The Nehari manifold for a semilinear elliptic
system involving sign-changing weight
functions
Tsung-fang Wu
Department of Applied Mathematics,
National University of Kaohsiung, Kaohsiung 811, Taiwan
Abstract
In this paper we study the combined effect of concave and convex nonlinearities on the number of solutions for a semilinear elliptic system (Eλ,µ) with sign-changing weight function. With the help of the Nehari manifold, we prove that system has at least two nontrivial nonnegative solutions when the pair of the parameters (λ, µ) belongs to a certain subset of R2.
Key words: Semilinear elliptic systems; Nehari manifold; Concave-convex
nonlinearities; Sign-changing weight functions
1 Introduction
In this paper we obtain multiplicity results of nontrivial nonnegative solu-tions 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 + β α+βh (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 satisfy the following conditions:
(A) f, g ∈ Lp∗
(Ω), where p∗ = α+β
α+β−q, and either f± = max {±f, 0} 6≡ 0 or
g± = max {±g, 0} 6≡ 0;
(B) h ∈ C³Ω´ with khk∞= 1 and h ≥ 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 a weak solution of problem (Eλ,µ) if Z Ω∇u∇ϕ1+ Z Ω∇v∇ϕ2− λ Z Ωf |u| q−2uϕ 1 − µ Z Ωg |v| q−2vϕ 2 − α α + β Z Ωh |u| α−2u |v|βϕ 1− β α + β Z Ωh |u| α|v|β−2vϕ 2 = 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 Ωf |u| q+ µZ Ωg |v| q¶− 1 α + β Z Ωh |u| α|v|β for (u, v) ∈ H.
Set α = β = p/2, u = v, λ = µ and f = g. Then problem (Eλ,µ) reduces to
the semilinear scalar elliptic equations with concave-convex nonlinearities:
−∆u = λf (x) |u|q−2u + h (x) |u|p−2u in Ω,
u = 0 in ∂Ω. (Eλ)
For 2 < p < 2∗ and the weight functions f ≡ h ≡ 1, the authors
Ambrosetti-Brezis-Cerami [3] have been investigated equation (Eλ) . They found that there
exists λ0 > 0 such that equation (Eλ) admits at least two positive solutions for
λ ∈ (0, λ0) , has a positive solution for λ = λ0 and no positive solution exists for λ > λ0. For more general case; we refer the reader to Ambrosetti-Azorezo-Peral [2], de Figueiredo-Gossez-Ubilla [10], EL Hamidi [11] and Wu [18,19], etc.. Recently, in [18,19] the author has considered a semilinear scalar ellip-tic equation involving concave-convex nonlinearities and sign-changing weight function, and showed multiplicity results with respect to the parameter λ via the extraction of Palais-Smale sequences in the Nehari manifold, where the definition of Nehari manifold we refer the reader to see Nehari [13] or Willem [17].
authors Adriouch-EL Hamidi [4] considered the following problems: −∆u = λu + α α+β|u| α−2u |v|β in Ω, −∆v = µ |v|q−2v + α+ββ |u|α|v|β−2v in Ω, u = v = 0 on ∂Ω,
they proved system has at least two positive solutions when the pair of the parameters (λ, µ) belongs to a certain subset of R2. For more similar problems, we refer the reader to Ahammou [1], Alves-de Morais Filho-Souto [5], Bozhkov-Mitidieri [6], Cl´ement-de Figueiredo -Bozhkov-Mitidieri [7], de Figueiredo-Felmer [9], EL Hamidi [12], Squassina [14] and V´elin [16], etc..
By the above results we know that the existence and multiplicity of positive solution of semilinear elliptic problems depends on the nonlinearity term. In this paper, we extend the method of [18,19] 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
con-ditions (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
³
u+0, v0+´and³u−0, v−0´such that u±
0 ≥ 0, v±0 ≥ 0 in Ω and u±0 6= 0, v±0 6= 0.
Furthermore, if f is a nonnegative function (or nonpositive function), then we have the following result.
