The Nehari manifold for a semilinear elliptic
system involving concave-convex nonlinearities
∗
Tsung-fang Wu
Department of Applied Mathematics
National University of Kaohsiung
Kaohsiung 811, Taiwan
e-mail: [email protected]
Abstract
In this paper, we study the combined effect of concave and convex non-linearities on the number of solutions for a semilinear elliptic system with sign-changing weight function. With the help of the Nehari manifold, we prove that system (Eλ,µ) has at least two nontrivial nonnegative solutions
when the pair of the parameters (λ, µ) belongs to a subset of R2.
1
Introduction
1In this paper, we consider the multiplicity results of nontrivial nonnegative
so-lutions of the following semilinear elliptic system: −∆u = λf (x) |u|q−2u + α α+βh (x) |u| α−2u |v|β in Ω, −∆v = µg (x) |v|q−2v +α+ββ h (x) |u|α|v|β−2v in Ω, u = v = 0 on ∂Ω, (Eλ,µ)
where Ω is a bounded domain in RN, α > 1, β > 1 satisfying 2 < α + β < 2∗
(2∗ = 2N
N −2 if N ≥ 3, 2∗ = ∞ if N = 2), 1 < q < 2, the pair of parameters
(λ, µ) ∈ R2\ {(0, 0)} and the weight functions f, g, h are satisfying the following
conditions: (A) f, g ∈ Lp∗
(Ω), where p∗ = α+β
α+β−q, and either f± = max {±f, 0} 6≡ 0 or
∗2000 Mathematics Subject Classification. 35J55, 35J60
g± = max {±g, 0} 6≡ 0;
(B) h ∈ C¡Ω¢ with khk∞= 1 and h+= max {h, 0} 6≡ 0.
Problem (Eλ,µ) is posed in the framework of the Sobolev space H = H01(Ω) ×
H1
0(Ω) with the standard norm
k(u, v)kH = µZ Ω |∇u|2+ Z Ω |∇v|2 ¶1 2 .
Moreover, a pair of functions (u, v) ∈ H is said to be weak solution of problem (Eλ,µ) if Z Ω ∇u∇ϕ1 + Z Ω ∇v∇ϕ2− λ Z Ω f |u|q−2uϕ1− µ Z Ω g |v|q−2vϕ2 − α α + β Z Ω h |u|α−2u |v|βϕ1− β α + β Z Ω h |u|α|v|β−2vϕ2 = 0 ∀ (ϕ1, ϕ2) ∈ H.
Thus, the corresponding energy functional of problem (Eλ,µ) is defined by
Jλ,µ(u, v) = 1 2k(u, v)k 2 H − 1 q µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶ − 1 α + β Z Ω h |u|α|v|β for (u, v) ∈ H.
Semilinear scalar elliptic equations with concave and convex nonlinearities are widely studied; we refer the reader to Brezis-Cerami [3], Ambrosetti-Azorezo-Peral [2], Bartsch-Willem [6], de Figueiredo-Gossez-Ubilla [10], EL Hamidi [11] and Wu [16] etc.. For the semilinear elliptic systems, we refer to Ahammou [1], Alves-de Morais Filho-Souto [4], Bozhkov-Mitidieri [5], Cl´ement-de Figueiredo -Mitidieri [7], de Figueiredo-Felmer [9], EL Hamidi [12], Squassina [13] and V´elin [15]. Recently, in [16] the author has consider a semilinear elliptic equation involv-ing concave-convex nonlinearities and sign-changinvolv-ing weight function, and showed multiplicity results with respect to the parameter via the extraction of Palais-Smale sequences in the Nehari manifold.
In this paper, we extend this method to consider the multiplicity of nontrivial nonnegative solutions of problem (Eλ,µ). Let S be the best Sobolev constant for
the embedding of H1
0(Ω) in Lα+β(Ω) . Then we have the following result.
Theorem 1.1 Suppose that the weight functions f, g, h are satisfying the
condi-tions (A) , (B) . Then there exists an explicit number C (α, β, q, S) > 0 such that if the parameters λ, µ satisfy
0 < (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q < C (α, β, q, S) ,
then problem (Eλ,µ) has at least two solutions
¡ u+ 0, v0+ ¢ and ¡u− 0, v0− ¢ such that u±0 ≥ 0, v0±≥ 0 in Ω and u±0 6= 0, v0±6= 0.
Furthermore, if either f or g is a nonnegative function (or nonpositive func-tion), then we have the following results.
