vis said to be positive inΩifu >0 andv >0 inΩ;

|Ω|is the Lebesgue measure ofΩ;

Bx0, r {x∈RN| |x−x0|< r}is the ball inRN;

Oεt_denotes_|_O_εt_|_/εt_≤_C_as_ε _→ _{0 for}_t_≥_0; on1denoteson1 → 0 asn → ∞;

C, Ci will denote various positive constants, the exact values of which are not

important;

p∗pN/N−p1< p < Nis the critical Sobolev exponent;

LetJ, Kλ,μ:W → Rbe the functionals defined by

Jz

Ωhx|u|

α_|_v_|β_dx,

Kλ,μz

Ω

λfx|u|qμgx|v|qdx

2.1

for allz u, v∈W.

As the energy functionalIλ,μis not bounded below onW, it is useful to consider the

functional on the Nehari manifold

Nλ,μ

z∈W\ {0} |I_λ,μ z, z0. 2.2

Thus,z u, v∈ Nλ,μif and only if

I_λ,μ z, zzp−Kλ,μz−Jz 0. 2.3

Note thatNλ,μcontains every nonzero solution of SystemSλf,μg,h. Moreover, we have

the following results

Lemma 2.1. The energy functionalIλ,μis coercive and bounded onNλ,μ.

Proof. Ifz u, v ∈ Nλ,μ, then by2.3, the H ¨older inequality, and the Sobolev embedding

theorem,

Iλ,μz

αβ−p

pαβz

p₋αβ−q

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≥ αβ−p

pαβz

p₋αβ−q

qαβS

−q/p_|_Ω_|αβ−q/αβ_λp/p−q_μp/p−qp−q/p_zq_. _2.5

Thus,Iλ,μis coercive and bounded onNλ,μ.

Define

Φλ,μz I_λ,μ z, z. 2.6

Then foru∈ Nλ,μ,

Φ

λ,μz, z

pzp−qKλ,μz−

αβJz

p−qzp−αβ−qJz

2.7

αβ−qKλ,μz−

αβ−pzp. 2.8

Similar to the method used in Tarantello23 , we splitNλ,μinto three parts:

N

λ,μ

z∈ Nλ,μ:

Φ

λ,μz, z

>0,

N0

λ,μ

z∈ Nλ,μ:

Φ

λ,μz, z

0,

N−

λ,μ

z∈ Nλ,μ:

Φ

λ,μz, z

<0.

2.9

Then, we have the following results.

Lemma 2.2. Assume thatz0is a local minimizer forIλ,μonNλ,μandz0 ∈ N/ 0_λ,μ. ThenI_λ,μ z0 0

inW−1.

Proof. Our proof is almost the same as that in24, Theorem 2.3 .

Lemma 2.3. One has the following:

iifz∈ N_λ,μ, thenKλ,μz>0;

iiifz∈ N0_λ,μ, thenKλ,μz>0 andJz>0;

iiiifz∈ N−_λ,μ, thenJz>0.

Proof. The proof is immediate from2.7and2.8.

Moreover, we have the following result.

Lemma 2.4. If 0< λp/p−qμp/p−q<Λ1, thenN0_λ,μ∅whereΛ1is the same as in1.6.

Proof. We argue by contradiction. Assume that there existλ, μ >0 with

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such thatN0_λ,μ/∅. Then by2.7and2.8, foru∈ N0_λ,μ, on has the following:

zp αβ−q

p−q Jz,

zp αβ−q

αβ−pKλ,μz.

2.11

By|f|_∞ |g|_∞ |h|_∞ 1, the H ¨older inequality, and the Sobolev embedding theorem, we have

z ≥

_p₋_q

αβ−qS

αβ/p1/αβ−p_,

z ≤

_α_β₋_q

αβ−pS

−q/p_|_Ω_|αβ−q/αβ1/p−q_λp/p−q_μpp−q1/p_.

2.12

This implies

λp/p−qμp/p−qλp/p−qμp/p−q≥

p−q

αβ−q

_p/αβ−p

αβ−q

αβ−p|Ω|

αβ−q/αβ−p/p−q

×Sαβ/αβ−pq/p−q Λ1,

2.13

which contradicts 0< λp/p−qμp/p−q<Λ1.

By Lemma2.4, we writeNλ,μN_λ,μ∪ N−_λ,μand define

θλ,μ inf

z∈Nλ,μ

Iλ,μz; θ_λ,μ inf

z∈N

λ,μ

Iλ,μz; θ_λ,μ− inf

z∈N−

λ,μ

Iλ,μz. _2.14

Then we get the following result.