Theorem 1.2 Suppose that the weight functions f, g, h are satisfying the
con-ditions (A) , (B) and in addition f ≥ 0 (≤ 0) . Then there exists an explicit number C (α, β, q, S, kgkLp∗) > 0 such that if λ ≤ 0 (≥ 0) and µ satisfies
0 < |µ| < C (α, β, q, S, kgkLp∗) ,
then problem (Eλ,µ) has at least two solutions
³ u+ 0, v0+ ´ and³u− 0, v−0 ´ such that u±0 ≥ 0, v±0 ≥ 0 in Ω and u±0 6= 0, v±0 6= 0.
The proof of Theorem 1.2 is similar to that of Theorem 1.1, for this reason it well be 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 = n (u, v) ∈ H\ {(0, 0)} | DJ0 λ,µ(u, v) , (u, v) E = 0o is the Nehari manifold and D J0 λ,µ(u, v) , (u, v) E = k(u, v)k2H − µ λ Z Ωf |u| q+ µZ Ωg |v| q¶−Z Ωh |u| α|v|β.
Note that Nλ,µ contains every nonzero solution of problem (Eλ,µ) .
Define Φλ,µ(u, v) = D J0 λ,µ(u, v) , (u, v) E . Then D Φ0λ,µ(u, v) , (u, v)E = 2 k(u, v)k2H − q µ λ Z Ωf |u| q+ µZ Ωg |v| q¶− (α + β)Z Ωh |u| α|v|β.
Moreover, for (u, v) ∈ Nλ,µ, if λ
R Ωf |u|q+ µ R Ωg |v|q 6= 0, then D Φ0 λ,µ(u, v) , (u, v) E = (2 − q) k(u, v)k2H − (α + β − q) Z Ωh |u| α|v|β. (2.2)
Similarly to the method used in Tarantello[15], we split Nλ,µ into three parts:
N+ λ,µ= n (u, v) ∈ Nλ,µ | D Φ0 λ,µ(u, v) , (u, v) E > 0o; N0 λ,µ= n (u, v) ∈ Nλ,µ | D Φ0 λ,µ(u, v) , (u, v) E = 0o; N− λ,µ= n (u, v) ∈ Nλ,µ | D Φ0 λ,µ(u, v) , (u, v) E < 0o.
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) , (2.3) 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 Ωf |u| q+ µZ Ωg |v| q = k(u, v)k2 H − Z Ωh |u| α|v|β > 0.
Thus, D Φ0 λ,µ(u, v) , (u, v) E = (2 − q) k(u, v)k2H − (α + β − q) Z Ωh |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,µ6= ∅ for all (λ, µ) ∈ R2\ {(0, 0)}. Then for each (u, v) ∈ N0λ,µ
we have λ Z Ωf |u| q+ µZ Ωg |v| q > 0 and 0 =DΦ0 λ,µ(u, v) , (u, v) E = (2 − q) k(u, v)k2H − (α + β − q) Z Ωh |u| α|v|β. Thus, k(u, v)k2H = α + β − q 2 − q Z Ωh |u| α|v|β (2.4) and λ Z Ωf |u| q+ µZ Ωg |v| q= k(u, v)k2 H − Z Ωh |u| α|v|β (2.5) =α + β − 2 2 − q Z Ωh |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 Ωh |u| α|v|β = λ Z Ωf |u| q+ µZ Ωg |v| q ≤ |λ| kf kLp∗ kukqLα+β + |µ| kgkLp∗ kvkqLα+β ≤ Sq³(|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 ≤ Ã (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 . (2.6) Let Iλ,µ: Nλ,µ→ R be given by Iλ,µ(u, v) = K (α, β, q) k(u, v)k 2(α+β−1) H R Ωh |u|α|v|β 1 α+β−2 − µ λ Z Ωf |u| q+ µZ Ωg |v| q¶,
where K (α, β, q) = ³α+β−q2−q ´
α+β−1
α+β−2 ³α+β−2
2−q
´
. Then from (2.4) and (2.5) it fol-lows that
Iλ,µ(u, v) = 0 for all (u, v) ∈ N0λ,µ. (2.7)
However, by (2.6) and the H¨older and Sobolev inequalities, for (u, v) ∈ N0
λ,µ Iλ,µ(u, v) ≥ K (α, β, q) k(u, v)k 2(α+β−1) H R Ωh |u|α|v|β 1 α+β−2 −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 ½ K (α, β, q) S−(α+β)α+β−2 k(u, v)k−(q+1) H −Sq³(|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´2−q 2 ) ≥ 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 ) , 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) =³D (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 (2.7). This
completes the proof. ¤
Lemma 2.1 suggests that we introduce 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) and θ
−
λ,µ= inf
(u,v)∈N−λ,µJλ,µ(u, v) .