Theorem 1.2 Suppose that the weight functions f, g, h are satisfying the
condi-tions (A) , (B) with in addition f ≥ 0 (≤ 0) . Then there exists an explicit number C (α, β, q, S, kgkLp∗) > 0 such that if the parameter λ ≤ 0 (≥ 0) and the parameter
µ satisfies
0 < |µ| < C (α, β, q, S, kgkLp∗) ,
then problem (Eλ,µ) has at least two solutions
¡ u+ 0, v0+ ¢ and ¡u− 0, v0− ¢ such that u±0 ≥ 0, v0±≥ 0 in Ω and u±0 6= 0, v0±6= 0.
Proof. Similar to the argument in Theorem 1.1 and omitted here. ¤ Theorem 1.3 Suppose that the weight functions f, g, h are satisfying the
condi-tions (A) , (B) with in addition g ≥ 0 (≤ 0) . Then there exists an explicit number
e
C (α, β, q, S, kf kLp∗) > 0 such that if the parameter µ ≤ 0 (≥ 0) and the parameter
λ satisfies
0 < |λ| < eC (α, β, q, S, kf kLp∗) ,
then problem (Eλ,µ) has at least two solutions
¡ u+ 0, v0+ ¢ and ¡u− 0, v0− ¢ such that u±0 ≥ 0, v0±≥ 0 in Ω and u±0 6= 0, v0±6= 0.
Proof. Similar to the argument in Theorem 1.1 and omitted here. ¤ This paper is organized as follows. In section 2, we give some notations and preliminaries. In section 3, we prove Theorem 1.1.
2
Notations and Preliminaries
First, we consider the Nehari minimization problem: for (λ, µ) ∈ R2\ {(0, 0)} ,
θλ,µ= inf {Jλ,µ(u, v) | (u, v) ∈ Nλ,µ} ,
where Nλ,µ= © (u, v) ∈ H\ {(0, 0)} | J0 λ,µ(u, v) , (u, v) ® = 0ª. Define Φλ,µ(u, v) = J0 λ,µ(u, v) , (u, v) ® = k(u, v)k2H − µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶ − Z Ω h |u|α|v|β.
Then for (u, v) ∈ Nλ,µ,
Φ0 λ,µ(u, v) , (u, v) ® = 2 k(u, v)kH − q µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶ − (α + β) Z Ω h |u|α|v|β.
Moreover, if λRΩf |u|q+ µRΩg |v|q 6= 0, then Φ0 λ,µ(u, v) , (u, v) ® = (2 − q) k(u, v)k2H − (α + β − q) Z Ω h |u|α|v|β.
Similarly to the method used in Tarantello[14], we split Nλ,µ into three parts:
N+ λ,µ = © (u, v) ∈ Nλ,µ | Φ0 λ,µ(u, v) , (u, v) ® > 0ª; N0λ,µ = ©(u, v) ∈ Nλ,µ | Φ0λ,µ(u, v) , (u, v)® = 0ª; N−λ,µ = ©(u, v) ∈ Nλ,µ | Φ0λ,µ(u, v) , (u, v)® < 0ª.
Then, we have the following results.
Lemma 2.1 There exists an explicit number C (α, β, q, S) > 0 such that if 0 < (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q < C (α, β, q, S) , (2) then N0 λ,µ= ∅.
Proof. We consider the following two cases. Case (a) : (u, v) ∈ Nλ,µ and
R Ωh |u| α|v|β ≤ 0. We have λ Z Ω f |u|q+ µ Z Ω g |v|q = k(u, v)k2H − Z Ω h |u|α|v|β > 0. Thus,
Φ0λ,µ(u, v) , (u, v)®= (2 − q) k(u, v)k2H − (α + β − q)
Z
Ω
h |u|α|v|β > 0
and so (u, v) /∈ N0 λ,µ.
Case (b) : (u, v) ∈ Nλ,µ and
R
Ωh |u|
α|v|β > 0.
Suppose that N0
λ,µ 6= ∅ for all (λ, µ) ∈ R2\ {(0, 0)}. Then for each (u, v) ∈ N0λ,µ
we have λ Z Ω f |u|q+ µ Z Ω g |v|q > 0 and 0 = Φ0 λ,µ(u, v) , (u, v) ® = (2 − q) k(u, v)k2H − (α + β − q) Z Ω h |u|α|v|β. Thus, k(u, v)k2H = α + β − q 2 − q Z Ω h |u|α|v|β (3)
and λ Z Ω f |u|q+ µ Z Ω g |v|q = k(u, v)k2H − Z Ω h |u|α|v|β (4) = α + β − 2 2 − q Z Ω h |u|α|v|β.