Theorem 2.5. iIf 0< λp/p−qμp/p−q<Λ1, then one hasθλ,μ≤θ_λ,μ <0;

iiif 0< λp/p−qμp/p−q < q/pp/p−qΛ1, thenθ−_λ,μ> d0for some positive constantd0

depending onλ, μ, p, q, N, S,|Ω|,|f|_∞,|g|_∞and|h|_∞.

Proof. iLetz u, v∈ N_λ,μ. By2.7

p−q

αβ−qz

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and so

Iλ,μz

1

p−

1

q

zp

1

q−

1

αβ

Jz

<

1

p−

1

q

1

q−

1

αβ

_p₋_q

αβ−q

zp

−

p−qαβ−p

pqαβ z

p_<₀_.

2.16

Therefore, from the definitions ofθλ,μ,θ_λ,μ, we can deduce thatθλ,μ≤θ_λ,μ<0.

iiLetz∈ N−_λ,μ. By2.7

p−q

αβ−qz

p_{< J}_z_.

2.17

Moreover, by|h|_∞1, the H ¨older inequality, and the Sobolev embedding theorem,

Jz≤S−αβ/pzαβ. 2.18

This implies

z>

_p₋_q

αβ−q

1/αβ−p

Sαβ/pαβ−p ∀z∈ N−_λ,μ. 2.19

By2.5, one has the following:

Iλ,μz≥ zq

αβ−p

pαβz

p−q₋ αβ−q

qαβS

−q/p_|_Ω_|αβ−q/αβ_λp/p−q_μp/p−qp−q/p

>

_p₋_q

αβ−q

_q/αβ−p

Sqαβ/pαβ−p

×

αβ−p

pαβS

p−qαβ/pαβ−p p−q

αβ−q

p−q/αβ−p

−αβ−q

qαβS

−q/p_|_Ω_|αβ−q/αβ_λp/p−q_μp/p−qp−q/p

.

2.20

Thus, if 0< λp/p−qμp/p−q<q/pp/p−qΛ1, then

Iλ,μz> d0 ∀z∈ N−_λ,μ, 2.21

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For eachz∈WwithJz>0, we write

tmax p−q

zp

αβ−qJz

_1/αβ−p

>0. 2.22

Then the following lemma holds.

Lemma 2.6. Letλp/p−qμp/p−q∈0,Λ1. For eachz∈WwithJz>0, one has the following:

iifKλ,μz≤0, then there exists a uniquet−> tmaxsuch thatt−z∈ N−_λ,μand

Iλ,μ

t−zsup

t≥0 Iλ,μtz; 2.23

iiifKλ,μz>0, then there exist unique 0< t< tmax< t−such thattz∈ N_λ,μ,t−z∈ N−_λ,μ

and

Iλ,μtz inf

0≤t≤tmaxIλ,μtz; Iλ,μ

t−zsup

t≥0 Iλ,μtz. 2.24

Proof. The proof is almost the same as that in25, Lemma 2.6 and is omitted here.

For eachz∈WwithKλ,μz>0, we write

tmax

αβ−qKλ,μz

αβ−pzp

_1/p−q

>0. 2.25

Then we have the following lemma.

Lemma 2.7. Let λp/p−q μp/p−q ∈ 0,Λ1. For each z ∈ W with Kλ,μz > 0, one has the

following:

iifJz≤0, then there exists a unique 0< t< tmaxsuch thattz∈ N_λ,μand

Iλ,μtz inf

t≥0Iλ,μtz; 2.26

iiifJz>0, then there exist unique 0< t < tmax < t−such thattz∈ N_λ,μ,t−z∈ N−_λ,μ

and

Iλ,μtz inf

0≤t≤tmaxIλ,μtz; Iλ,μ

t−zsup

t≥0 Iλ,μtz. 2.27

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3. Proofs of Theorems

1.1 and

1.2

First, we give the following definitions aboutPS_c-sequence.

Definition 3.1. Letc∈R,Wbe a Banach space andI∈C1_W,_R_.

i{zn}is aPS_c-sequence inWforIifIzn con1andIzn on1strongly inW−1asn → ∞.

iiWe say thatIsatisfies thePS_ccondition if anyPS_c-sequence{zn}inWforIhas a convergent subsequence.

Now, we use the idea of Tarantello23 to get the following results.

Proposition 3.2. Let 1≤q < p < Nandp < αβ≤p∗, we have

iif 0< λp/p−qμp/p−q<Λ1, then there exists aP Sθλ,μ-sequence{zn} ⊂ Nλ,μinWfor

Iλ,μ;

iiif 0 < λp/p−qμp/p−q <q/pp/p−qΛ1, then there exists aP Sθ−_λ,μ-sequence{zn} ⊂

Nλ,μinWforIλ,μ,

whereΛ1is the positive constant given in1.6.

Proof. The proof is almost the same as that in8, Proposition 9 .

Now, we establish the existence of a local minimum forIλ,μonN_λ,μ.