The following lemma shows that the minimizers on Nλ,µare “usually” critical
points for Jλ,µ.
Lemma 2.2 For each (λ, µ) ∈ Θ and (u0, v0) 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, D Jλ,µ0 (u0, v0) , (u0, v0) E = ΛDΦ0λ,µ(u0, v0) , (u0, v0) E = 0. (2.8) But DΦ0 λ,µ(u0, v0) , (u0, v0) E 6= 0, since (u0, v0) /∈ N0λ,µ. Thus Λ = 0 by (2.8) .
This completes the proof. ¤
Lemma 2.3 We have (i) if (u, v) ∈ N+ λ,µ, then λ R Ωf |u|q+ µ R Ωg |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 Ωf |u| q+ µZ Ωg |v| q = k(u, v)k2 H − Z Ωh |u| α|v|β > 0.
Case (i − b) :RΩh |u|α|v|β > 0. Since k(u, v)k2H − µ λ Z Ωf |u| q+ µZ Ωg |v| q¶−Z Ωh |u| α|v|β = 0 and D Φ0 λ,µ(u, v) , (u, v) E = 2 k(u, v)k2H − q µ λ Z Ωf |u| q+ µZ Ωg |v| q¶− (α + β)Z Ωh |u| α|v|β > 0 it follows that (2 − q) µ λ Z Ωf |u| q+ µZ Ωg |v| q¶− (α + β − 2)Z Ωh |u| α|v|β > 0, which implies λ Z Ωf |u| q+ µZ Ωg |v| q> α + β − 2 2 − q Z Ωh |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
Ωh |u|
α|v|β = k(u, v)k2
Case (ii − b) : λRΩf |u|q+ µRΩg |v|q6= 0. We have (2 − q) k(u, v)k2H − (α + β − q) Z Ωh |u| α|v|β =DΦ0 λ,µ(u, v) , (u, v) E < 0. ThusRΩh |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 the following lemma holds.
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 ³ t+u, t+v´= inf
0≤t≤tmaxJλ,µ(tu, tv) , Jλ,µ(u, v) = supt≥tmaxJλ,µ(tu, tv) .
Proof. Fix (u, v) ∈ N− λ,µ. Let
m (t) = t2−qk(u, v)k2H − tα+β−q Z
Ωh |u|
α|v|β for t ≥ 0.
We have m(0) = 0, m(t) → −∞ as t → ∞, m (t) is achieves its maximum at
tmax, increasing for t ∈ [0, tmax) and decreasing for t ∈ (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 Ωh |u| α|v|β = k(u, v)kqH Ã 2 − q α + β − q ! 2−q α+β−2 − Ã 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 Ã α + β − 2 α + β − q ! Ã 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 Ã α + β − 2 α + β − q ! Ã 2 − q α + β − q ! 2−q α+β−2 µ 1 Sα+β ¶ 2−q α+β−2 . (2.9)
(i) λRΩf |u|q+ µRΩg |v|q ≤ 0.