Moreover, by the H¨older and Sobolev inequalities
α + β − 2 α + β − qk(u, v)k 2 H = k(u, v)k 2 H − Z Ω h |u|α|v|β = λ Z Ω f |u|q+ µ Z Ω g |v|q ≤ |λ| kf kLp∗kukqLα+β + |µ| kgkLp∗ kvkqLα+β ≤ Sq ³ (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´2−q 2 k(u, v)kqH . This implies k(u, v)kH ≤ µ (α + β) − q (α + β) − 2 ¶ 1 2−q S2−qq ³ (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´1 2 . (5) Let Iλ,µ : Nλ,µ → R be given by Iλ,µ(u, v) = K (α, β, q) Ã k(u, v)k2(α+β−1)H R Ωh |u| α|v|β ! 1 α+β−2 − µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶ , where K (α, β, q) = ³ 2−q α+β−q ´α+β−1 α+β−2 ³α+β−2 2−q ´
. Then from (3) and (4) it follows that
However, by (5) and the H¨older and Sobolev inequalities, for (u, v) ∈ N0 λ,µ Iλ,µ(u, v) ≥ K (α, β, q) Ã k(u, v)k2(α+β−1)H R Ωh |u| α|v|β ! 1 α+β−2 −Sq ³ (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´2−q 2 k(u, v)kqH ≥ k(u, v)kqH n K (α, β, q) S−(α+β)α+β−2 k(u, v)k−(q+1) H −Sq³(|λ| kf k Lp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´2−q 2 ¾ ≥ k(u, v)kqH ( D (α, β, q, S) ³ (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´−(q+1) 2 −Sq³(|λ| kf k Lp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´2−q 2 ¾ , where D (α, β, q, S) = K (α, β, q) S−(α+β)α+β−2− q(q+1) 2−q ³ α+β−q α+β−2 ´−(q+1) 2−q
. This implies that
there exists an explicit number
C (α, β, q, S) = ¡D (α, β, q, S) S−q¢23 > 0 such that if 0 < (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q < C (α, β, q, S) ,
then we have Iλ,µ(u, v) > 0 for all (u, v) ∈ N0λ,µ, this contradicts (6). This
completes the proof. ¤
By Lemma 2.1, we denote the set Θ = n (λ, µ) ∈ R2\ {(0, 0)} | (|λ| kf k Lp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q < C (α, β, q, S) o .
Then for each (λ, µ) ∈ Θ, we write Nλ,µ= N+λ,µ∪ N−λ,µ and define
θ+ λ,µ= inf (u,v)∈N+λ,µJλ,µ(u, v) ; θ − λ,µ= inf (u,v)∈N−λ,µJλ,µ(u, v) .
The following lemma shows that the minimizers on Nλ,µ are ”usually” critical
points for Jλ,µ.
Lemma 2.2 For each (λ, µ) ∈ Θ and (u0, v0) is a local minimizer for Jλ,µ on
Proof. If (u0, v0) is a local minimizer for Jλ,µ on Nλ,µ, then (u0, v0) is a solution
of the optimization problem
minimize Jλ,µ(u, v) subject to Φλ,µ(u, v) = 0.
Hence, by the theory of Lagrange multipliers, there exists Λ ∈ R such that
Jλ,µ0 (u0, v0) = ΛΦ0λ,µ(u0, v0) in H−1. Thus, J0 λ,µ(u0, v0) , (u0, v0) ® = ΛΦ0 λ,µ(u0, v0) , (u0, v0) ® = 0. (7) Since (u0, v0) /∈ N0λ,µ. Thus, Φ0 λ,µ(u0, v0) , (u0, v0) ® 6= 0 and so by (7) Λ = 0. This
completes the proof. ¤
Lemma 2.3 We have
(i) if (u, v) ∈ N+λ,µ, then λRΩf |u|q+ µRΩg |v|q > 0;
(ii) if (u, v) ∈ N−λ,µ, then RΩh |u|α|v|β > 0.
Proof. (i) We consider the following two cases. Case (i − a) :RΩh |u|α|v|β ≤ 0. We have
λ Z Ω f |u|q+ µ Z Ω g |v|q = k(u, v)kH − Z Ω h |u|α|v|β > 0.
Case (i − b) :RΩh |u|α|v|β > 0. Since
k(u, v)kH − µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶ − Z Ω h |u|α|v|β = 0 and Φ0λ,µ(u, v) , (u, v)® = 2 k(u, v)kH − q µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶ − (α + β) Z Ω h |u|α|v|β > 0. Thus, (2 − q) µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶ − (α + β − 2) Z Ω h |u|α|v|β > 0, this implies λ Z Ω f |u|q+ µ Z Ω g |v|q > α + β − 2 2 − q Z Ω h |u|α|v|β > 0.