Theorem 3.3. LetΛ1 be the same positive constant as in1.6. If 1≤q < p < N, p < αβ ≤p∗,

and 0< λp/p−qμp/p−q<Λ1, thenIλ,μhas a minimizerz1_λ,μinN_λ,μand it satisfies the following:

iIλ,μz_λ,μ1 θλ,μθ_λ,μ<0;

iiz1

λ,μis a positive solution of SystemSλf,μg,h;

iiiIλ,μz1_λ,μ → 0 asλ → 0,μ → 0.

Proof. By Proposition3.2i, there exists a minimizing sequence {zn}forIλ,μ onNλ,μsuch

that

Iλ,μzn θλ,μon1, Iλ,μ zn on1 inW−1. 3.1

SinceIλ,μis coercive onNλ,μsee Lemma2.1, we get that{zn}is bounded inW. Then there

exist a subsequence{zn un, vn}andz1_λ,μ u1_λ,μ, v1_λ,μ∈Wsuch that

un u1λ,μ, vn v1λ,μ weakly in W 1,p

0 Ω,

un−→u1_λ,μ, vn−→v1_λ,μ almost everywhere inΩ, un−→u1_λ,μ, vn−→v_λ,μ1 strongly inLsΩ∀1≤s < p∗.

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This implies

Kλ,μzn Kλ,μ

z1

λ,μ

on1 asn−→ ∞. 3.3

First, we claim thatz1

λ,μis a nontrivial solution of SystemSλf,μg,h. By3.1and 3.2, it is

easy to verify thatz1_λ,μis a weak solution of SystemSλf,μg,h. Fromzn ∈ Nλ,μand2.4, we

deduce that

Kλ,μzn

qαβ−p

pαβ−qzn

p₋q

αβ

αβ−qIλ,μzn. 3.4

Letn → ∞in3.4, by3.1,3.3, andθλ,μ<0, we get

Kλ,μz1_λ,μ≥ −q

αβ

αβ−qθλ,μ>0. 3.5

Thus,z1

λ,μ ∈ Nλ,μis a nontrivial solution of SystemSλf,μg,h. Now we prove thatzn → z1λ,μ

strongly inWandIλ,μz1_λ,μ θλ,μ. By3.4, ifz∈ Nλ,μ, then

Iλ,μz

αβ−p

pαβz

p₋αβ−q

qαβKλ,μz. 3.6

In order to prove thatIλ,μz1_λ,μ θλ,μ, it suﬃces to recall thatz1_λ,μ∈ Nλ,μ, by3.6, and apply

Fatou’s lemma to get

θλ,μ≤Iλ,μ

z1_λ,μ αβ−p

pαβ

z1_λ,μp−αβ−q

qαβKλ,μ

z1_λ,μ

≤lim inf

n→ ∞

αβ−p

pαβzn

p₋αβ−q

qαβKλ,μzn

≤lim inf

n→ ∞ Iλ,μzn θλ,μ.

3.7

This implies thatIλ,μz1

λ,μ θλ,μand limn→ ∞zn

p _z1

λ,μ

p

. Letzn zn−z1

λ,μ, then by Br´ezis

and Lieb lemma26 implies

znpznp−z1λ,μ p

. 3.8

Therefore,zn → z1_λ,μstrongly inW. Moreover, we havez_λ,μ1 ∈ N_λ,μ. Thusθλ,μθ_λ,μ . On the

contrary, ifz1

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from Lemma2.6ii, there exist uniquet₁andt−₁ such thatt₁z1_λ,μ∈ N_λ,μandt−₁z1_λ,μ∈ N−_λ,μ. In particular, we havet₁ < t−₁ 1. Since

d

dtIλ,μ

t₁z1_λ,μ0, d

2

dt2Iλ,μ

t₁z1_λ,μ>0, 3.9

there existst₁ < t≤t₁−such thatIλ,μt₁z1

λ,μ< Iλ,μtz1λ,μ. By Lemma2.6,

Iλ,μt₁z1_λ,μ< Iλ,μtz1_λ,μ≤Iλ,μt−₁z1_λ,μIλ,μz1_λ,μθλ,μ, 3.10

which is a contradiction. SinceIλ,μz1_λ,μ Iλ,μ|z_λ,μ1 |and|z1_λ,μ| ∈ N_λ,μ, by Lemma2.2we may

assume thatz1_λ,μis a nontrivial nonnegative solution of SystemSλf,μg,h. In particularu1