There is a unique t− > t
max such that m (t−) = λ
R Ωf |u|q + µ R Ωg |v|q and h0(t−) < 0. Now, (2 − q)³t−´2k(u, v)k2H − (α + β − q)³t−´α+β Z Ωh |u| α|v|β =³t−´1+q · (2 − q)³t−´1−qk(u, v)k2 H − (α + β − q) ³ t−´α+β−q−1 Z Ωh |u| α|v|β¸ =³t−´1+qm0³t−´< 0, and D Jλ,µ0 ³t−u, t−v´,³t−u, t−v´E =³t−´2k(u, v)k2 H − ³ t−´q µ λ Z Ωf |u| q+ µZ Ωg |v| q¶−³t−´α+βZ Ωh |u| α|v|β =³t−´q · m³t−´− µ λ Z Ωf |u| q+ µZ Ωg |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 Ωh |tu| α|tv|β < 0, d2 dt2Jλ,µ(tu, tv) < 0 and d dtJλ,µ(tu, tv) = t k(u, v)k2H − tq µ λ Z Ωf |u| q+ µZ Ωg |v| q¶− tα+β Z Ωh |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 Ωh |u| α|v|β for all t ≥ 0. Similarly 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 (2.9) and
m (0) = 0 < λ Z Ωf |u| q+ µZ Ωg |v| q ≤ Sq³(|λ| kf kLp∗) 2 2−q + (|µ| kgk Lp∗) 2 2−q ´2−q 2 k(u, v)kqH < k(u, v)kqH Ã α + β − 2 α + β − q ! Ã 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−, m³t+´= λ Z Ωf |u| q+ µZ Ωg |v| q= m³t−´ and m0³t+´> 0 > m0³t−´. 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 ³ 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
equa-tion −∆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 Ωb |u| q
and the Nehari minimization problem
γσ,b = inf {Kσ,b(u) | u ∈ Mσ,b} , where Mσ,b = n u ∈ H1 0(Ω) \ {0} | D K0 σ,b(u) , u E
= 0o. Then we have the
fol-lowing results.
Theorem 2.5 (i) Suppose that the parameter σ 6= 0 and b ∈ Lp∗
(Ω) with
b± = max {±b, 0} 6≡ 0. Then for 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∗
0. Then for 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. For u ∈ Mσ,b, Z Ω|∇u| 2 = σZ Ωb |u| q ≤ |σ| kbk Lp∗ Sp µZ Ω|∇u| 2¶ q 2 . This implies Z Ω|∇u| 2 ≤ (|σ| kbk Lp∗ Sp) 2 2−q . (2.10) Hence Kσ,b(u) = 1 2 Z Ω|∇u| 2−σ q Z Ωb |u| q ≤ Ã 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 (2.10) 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). (2.11)
We claim that RΩb |wσ,b|q> 0. Otherwise, by (2.11) we can conclude that
Z Ωb |wn| qdx → 0 as n → ∞. Thus, kwnk2H1 = o (1) and Kσ,b(wn) = 1 2 Z Ω|∇wn| 2−σ q Z Ωb |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 shall prove that wn → wσ,b strongly in H01(Ω). Suppose the opposite that
Z Ω|∇wσ,b| 2 < lim inf n→∞ Z Ω|∇wn| 2 and hence Z Ω|∇wσ,b| 2 − σZ Ωb |wσ,b| q < lim inf n→∞ µZ Ω|∇wn| 2− σZ Ωb |vn| q¶= 0.
ByRΩb |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) < limn→∞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) The argument is similar to that in part (i) and is omitted here. ¤ Moreover, we have the following results.
Lemma 2.6 If the pair of parameters (λ, µ) ∈ Θ, then (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
by condition (B)
Jλ,µ(wλ,f, wµ,g) < min {γλ,f, γµ,g} . (2.12)
Similarly to the argument in Lemma 2.4 (ii) there exists t+= t+(w
λ,f, wµ,g) > 0 such that ³ t+w λ,f, t+wµ,g ´ ∈ N+ λ,µ (2.13) and Jλ,µ(wλ,f, wµ,g) ≥ inft≥0Jλ,µ(twλ,f, twµ,g) = Jλ,µ ³ t+wλ,f, t+wµ,g ´ . (2.14) Thus, by (2.12) − (2.14) we obtain θλ,µ≤ θ+λ,µ< min {γλ,f, γµ,g} < 0.
Jλ,µ(u, v) =α + β − 2 2 (α + β) k(u, v)k 2 H − à α + β − q q (α + β) ! µZ Ωλf |u| q+Z Ωµg |v| q¶ ≥α + β − 2 2 (α + β) k(u, v)k 2 H −Sq à 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−q2 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 (i) there exists a minimizing sequence {(un, vn)} ⊂ Nλ,µ such that
Jλ,µ(un, vn) = θλ,µ+ o (1) ,
J0
λ,µ(un, vn) = o (1) in H−1;
(ii) there exists a minimizing sequence {un} ⊂ N−λ,µ such that
Jλ,µ(un, vn) = θ−λ,µ+ o (1) ,
J0
λ,µ(un, vn) = o (1) in H−1.