(ii) We consider the following two cases.
Case (ii − a) : λRΩf |u|q+ µRΩg |v|q= 0. We have Z
Ω
h |u|α|v|β = k(u, v)kH > 0.
Case (ii − b) : λRΩf |u|q+ µRΩg |v|q 6= 0. We have
(2 − q) k(u, v)k2H − (α + β − q)
Z
Ω
h |u|α|v|β =ψλ,µ0 (u, v) , (u, v)®< 0.
Thus, RΩh |u|α|v|β > 0. ¤
For each (u, v) ∈ N−λ,µ, we write
tmax = Ã (2 − q) k(u, v)k2H (α + β − q)RΩh |u|α|v|β ! 1 α+β−2 > 0.
Then we have the following lemma.
Lemma 2.4 For each (λ, µ) ∈ Θ and (u, v) ∈ N−λ,µ, we have
(i) if λRΩf |u|q+ µRΩg |v|q ≤ 0, then Jλ,µ(u, v) = supt≥0Jλ,µ(tu, tv) > 0;
(ii) if λRΩf |u|q+ µRΩg |v|q > 0, then there is a unique 0 < t+ = t+(u) < t max such that (t+u, t+v) ∈ N+ λ,µ and Jλ,µ ¡ t+u, t+v¢= inf 0≤t≤tmax
Jλ,µ(tu, tv) , Jλ,µ(u, v) = sup t≥tmax
Jλ,µ(tu, tv) .
Proof. Fix (u, v) ∈ N− λ,µ. Let m (t) = t2−qk(u, v)k2 H − tα+β−q Z Ω h |u|α|v|β for t ≥ 0.
We have m(0) = 0, m(t) → −∞ as t → ∞, m (t) is concave and achieves its maximum at tmax. Moreover,
m (tmax) = Ã (2 − q) k(u, v)k2H (α + β − q)RΩh |u|α|v|β ! 2−q α+β−2 k(u, v)k2H − Ã (2 − q) k(u, v)k2H (α + β − q)RΩh |u|α|v|β ! p−q α+β−2 Z Ω h |u|α|v|β = k(u, v)kqH "µ 2 − q α + β − q ¶ 2−q α+β−2 − µ 2 − q α + β − q ¶ p−q α+β−2 # Ã k(u, v)kα+βH R Ωh |u| α|v|β ! 2−q α+β−2 ≥ k(u, v)kqH µ α + β − 2 α + β − q ¶ µ 2 − q α + β − q ¶ 2−q α+β−2 µ 1 Sα+β ¶ 2−q α+β−2 ,
or m (tmax) ≥ k(u, v)kqH µ α + β − 2 α + β − q ¶ µ 2 − q α + β − q ¶ 2−q α+β−2 µ 1 Sα+β ¶ 2−q α+β−2 . (8) (i) λRΩf |u|q+ µRΩg |v|q ≤ 0. There is a unique t− > t
max such that m (t−) = λ
R Ωf |u| q + µR Ωg |v| q and h0(t−) < 0. Now, (2 − q)¡t−¢2k(u, v)k2 H − (α + β − q) ¡ t−¢α+β Z Ω h |u|α|v|β = ¡t−¢1+q · (2 − q)¡t−¢1−qk(u, v)k2 H − (α + β − q) ¡ t−¢α+β−q−1 Z Ω h |u|α|v|β ¸ = ¡t−¢1+qm0¡t−¢< 0, and J0 λ,µ ¡ t−u, t−v¢,¡t−u, t−v¢® = ¡t−¢2k(u, v)k2H −¡t−¢q µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶ −¡t−¢α+β Z Ω h |u|α|v|β = ¡t−¢q · m¡t−¢− µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶¸ = 0. Thus, (t−u, t−v) ∈ N−
λ,µ or t− = 1. Since for t > tmax, we have
(2 − q) k(tu, tv)k2H − (α + β − q) Z Ω h |tu|α|tv|β < 0, d 2 dt2Jλ,µ(tu, tv) < 0 and d dtJλ,µ(tu, tv) = t k(u, v)k2H − tq µ λ Z Ω f |u|q+ µ Z Ω g |v|q ¶ − tα+β Z Ω h |u|α|v|β = 0 for t = t−.
Thus, Jλ,µ(u, v) = supt≥0Jλ,µ(tu, tv) . Moreover,
Jλ,µ(u, v) ≥ Jλ,µ(tu, tv) ≥ t2 2 k(u, v)k 2 H − tα+β α + β Z Ω
h |u|α|v|β for all t ≥ 0.