λ,μ/≡0, vλ,μ1 /≡0. Indeed, without loss of generality, we may assume that

v1

λ,μ≡0. Then asu1λ,μis a nontrivial nonnegative solution of

−Δpuλfx|u|q−2u inΩ,

u0 on∂Ω. 3.11

By the standard regularity theory, we haveu1

λ,μ>0 inΩand

u1_λ,μ,0pKλ,μu1_λ,μ,0>0. 3.12

Moreover, by conditionsA1,A2andu1

λ,μ>0 inΩ, we may choosew∈W 1,p

0 Ω\ {0}such

that

0, wpKλ,μ0, w>0,

Ju1_λ,μ, w≥0. 3.13

Now

Kλ,μ

u1_λ,μ, wKλ,μ

u1_λ,μ,0Kλ,μ0, w>0, 3.14

and so by Lemma2.7there is unique 0< t< tmaxsuch thattu1_λ,μ, tw∈ N_λ,μ. Moreover,

tmax

⎛ ⎜ ⎝

αβ−qKλ,μu1_λ,μ, w

αβ−pu1

λ,μ, w

p

⎞ ⎟

⎠

_α_β₋_q

αβ−p

>1,

Iλ,μ

tu1_λ,μ, tw inf

0≤t≤tmaxIλ,μ

tu1_λ,μ, tw.

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This implies

θ_λ,μ≤Iλ,μtu1_λ,μ, tw≤Iλ,μu1_λ,μ, w< Iλ,μu1_λ,μ,0θ_λ,μ 3.16

which is a contradiction.

By a standard bootstrap argument, it is proved that a weak solution for System

Sλf,μg,h is in the space C2_Ω _×_C2_Ω_{, and it is really a classical solution. Finally, by}

the Harnack inequality27 we deduce thatz1

λ,μ is a positive solution of SystemSλf,μg,h.

Moreover, by Theorem2.5iand2.5we have

0> θ_λ,μ Iλ,μ

z1_λ,μ>−αβ−q

qαβS

−q/p_|_Ω_|αβ−q/αβ_λp/p−q_μp/p−qp−q/p_z1

λ,μ

q .

3.17

This implies thatIλ,μz1_λ,μ → 0 asλ → 0,μ → 0.

Next, we establish the existence of a local minimum forIλ,μonN−_λ,μin the subcritical

casep < αβ < p∗. This implies that there exists the second positive solution in the subcritical

casep < αβ < p∗.

Theorem 3.4. LetΛ1 be the same positive constant as in1.6. If 1≤q < p < N, p < αβ < p∗,

and 0< λp/p−qμp/p−q <q/pp/p−qΛ1, thenIλ,μhas a minimizerz2_λ,μinN−_λ,μand it satisfies

the following:

iIλ,μz2_λ,μ θ−_λ,μ;

iiz2_λ,μis a positive solution of SystemSλf,μg,h;

Proof. Let {zn} be a minimizing sequence for Iλ,μ on N−_λ,μ. Then by Iλ,μ coercive on Nλ,μ

and the compact imbedding theorem, there exist a subsequence{zn un, vn}andz2_λ,μ

u2

λ,μ, v2λ,μ∈Wsuch that

un u2_λ,μ, vn v2_λ,μ weakly inW₀1,pΩ,

un−→u2_λ,μ, vn−→v2_λ,μ strongly inLqΩ, LαβΩ. 3.18

This implies

Kλ,μzn Kλ,μ

z2

λ,μ

on1, Jzn J

z2

λ,μ

on1. 3.19

By2.17and2.19there exists a positive numberCsuch that

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This implies

Jz2

λ,μ

≥C. 3.21

Now, we prove that zn → z2_λ,μ strongly in W. Suppose otherwise, then z2_λ,μ <

lim infn→ ∞zn. By Lemma 2.6, there exists a unique t2− such that t−2z2λ,μ ∈ N−λ,μ. Since

zn∈ N−_λ,μ, Iλ,μzn≥Iλ,μtznfor allt≥0, we have

θ_λ,μ− ≤Iλ,μ

t−z2_λ,μ< lim

n→ ∞Iλ,μ

t−zn

≤ lim

n→ ∞Iλ,μzn θ

−

λ,μ, 3.22

and this is a contradiction. Hence

zn−→z2λ,μ strongly inW. 3.23

This implies

Iλ,μ

z2_λ,μ lim

−

λ,μ. 3.24

SinceIλ,μz2_λ,μ Iλ,μ|z2_λ,μ|and|z2_λ,μ| ∈ N−_λ,μ, by Lemma2.2and3.21we may assumez2_λ,μis

a nontrivial nonnegative solution of SystemSλf,μg,h. Finally, by using the same arguments as in the proof of Theorem3.3, for all 0< λp/p−qμp/p−q<q/pp/p−qΛ1, we have thatz2_λ,μ

is a positive solution of SystemSλf,μg,h.