Proof. The proof is almost the same as that in Wu [18, Proposition 9] and
is omitted 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
³ u+ 0, v+0 ´ in N+ λ,µ satisfying (i) Jλ,µ ³ u+0, v+0´= θλ,µ= θ+λ,µ< 0; (ii) ³u+ 0, v0+ ´
and u+0 6= 0, v0+ 6= 0; (iii) Jλ,µ ³ u+ 0, v+0 ´ → 0 as (λ, µ) → (0, 0) .
Proof. Let {(un, vn)} ⊂ Nλ,µbe a minimizing sequence for Jλ,µon Nλ,µsuch
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 sub-sequence {(un, vn)} and
³ u+ 0, v+0 ´ ∈ H such that ³u+ 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* v+0 weakly in H01(Ω), vn→ v+0 strongly in Lq(Ω) and in Lq 0 (Ω), where α p0 +qβ0 = 1. This implies Z Ωf |un| q→Z Ωf ¯ ¯ ¯u+0¯¯¯q as n → ∞, Z Ωg |vn| q→Z Ωf ¯ ¯ ¯v0+¯¯¯q as n → ∞, Z Ωh |un| α|v n|β→ Z Ωh ¯ ¯ ¯u+ 0 ¯ ¯ ¯α ¯ ¯ ¯v+ 0 ¯ ¯ ¯β as n → ∞. First, we claim that λRΩf¯¯¯u+
0 ¯ ¯ ¯q + µRΩg ¯ ¯ ¯v+ 0 ¯ ¯
¯q 6= 0. Otherwise, then we can
conclude that λ Z Ωf |un| q+ µZ Ωg |vn| q → 0 as n → ∞. Thus, k(u, v)k2H = Z Ωh |un| α|v n|β+ o (1) and Jλ,µ(un, vn) =1 2k(un, vn)k 2 H − 1 α + β Z Ωh |un| α|v n|β+ o (1) = Ã 1 2 − 1 α + β ! k(un, vn)k2H + o (1) .
This contradicts Jλ,µ(un) → θλ,µ < 0 as n → ∞. In particular,
³ u+0, v0+´ ∈ Nλ,µ and Jλ,µ ³ u+ 0, v+0 ´
≥ θλ,µ. Moreover, by Lemma 2.6 (i), we can conclude
u+
un→ u+0 strongly in H01(Ω),
vn→ v+0 strongly in H01(Ω).
Supposing the contrary, then either °°°u+ 0 ° ° ° H1 < lim infn→∞ kunkH1 or ° ° °v+ 0 ° ° ° H1 < lim inf n→∞ kvnkH1 and so ° ° ° ³ u+ 0, v+0 ´°° °2 H − µ λ Z Ωf ¯ ¯ ¯u+ 0 ¯ ¯ ¯q+ µ Z Ωg ¯ ¯ ¯v+ 0 ¯ ¯ ¯q ¶ − Z Ωh ¯ ¯ ¯u+ 0 ¯ ¯ ¯α ¯ ¯ ¯v+ 0 ¯ ¯ ¯β < lim inf n→∞ µ k(un, vn)k2H − µ λ Z Ωf |un| q+ µZ Ωg |vn| q¶−Z Ωh |un| α|v n|β ¶ = 0, which contradicts³u+ 0, v0+ ´ ∈ Nλ,µ. Hence un→ u+0 strongly in H01(Ω), vn→ v+0 strongly in H01(Ω). This implies Jλ,µ(un, vn) → Jλ,µ ³ 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 ³t0+u+0, t+0v+0´ ∈ N+λ,µ and ³t−0u+0, t−0v0+´ ∈ N−λ,µ,
we have t+0 < t−0 = 1. Since d dtJλ,µ ³ t+ 0u+0, t+0v0+ ´ = 0 and d 2 dt2Jλ,µ ³ t+ 0u+0, t+0v+0 ´ > 0, there exists t+ 0 < ¯t ≤ t−0 such that Jλ,µ ³ t+ 0u+0, t+0v+0 ´ < Jλ,µ ³ ¯tu+ 0, ¯tv+0 ´ . By Lemma 2.4, Jλ,µ ³ t+ 0u+0, t+0v+0 ´ < Jλ,µ ³ ¯tu+ 0, ¯tv0+ ´ ≤ Jλ,µ ³ t− 0u+0, t−0v0+ ´ = Jλ,µ ³ u+ 0, v+0 ´ ,