Similar to the argument in the function m (t), we obtain
Jλ,µ(u, v) ≥ α + β − 2 2 (α + β) Ã k(u, v)kα+βH R Ωh |u| α|v|β ! 2 α+β−2 > 0.
(ii) λRΩf |u|q+ µRΩg |v|q > 0. By (8) and m (0) = 0 < λ Z Ω f |u|q+ µ Z Ω g |v|q ≤ Sq³(|λ| kf k Lp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´2−q 2 k(u, v)kqH < k(u, v)kqH µ α + β − 2 α + β − q ¶ µ 2 − q α + β − q ¶ 2−q α+β−2 µ 1 Sα+β ¶ 2−q α+β−2 ≤ m (tmax) for (λ, µ) ∈ Θ,
there are unique t+ and t− such that 0 < t+< t
max < t−, m¡t+¢ = λ Z Ω f |u|q+ µ Z Ω g |v|q = m¡t−¢ and m0¡t+¢ > 0 > m0¡t−¢. We have (t+u, t+v) ∈ N+ λ,µ, (t−u, t−v) ∈ N−λ,µ, and Jλ,µ(t−u, t−v) ≥ Jλ,µ(tu, tv) ≥
Jλ,µ(t+u, t+v) for each t ∈ [t+, t−] and Jλ,µ(t+u, t+v) ≤ Jλ,µ(tu, tv) for each
t ∈ [0, t+] . Thus, t− = 1 and Jλ,µ(u, v) = sup t≥0 Jλ,µ(tu, tv) , Jλ,µ ¡ t+u, t+v¢= inf 0≤t≤tmax Jλ,µ(tu, tv) .
This completes the proof. ¤
Next, we establish the existence of solutions for the semilinear elliptic equation −∆u = σb (x) |u|q−2u in Ω, u ≥ 0, u 6= 0 u = 0 on ∂Ω, (Eσ,b)
where the parameter σ 6= 0 and b ∈ Lp∗
(Ω) with b+ = max {b, 0} 6≡ 0. Associated
with equation (Eσ,b) , we consider the energy functional
Kσ,b(u) = 1 2 Z Ω |∇u|2− σ q Z Ω b |u|q
and the Nehari minimization problem
γσ,b = inf {Kσ,b(u) | u ∈ Mσ,b} , where Mσ,b = © u ∈ H1 0 (Ω) \ {0} | K0 σ,b(u) , u ®
= 0ª. Then we have the
Theorem 2.5 (i) Suppose that the parameter σ 6= 0 and b ∈ Lp∗
(Ω) with
b± = max {±b, 0} 6≡ 0. Then equation (E
σ,b) there exists a solution wσ,b such
that Kσ,b(wσ,b) = γσ,b < 0;
(ii) Suppose that the parameter σ > 0 and b ∈ Lp∗
(Ω) with b+ = max {b, 0} 6≡ 0.
Then equation (Eσ,b) there exists a solution wσ,b such that Kσ,b(wσ,b) = γσ,b < 0.
Proof. (i) First, we need to show that Kσ,b is bounded below on Mσ,b and
γσ,b < 0. Then for u ∈ Mσ,b, Z Ω |∇u|2 = σ Z Ω b |u|q ≤ |σ| kbkLp∗Sp µZ Ω |∇u|2 ¶q 2 . This implies Z Ω |∇u|2 ≤ (|σ| kbkLp∗ Sp) 2 2−q . (9) Hence Kλ(u) = 1 2 Z Ω |∇u|2−σ q Z Ω b |u|q ≤ µ 1 2− 1 q ¶ (|σ| kbkLp∗ Sp) 2 2−q for all u ∈ M σ,b.
This implies γσ,b < 0. Let {wn} be a minimizing sequence for Kσ,b on Mσ,b, then
by (9) and the compact imbedding theorem, there exist a subsequence {wn} and
wσ,b in H01(Ω) such that
wn * wσ,b weakly in H01(Ω)
and
wn→ wσ,b strongly in Lq(Ω). (10)
First, we claim that RΩb |wσ,b|q > 0. If not, by (10) we can conclude that
Z Ω b |wn|qdx → 0 as n → ∞. Thus, kwnk2H1 = o (1) and Kσ,b(wn) = 1 2 Z Ω |∇wn|2− σ q Z Ω b |wn|q → 0 as n → ∞,
this contradicts Kσ,b(wn) → γσ,b < 0 as n → ∞. Thus,
R
Ωb |wλ,b|
q > 0. In
particular, wλ,b 6≡ 0. Now, we prove that wn → wσ,b strongly in H01(Ω). Suppose
otherwise, then Z Ω |∇wσ,b|2 < lim inf n→∞ Z Ω |∇wn|2
and so Z Ω |∇wσ,b|2− σ Z Ω b |wσ,b|q< lim inf n→∞ µZ Ω |∇wn|2− σ Z Ω b |vn|q ¶ = 0. By RΩb |wσ,b|q > 0, there is a unique t0 6= 1 such that t0wσ,b ∈ Mσ,b. Thus,
t0wn* t0wσ,b weakly in H01(Ω).