Now, we complete the proofs of Theorems 1.1 and 1.2: Theorem 1.1 follows from Theorem3.3. By Theorems3.3and3.4, we obtain that for all 1≤ q < p < N, p < αβ < p∗,

λ, μ > 0, and 0 < λp/p−q μp/p−q < q/pp/p−qΛ1. SystemSλf,μg,h has two positive

solutions z1

λ,μ, z2λ,μ with z1λ,μ ∈ Nλ,μ,z2λ,μ ∈ N−λ,μ. Since Nλ,μ∩ N−λ,μ ∅, this implies that

z1

λ,μandz2λ,μare distinct. This completes the proofs of Theorems1.1and1.2.

4. Proof of Theorem

1.3

For the existence of a second positive solution of SystemSλf,μg,hin the critical caseαβp∗,

we will however need here a stronger restriction onhx, namely,hx>0 inΩbutfxand

gxmay be also allowed to change sign inΩ. Now, we will establish the existence of a local minimum forIλ,μonN−_λ,μin the critical caseαβp∗to obtain a second positive solution of

SystemSλf,μg,h.

Lemma 4.1. If{zn} ⊂Wis aPSc-sequence forIλ,μ, then{zn}is bounded inW.

Proof. Letzn un, vn. We argue by contradiction. Assume thatzn → ∞. Let

zn un, vn zn

zn

un

zn

, vn

zn

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We may assume thatzn z u, vinW. This implies thatun → u, vn → vstrongly in

Ls_Ω_{for all 1}_≤_{s < p}∗_and

Kλ,μzn Kλ,μz on1. 4.2

Since{zn}is aPSc-sequence forIλ,μandzn → ∞, there hold

1

pzn

p₋znq−p

q Kλ,μzn −

znαβ−p

αβ Jzn on1, 4.3

zn p− znq−pKλ,μzn − znαβ−pJzn on1. 4.4

From4.2–4.4, we can deduce that

znp

pαβ−q

qαβ−pzn

q−p_K

λ,μzn on1. 4.5

Since 1≤q < pandzn → ∞,4.5implies

znp−→0, asn−→ ∞, 4.6

which is contrary to the factzn1.

Denote

Sα,β inf

u,v∈W1,p

0 Ω\{0}

Ω

|∇u|p|∇v|pdx

Ω|u|α|v|βdx

_p/αβ. 4.7

Modifying the proof of Alves et al.11, Theorem 5 , we can easily deduce that

Sα,β

_α

β

β/αβ

_β

α

α/αβ

S, 4.8

whereSis the best constant defined by

S inf

u∈W1,p

0 Ω\{0}

Ω|∇u|pdx

Ω|u|αβdx

_p/αβ. _4.9

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Uεx CN

ε1/p

ε|x|p/p−1

_N−p/p

, ε >0, 4.10

whereCN N−p/p−1p−1N

N−p/p2

.

Lemma 4.2. Letc_∞ 1/NSN/p_α,β |h|−N−p/p_∞ . If{zn} ⊂ W is aPSc-sequence forIλ,μwith c ∈

0, c∞, then there exists a subsequence of{zn}converging weakly to a nontrivial solution of System

Sλf,μg,h.

Proof. Let{zn} ⊂Wbe aPSc-sequence forIλ,μwithc∈0, c∞. Writezn un, vn. We know

from Lemma4.1that{zn}is bounded inW, and thenzn z u, vup to a subsequence,

whenzis a critical point ofIλ,μ. Furthermore, we may assumeun u,vn vinW₀1,pΩ

andun → u,vn → vinLsΩfor all 1≤s < p∗andun → u,vn → va.e. onΩ. Hence we

have thatI_λ,μ z 0 and

Kλ,μzn Kλ,μz on1. 4.11

Next we verify thatu /≡0 orv /≡0. Arguing by contradiction, we assumeu≡0 andv≡0. Set

l lim

n→ ∞Jzn. 4.12

SinceI_λ,μ zn on1and{zn}is bounded inW, then by4.11, we can deduce that

0

lim

n→ ∞I

λ,μzn, zn

lim

n→ ∞

znp−Jzn lim

n→ ∞zn

p₋_l, _4.13

that is,

lim

n→ ∞zn

p_l.

4.14

Ifl0, then we getclimn→ ∞Iλ,μzn 0, which contradicts withc >0. Thus we conclude

thatl >0. By4.7and4.12, we obtain

Sα,βlp/p∗Sα,βlim

n→ ∞

Ωhx|un|

α_|_vn_|β_dxp/p

∗

≤Sα,β_nlim_{→ ∞}

Ω|h|∞|un|

α_|_v

n|βdx

p/p∗

≤ |h|p/p∞ ∗_nlim_{→ ∞}znp

|h|p/p∞ ∗l,

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which implies that

l≥SN/p_α,β |h|−N−p/p∞ . 4.16

Hence, from4.11–4.16we get

c lim

n→ ∞Iλ,μzn

lim

n→ ∞

1

pzn

p₋1

qKλ,μzn−

1

αβJzn

1

p−

1

αβ

l

≥ 1

NS

N/p

α,β |h|

−N−p/p

∞ c∞.