which is a contradiction. Since Jλ,µ
³ u+ 0, v0+ ´ = Jλ,µ ³¯ ¯ ¯u+ 0 ¯ ¯ ¯, ¯ ¯ ¯v+ 0 ¯ ¯ ¯ ´ and³¯¯¯u+ 0 ¯ ¯ ¯, ¯ ¯ ¯v+ 0 ¯ ¯ ¯ ´ ∈ N+
λ,µ, by Lemma 2.2 we may assume that
³ u+
0, v+0
´
is a solution of problem (Eλ,µ), such that u+0 ≥ 0, v0+ ≥ 0 in Ω and u+0 6= 0, v0+ 6= 0. Moreover, by Lemmas 2.6, Jλ,µ ³ u+ 0, v+0 ´ < 0 and Jλ,µ ³ u+ 0, v+0 ´ ≥ 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 ³ u+ 0, v+0 ´ → 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
³ u−0, v−0´ in N−λ,µ which satisfies (i) Jλ,µ ³ u− 0, v−0 ´ = θ− λ,µ;
(ii) ³u−0, v0−´ is a solution of problem (Eλ,µ), such that u−0 ≥ 0, v0− ≥ 0 in Ω
and u−0 6= 0, v0− 6= 0.
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) = θ−λ + 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
Ωh |un|
α|v
n|β < 0
there exist a subsequence {(un, vn)} and
³
u−0, v−0´∈ 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* v−0 weakly in H01(Ω), vn→ v−0 strongly in Lq(Ω) and in Lq 0 (Ω), where α p0 +qβ0 = 1. This implies Z Ωf |un| q→Z Ωf ¯ ¯ ¯u−0¯¯¯q as n → ∞, Z Ωg |vn| q→Z Ωf ¯ ¯ ¯v− 0 ¯ ¯ ¯q as n → ∞, Z Ωh |un| α|v n|β→ Z Ωh ¯ ¯ ¯u−0¯¯¯α ¯ ¯ ¯v0−¯¯¯β as n → ∞. Now we prove that
un→ u−0 strongly in H01(Ω),
vn→ v−0 strongly in H01(Ω). Otherwise, then either °°°u−0°°°
H1 < lim infn→∞ kunkH1 or ° ° °v0−°°° H1 < lim infn→∞ kvnkH1, and so
° ° ° ³ u− 0, v−0 ´° ° °2 H − µ λ Z Ωf ¯ ¯ ¯u− 0 ¯ ¯ ¯q+ µ Z Ωg ¯ ¯ ¯v− 0 ¯ ¯ ¯q ¶ − Z Ωh ¯ ¯ ¯u− 0 ¯ ¯ ¯α ¯ ¯ ¯v− 0 ¯ ¯ ¯β < lim inf n→∞ µ k(un, vn)k2H − µ λ Z Ωf |un| q+ µZ Ωg |vn| q¶−Z Ωh |un| α|v n|β ¶ = 0, which contradicts³u− 0, v0− ´ ∈ N− λ,µ. Hence un→ u−0 strongly in H01(Ω), vn→ v−0 strongly in H01(Ω). This implies Jλ,µ(un, vn) → Jλ,µ ³ u− 0, v0− ´ = θ− λ,µ as n → ∞. Since Jλ,µ ³ u− 0, v−0 ´ = Jλ,µ ³¯ ¯ ¯u− 0 ¯ ¯ ¯, ¯ ¯ ¯v− 0 ¯ ¯ ¯ ´ and ³¯¯¯u− 0 ¯ ¯ ¯, ¯ ¯ ¯v− 0 ¯ ¯ ¯ ´ ∈ N− λ,µ, by Lemma
2.2 we may assume that ³u−
0, v−0
´
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. ¤
Now, we complete the proof of Theorem 1.1: By Theorems 3.2, 3.3 problem (Eλ,µ) there exist two solutions
³
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