Moreover,
Kσ,b(t0wσ,b) < Kσ,b(wσ,b) < lim
n→∞Kσ,b(wn) = γσ,b,
which is a contradiction. Hence wn → wσ,b strongly in H01(Ω). This implies
wσ,b ∈ Mσ,b and
Kσ,b(wσ,b) → Kσ,b(wσ,b) = γσ,b as n → ∞.
Since Kσ,b(wσ,b) = Kσ,b(|wσ,b|) and |wσ,b| ∈ Mσ,b, without loss of generality, we
may assume that wσ,b is a solution of equation (Eσ,b) .
(ii) Similar to the argument in part (i) and omitted here. ¤ Moreover, we have the following results.
Lemma 2.6 If the pair of parameters (λ, µ) ∈ Θ, then we have (i) θλ,µ≤ θλ,µ+ < min {γλ,f, γµ,g} < 0;
(ii) Jλ,µ is coercive and bounded below on Nλ,µ.
Proof. (i) Let wλ,f and wµ,g be a solution of equation (Eλ,f) and equation (Eµ,g) ,
respectively, such that Kλ,f(wλ,f) = γλ,f and Kµ,g(wµ,g) = γµ,g. Then
Jλ,µ(wλ,f, wµ,g) < min {γλ,f, γλ,g} . (11)
Similar to the argument in Lemma 2.4 (ii) there exists t+ = t+(w
λ,f, wµ,g) > 0 such that ¡ t+w λ,f, t+wµ,g ¢ ∈ N+ λ,µ (12) and Jλ,µ(wλ,f, wµ,g) ≥ inf t≥0Jλ,µ(twλ,f, twµ,g) = Jλ,µ ¡ t+w λ,f, t+wµ,g ¢ . (13) Thus, by (11) − (13) we obtain θλ,µ≤ θλ,µ+ < min {γλ,f, γµ,g} < 0.
(ii) If (u, v) ∈ Nλ,µ, then by the H¨older inequality Jλ,µ(u, v) = α + β − 2 2 (α + β) k(u, v)k 2 H − µ α + β − q q (α + β) ¶ µZ Ω λf |u|q+ Z Ω µg |v|q ¶ ≥ α + β − 2 2 (α + β) k(u, v)k 2 H −Sq µ α + β − q q (α + β) ¶ ³ (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´2−q 2 k(u, v)kqH ≥ S 2q 2−q (q − 2) (α + β − q) 2 2−q 2q (α + β) (α + β − 2)2−qq ³ (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´ .
Thus, Jλ,µ is coercive on Nλ,µ and
Jλ,µ(u, v) ≥ S2−q2q (q − 2) (α + β − q) 2 2−q 2q (α + β) (α + β − 2)2−qq ³ (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´ .
This completes the proof. ¤
3
Proof of Theorem 1
Proposition 3.1 If the pair of parameters (λ, µ) ∈ Θ, then we have (i) there exists a minimizing sequence {(un, vn)} ⊂ Nλ,µ such that
Jλ,µ(un, vn) = θλ,µ+ o (1) ,
Jλ,µ0 (un, vn) = o (1) in H−1;
(ii) there exists a minimizing sequence {un} ⊂ N−λ,µ such that
Jλ,µ(un, vn) = θλ,µ− + o (1) ,
J0
λ,µ(un, vn) = o (1) in H−1.
Proof. Our proof is almost the same as that in Wu [16, Proposition 9]. We will
omit detailed proof here. ¤
Now, we establish the existence of a local minimum for Jλ,µ on N+λ,µ.
Theorem 3.2 If the pair of parameters (λ, µ) ∈ Θ, then Jλ,µ has a minimizer
¡ u+ 0, v0+ ¢ in N+ λ,µ and it satisfies (i) Jλ,µ ¡ u+ 0, v0+ ¢ = θλ,µ= θ+λ,µ< 0; (ii) ¡u+ 0, v0+ ¢
is a solution of problem (Eλ,µ), such that u+0 ≥ 0, v+0 ≥ 0 in Ω and
u+
0 6= 0, v+0 6= 0;
(iii) Jλ,µ
¡
Proof. Let {(un, vn)} ⊂ Nλ,µ be a minimizing sequence for Jλ,µ on Nλ,µ such
that
Jλ,µ(un, vn) = θλ+ o (1) and Jλ,µ0 (un, vn) = o (1) in H−1.