4.17

This is a contradiction toc < c_∞. Thereforezis a nontrivial solution of SystemSλf,μg,h.

Lemma 4.3. Assume thatA1–A4hold. Then for anyλ, μ >0, there exist a nonnegative function

zλ,μ∈W\ {0}such that

sup

t≥0 Iλ,μ

tzλ,μ< c∞, _4.18

wherec∞is the constant given in Lemma4.2.

In particular,θ_λ,μ− < c_∞for all 0< λp/p−qμp/p−q<Λ1, whereΛ1is as in1.6.

Proof. FromA4, we know that there exists 0< ρ0≤r0such that, for allx∈B0,2ρ0,

hx h0 o|x|δ0 for someδ0 > N

p−1. 4.19

Now, we consider the functionalQ:W → Rdefined by

Qz 1

pz

p₋ 1

αβJz ∀z u, v∈W 4.20

and define a cut-oﬀfunctionηx ∈ C∞₀Ω such thatηx 1 for|x| < ρ0, ηx 0 for

|x|>2ρ0, 0≤η≤1, and|∇η| ≤C. Forε >0, let

uεx ε

N−p/p2

ηx

ε|x|p/p−1N−p/p

. _4.21

From Hsu7, Lemma 4.3 , we have

|∇uε|pp

RN|∇

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Ω|∇uε|pdx

Ωhx|uε|p

∗p/p∗ SO

εN−p/p, 4.23

whereUx 1|x|p/p−1−N−p/p. Setu0 p

√

αuε, v0 p

βuε, andz0 u0, v0∈W. Then, from4.8,4.23, and|h|∞1,

we conclude that

sup

t≥0Qtz0≤

1 N ⎛ ⎜ ⎝

αβ _Ω|∇uε|pdx

αα/p_ββ/p

Ωhx|uε|p

∗p/p∗

⎞ ⎟ ⎠ N/p ≤ 1 N _α β β/αβ _β α α/αβN/p

SOεN−p/pN/p

1 N _α β α/αβ _β α α/αβN/p

SN/pOεN−p/p

1

NS

N/p

α,β O

εN−p/p,

4.24

where the following fact has been used:

sup

t≥0

tp

pA−

tαβ

αβB

1 NA A B N−p/p 1 N A

Bp/p∗

N/p

, A, B >0. 4.25

Using the definitions ofIλ,μ,z0and byA3andA4, we get

Iλ,μtz0≤ t

p

pz0

p αβ

p t

p_|∇_uε_|p

p ∀t≥0, λ, μ >0. 4.26

Combining this with4.22, letε ∈0,1, then there existst0 ∈0,1independent ofεsuch

that

sup

0≤t≤t0Iλ,μtz0< c∞, ∀λ, μ >0, ∀ε∈0,1. 4.27

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sup

t≥t0Iλ,μtz0 supt≥t0

Qtz0−t

q

qKλ,μtz0

≤ 1

NS

N/p

α,β O

εN−p/p−t

q 0 q

a0αq/pλb0βq/pμ

B0,ρ0|uε|

q_dx

≤ 1

NS

N/p

α,β O

εN−p/p−t

q 0

qγ0

λμ

B0,ρ0|uε|

q_dx,

4.28

whereγ0 min{a0αq/p_{, b0β}q/p_}_.

Let 0< ε≤ρp/p−1₀ ; we have

B0,ρ0|uε|

q_dx

B0,ρ0

εqN−p/p2

ε|x|p/p−1N−p/pq

dx

≥

B0,ρ0

εqN−p/p2

2ρp/p−1₀ N−p/pq

dx

C1N, p, q, ρ0εqN−p/p2.

4.29

Combining with4.28and4.29, for allε∈0, ρ₀p/p−1, we get

sup

t≥t0Iλ,μtz0≤

1

NS

N/p

α,β O

εN−p/p−t

q 0

qC1γ0

λμεqN−p/p2. 4.30

Hence, for anyλ, μ >0, we can choose small positive constantελ,μ<min{1, ρp/p−10 }such that

OεN−p/p_λ,μ −t

q 0

qC1γ0

λμεqN−p/p_λ,μ 2<0. 4.31

Now, we fixελ,μand letzλ,μ √p_αuε λ,μ,

p

βuελ,μ. From4.27,4.30,4.31, we can deduce

that, for anyλ, μ >0, there exists a nonnegative functionzλ,μ∈W\ {0}such that

sup

t≥0Iλ,μ

tzλ,μ< c∞. _4.32

Finally, we prove thatθ−_λ,μ< c∞for all 0< λp/p−qμp/p−q<Λ1. Recall thatzλ,μ uλ,μ, vλ,μ

√p_αuε λ,μ,

p

βuελ,μ. ByA3,A4, and the definition ofuελ,μ, we have

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Combining this with Lemma2.6ii, from the definition ofθ_λ,μ− and4.32, for all 0< λp/p−q

μp/p−q<Λ1, we obtain that there existstλ,μ>0 such thattλ,μzλ,μ∈ N−_λ,μand

θ−_λ,μ≤Iλ,μtλ,μzλ,μ≤sup

t≥0Iλ,μ

tzλ,μ< c∞. _4.34

This completes the proof.

Theorem 4.4. Assume thatA1–A4hold. If 0< λp/p−qμp/p−q <q/pp/p−qΛ1, thenIλ,μ

satisfies theP S_θ−

λ,μcondition. Moreover,Iλ,μhas a minimizerz

2

λ,μinN−λ,μand satisfies the following:

iIλ,μz2_λ,μ θ−_λ,μ;

iiz2_λ,μis a positive solution of SystemSλf,μg,h, whereΛ1is the same as in1.6.

Proof. If 0< λp/p−q μp/p−q <q/pp/p−qΛ1, then by Theorem2.5ii, Proposition3.2ii,

and Lemma4.3, there exists aPS_θ−

λ,μ-sequence{zn} ⊂ N

−

λ,μinWforIλ,μwithθ−λ,μ∈0, c∞.

From Lemma4.2, there exist a subsequence still denoted by{zn}and a nontrivial solution

z2_λ,μ u2_λ,μ, v2_λ,μ∈ Wof SystemSλf,μg,hsuch thatzn z2_λ,μweakly inW. Now we prove thatzn → z2_λ,μstrongly inWandIλ,μz2_λ,μ θ−_λ,μ. By3.4, ifz∈ Nλ,μ, then

Iλ,μz

p∗−p

p∗p z

p₋p∗−q

p∗q Kλ,μz. 4.35

First, we prove thatz2

λ,μ∈ N−λ,μ. On the contrary, ifz2λ,μ∈ Nλ,μ, then byN−λ,μclosed inW, we

havez2

λ,μ<limn→ ∞zn. By Lemma2.3iandA4, we obtain that

Kλ,μz2_λ,μ>0, Jz2_λ,μ>0. 4.36

By Lemma2.7, there exists a uniquet− such thatt−z2

λ,μ ∈ N−λ,μ. Since zn ∈ Nλ,μ− ,Iλ,μzn ≥ Iλ,μtznfor allt≥0 and by4.35, we have

θ_λ,μ− ≤Iλ,μt−z2_λ,μ< lim

n→ ∞Iλ,μ

t−zn≤ lim

−

λ,μ, 4.37

and this is a contradiction.

In order to prove thatIλ,μz2

λ,μ θλ,μ− , it suﬃces to recall thatzn, z2λ,μ∈ N−λ,μfor alln,

by4.35. and apply Fatou’s lemma to get

θ−_λ,μ≤Iλ,μz2_λ,μ p

∗₋_p

p∗p

z2

λ,μ

p

−p∗−q

p∗q Kλ,μ

z2_λ,μ

≤lim inf

n→ ∞

_p_∗₋_p

p∗p zn

p₋p∗−q

p∗q Kλ,μzn

≤lim inf

n→ ∞ Iλ,μzn θ

−

λ,μ.

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This implies thatIλ,μz2_λ,μ θ−_λ,μand limn→ ∞znp z2λ,μ p

. Letzn zn−z2_λ,μ, then by Br´ezis and Lieb lemma26 implies

zn pznp−z2_λ,μp. 4.39

Therefore,zn → z2_λ,μstrongly inW.

SinceIλ,μz2

λ,μ Iλ,μ|z2λ,μ|and|z2λ,μ| ∈ N−λ,μ, by Lemmas2.2, and 2.3iii, we may

assume thatz2_λ,μis a nontrivial nonnegative solution of SystemSλf,μg,h. Finally, by using the same arguments as in the proof of Theorem3.3, for all 0< λp/p−qμp/p−q<q/pp/p−qΛ1,

we have thatz2_λ,μis a positive solution of SystemSλf,μg,h.

Proof of Theorem1.3. By Theorems3.3, and4.4, we obtain that for allλ, μ >0 and 0< λp/p−q

μp/p−q<q/pp/p−qΛ1,Sλf,μg,hhas two positive solutionsz_λ,μ1 ,z2_λ,μwithz_λ,μ1 ∈ N_λ,μ,z2_λ,μ∈

N−

λ,μ. SinceNλ,μ∩ N−λ,μ ∅, this implies that z1λ,μand z2λ,μ are distinct. This completes the

proof of Theorem1.3.