Then by Lemma 2.6 and the compact imbedding theorem, there exist a subse-quence {(un, vn)} and
¡
u+0, v+0¢ ∈ H such that ¡u+0, v0+¢ is a solution of problem (Eλ,µ) and un * u+0 weakly in H01(Ω), un → u+0 strongly in Lq(Ω) and in Lp 0 (Ω), vn * v0+ weakly in H01(Ω), vn → v0+ strongly in Lq(Ω) and in Lq 0 (Ω), where α p0 +qβ0 = 1. This implies Z Ω f |un|q → Z Ω f¯¯u+ 0 ¯ ¯q as n → ∞, Z Ω g |vn|q → Z Ω f¯¯v+ 0 ¯ ¯q as n → ∞, Z Ω h |un|α|vn|β → Z Ω h¯¯u+0¯¯α¯¯v0+¯¯β as n → ∞.
First, we claim that λRΩf¯¯u+0¯¯q+ µRΩg¯¯v0+¯¯q 6= 0. If not, then we can conclude
that λ Z Ω f |un|q+ µ Z Ω g |vn|q → 0 as n → ∞. Thus, k(u, v)k2H = Z Ω h |un|α|vn|β + o (1) and Jλ,µ(un, vn) = 1 2k(un, vn)k 2 H − 1 α + β Z Ω h |un|α|vn|β + o (1) = µ 1 2 − 1 α + β ¶ k(un, vn)k2H + o (1) ,
this contradicts Jλ,µ(un) → θλ,µ < 0 as n → ∞. In particular,
¡ u+ 0, v+0 ¢ ∈ Nλ,µ and Jλ,µ ¡
u+0, v0+¢ ≥ θλ,µ. Moreover, by Lemma 2.6 (i), we can conclude u+0 6=
0, v+
0 6= 0. Now we prove that
un → u+0 strongly in H01(Ω),
Supposing the contrary, then either °°u+ 0 ° ° H1 < lim inf n→∞ kunkH1 or ° °v+ 0 ° ° H1 < lim inf n→∞ kvnkH1 and so ° °¡u+ 0, v0+ ¢°°2 H − µ λ Z Ω f¯¯u+ 0 ¯ ¯q + µ Z Ω g¯¯v+ 0 ¯ ¯q ¶ − Z Ω h¯¯u+ 0 ¯ ¯α¯¯v+ 0 ¯ ¯β < lim inf n→∞ µ k(un, vn)k2H − µ λ Z Ω f |un|q+ µ Z Ω g |vn|q ¶ − Z Ω h |un|α|vn|β ¶ = 0,
this contradicts ¡u+0, v0+¢ ∈ Nλ,µ. Hence
un → u+0 strongly in H01(Ω), vn → v0+ strongly in H01(Ω). This implies Jλ,µ(un, vn) → Jλ,µ ¡ u+ 0, v+0 ¢ = θλ as n → ∞. Moreover, we have u+
0 ∈ N+λ,µ. In fact, if u+0 ∈ N−λ,µ, by Lemma 2.4, there are
unique t+
0 and t−0 such that
¡ t+ 0u+0, t+0v+0 ¢ ∈ N+ λ,µ and ¡ t− 0u+0, t−0v0+ ¢ ∈ N− λ,µ, we have t+ 0 < t−0 = 1. Since d dtJλ,µ ¡ t+0u+0, t+0v+0¢= 0 and d 2 dt2Jλ,µ ¡ t+0u+0, t+0v0+¢> 0, there exists t+ 0 < ¯t ≤ t−0 such that Jλ,µ ¡ t+ 0u+0, t+0v0+ ¢ < Jλ,µ ¡ ¯tu+ 0, ¯tv0+ ¢ . By Lemma 2.4, Jλ,µ ¡ t+ 0u+0, t+0v+0 ¢ < Jλ,µ ¡ ¯tu+ 0, ¯tv+0 ¢ ≤ Jλ,µ ¡ t− 0u+0, t−0v0+ ¢ = Jλ,µ ¡ u+ 0, v0+ ¢ ,
which is a contradiction. Since Jλ,µ
¡ u+ 0, v0+ ¢ = Jλ,µ ¡¯ ¯u+ 0 ¯ ¯ ,¯¯v+ 0 ¯ ¯¢and¡¯¯u+ 0 ¯ ¯ ,¯¯v+ 0 ¯ ¯¢∈ N+
λ,µ, by Lemma 2.2 we may assume that
¡ u+ 0, v0+ ¢ is a solution of problem (Eλ,µ), such that u+
0 ≥ 0, v+0 ≥ 0 in Ω and u+0 6= 0, v0+6= 0. Moreover, by Lemmas 2.6,