References

1 A. Ambrosetti, H. Brezis, and G. Cerami, “Combined eﬀects of concave and convex nonlinearities in some elliptic problems,” Journal of Functional Analysis, vol. 122, no. 2, pp. 519–543, 1994.

2 D. G. de Figueiredo, J.-P. Gossez, and P. Ubilla, “Local “superlinearity” and “sublinearity” for the -Laplacian,” Journal of Functional Analysis, vol. 257, no. 3, pp. 721–752, 2009.

3 J. G. Azorero and I. P. Peral, “Some results about the existence of a second positive solution in a quasilinear critical problem,” Indiana University Mathematics Journal, vol. 43, no. 3, pp. 941–957, 1994. 4 J. P. G. Azorero, I. Peral, and J. J. Manfredi, “Sobolev versus H ¨older local minimizers and global

multiplicity for some quasilinear elliptic equations,” Communications in Contemporary Mathematics, vol. 2, no. 3, pp. 385–404, 2000.

5 A. El Hamidi, “Multiple solutions with changing sign energy to a nonlinear elliptic equation,” Communications on Pure and Applied Analysis, vol. 3, no. 2, pp. 253–265, 2004.

6 N. Hirano, C. Saccon, and N. Shioji, “Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities,” Advances in Diﬀerential Equations, vol. 9, no. 1-2, pp. 197–220, 2004.

7 T.-S. Hsu, “Multiplicity results for P-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions,” Abstract and Applied Analysis, Article ID 652109, 24 pages, 2009. 8 T.-F. Wu, “On semilinear elliptic equations involving concave-convex nonlinearities and

sign-changing weight function,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 253– 270, 2006.

9 K. Adriouch and A. El Hamidi, “The Nehari manifold for systems of nonlinear elliptic equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 10, pp. 2149–2167, 2006.

10 A. Ahammou, “Positive radial solutions of nonlinear elliptic systems,” New York Journal of Mathematics, vol. 7, pp. 267–280, 2001.

11 C. O. Alves, D. C. D. M. Filho, and M. A. S. Souto, “On systems of elliptic equations involving subcritical or critical Sobolev exponents,” Nonlinear Analysis. Theory, Methods & Applications, vol. 42, no. 5, pp. 771–787, 2000.

12 Y. Bozhkov and E. Mitidieri, “Existence of multiple solutions for quasilinear systems via fibering method,” Journal of Diﬀerential Equations, vol. 190, no. 1, pp. 239–267, 2003.

13 Ph. Cl´ement, D. G. de Figueiredo, and E. Mitidieri, “Positive solutions of semilinear elliptic systems,” Communications in Partial Diﬀerential Equations, vol. 17, no. 5-6, pp. 923–940, 1992.

(21)

15 A. El Hamidi, “Existence results to elliptic systems with nonstandard growth conditions,” Journal of Mathematical Analysis and Applications, vol. 300, no. 1, pp. 30–42, 2004.

16 T.-S. Hsu, “Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2688–2698, 2009. 17 T.-S. Hsu and H.-L. Lin, “Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities,” Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 139, no. 6, pp. 1163–1177, 2009.

18 M. Squassina, “An eigenvalue problem for elliptic systems,” New York Journal of Mathematics, vol. 6, pp. 95–106, 2000.

19 J. V´elin, “Existence results for some nonlinear elliptic system with lack of compactness,” Nonlinear Analysis. Theory, Methods & Applications, vol. 52, no. 3, pp. 1017–1034, 2003.

20 T.-F. Wu, “The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 6, pp. 1733–1745, 2008. 21 Z. Nehari, “On a class of nonlinear second-order diﬀerential equations,” Transactions of the American

Mathematical Society, vol. 95, pp. 101–123, 1960.

22 M. Willem, Minimax Theorems, Birkh¨auser Boston, Boston, Mass, USA, 1996.

23 G. Tarantello, “On nonhomogeneous elliptic equations involving critical Sobolev exponent,” Annales de l’Institut Henri Poincar´e. Analyse Non Lin´eaire, vol. 9, no. 3, pp. 281–304, 1992.

24 K. J. Brown and Y. Zhang, “The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,” Journal of Diﬀerential Equations, vol. 193, no. 2, pp. 481–499, 2003.

25 K. J. Brown and T.-F. Wu, “A semilinear elliptic system involving nonlinear boundary condition and sign-changing weight function,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1326–1336, 2008.

26 H. Br´ezis and E. Lieb, “A relation between pointwise convergence of functions and convergence of functionals,” Proceedings of the American Mathematical Society, vol. 88, no. 3, pp. 486–490, 1983. 27 N. S. Trudinger, “On Harnack type inequalities and their application to quasilinear elliptic equations,”

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