0 > Jλ,µ ¡ u+ 0, v0+ ¢ ≥ S 2q 2−q (q − 2) (α + β − q) 2 2−q 2q (α + β) (α + β − 2)2−qq ³ (|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´ . We obtain Jλ,µ ¡ u+0, v0+¢→ 0 as (λ, µ) → (0, 0) . ¤ Next, we establish the existence of a local minimum for Jλ,µ on N−λ,µ
Theorem 3.3 If the pair of parameters (λ, µ) ∈ Θ, then Jλ,µ has a minimizer
¡ u− 0, v0− ¢ in N− λ,µ and it satisfies (i) Jλ,µ ¡ u− 0, v0− ¢ = θ− λ,µ; (ii) ¡u− 0, v0− ¢
is a solution of problem (Eλ,µ), such that u−0 ≥ 0, v−0 ≥ 0 in Ω and
Proof. By Proposition 3.1 (ii), there exists a minimizing sequence {(un, vn)} for
Jλ,µ on N−λ,µ such that
Jλ,µ(un, vn) = θλ−+ o (1) and Jλ,µ0 (un, vn) = o (1) in H−1.
By Lemma 2.6, the compact imbedding theorem and (2 − q) k(un, vn)k2H − (α + β − q)
Z
Ω
h |un|α|vn|β < 0
there exist a subsequence {(un, vn)} and
¡ u− 0, v0− ¢ ∈ N− λ,µ is a nonzero solution of
problem (Eλ,µ) such that
un * u−0 weakly in H01(Ω), un → u−0 strongly in Lq(Ω) and in Lp 0 (Ω), vn * v0− weakly in H01(Ω), vn → v0− strongly in Lq(Ω) and in Lq 0 (Ω), where α p0 +qβ0 = 1. This implies Z Ω f |un|q → Z Ω f¯¯u− 0 ¯ ¯q as n → ∞, Z Ω g |vn|q → Z Ω f¯¯v− 0 ¯ ¯q as n → ∞, Z Ω h |un|α|vn|β → Z Ω h¯¯u− 0 ¯ ¯α¯¯v− 0 ¯ ¯β as n → ∞. Now we prove that
un → u−0 strongly in H01(Ω),
vn → v0− strongly in H01(Ω).
Suppose otherwise, then either°°u− 0 ° ° H1 < lim inf n→∞ kunkH1 or ° °v− 0 ° ° H1 < lim inf n→∞ kvnkH1, and so ° °¡u− 0, v0− ¢°°2 H − µ λ Z Ω f¯¯u− 0 ¯ ¯q + µ Z Ω g¯¯v− 0 ¯ ¯q ¶ − Z Ω h¯¯u− 0 ¯ ¯α¯¯v− 0 ¯ ¯β < lim inf n→∞ µ k(un, vn)k2H − µ λ Z Ω f |un|q+ µ Z Ω g |vn|q ¶ − Z Ω h |un|α|vn|β ¶ = 0,
this contradicts ¡u−0, v0−¢ ∈ N−λ,µ. Hence
un → u−0 strongly in H01(Ω),
This implies Jλ,µ(un, vn) → Jλ,µ ¡ u− 0, v0− ¢ = θ− λ,µ as n → ∞. Since Jλ,µ ¡ u− 0, v−0 ¢ = Jλ,µ ¡¯ ¯u− 0 ¯ ¯ ,¯¯v− 0 ¯ ¯¢ and ¡¯¯u− 0 ¯ ¯ ,¯¯v− 0 ¯ ¯¢ ∈ N− λ,µ, by Lemma 2.2
we may assume that ¡u− 0, v0−
¢
is a solution of problem (Eλ,µ), such that u−0 ≥
0, v0− ≥ 0 in Ω and u−0 6= 0, v−0 6= 0. ¤ Now, we complete the proof of Theorem 1.1: By Theorems 3.2, 3.3 problem (Eλ,µ) there exist two solutions
¡ u+ 0, v+0 ¢ ∈ N+ λ,µ and ¡ u− 0, v0− ¢ ∈ N− λ,µ such that u±
0 ≥ 0, v0±≥ 0 in Ω and u±0 6= 0, v0±6= 0. Since N+λ,µ∩ N−λ,µ= ∅, this implies that
¡ u+ 0, v0+ ¢ and ¡u− 0, v0− ¢ are different.
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