A semilinear elliptic system involving
nonlinear boundary condition and
sign-changing weight function
K. J. Brown
Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK
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) involving nonlinear boundary condition and 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; Nonlinear boundary conditions; Nehari manifold
1 Introduction
In this paper, we consider the multiplicity results of nontrivial nonnegative solutions of the following semilinear elliptic system:
−∆u + u = α α+βf (x) |u| α−2 u |v|β in Ω, −∆v + v = β α+βf (x) |u| α|v|β−2 v in Ω, ∂u ∂n = λg (x) |u| q−2 u,∂n∂v = µh (x) |v|q−2v on ∂Ω, (Eλ,µ)
Email addresses: [email protected] (K. J. Brown), [email protected] (Tsung-fang Wu).
where Ω is a bounded domain in RN with smooth boundary, α > 1, β > 1
satisfying 2 < α + β < 2∗ (2∗ = N −22N 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 ∈ CΩ with kf k∞= 1 and f+ = max {f, 0} 6≡ 0;
(B) g, h ∈ C (∂Ω) with kgk∞ = khk∞ = 1, and either g± = max {±g, 0} 6≡ 0 and h± = max {±h, 0} 6≡ 0.
Semilinear elliptic problems with nonlinear boundary condition are widely studied; we refer the reader to Garcia-Azorero, Peral and Rossi [5] and Wu [6,7]. Recently, in [6] the author considered a semilinear elliptic equation in-volving sign-changing weight function, and showed multiplicity results with respect to the parameter via the extraction of Palais–Smale sequences in the Nehari manifold.
Because the two sublinear boundary conditions in problem (Eλ,µ) are
ho-mogeneous of the same degree q − 1 and so the problem (Eλ,µ) is similar to
the Ambrosetti-Brezis-Cerami problem [1] (a semilinear elliptic equation in-volving concave and convex nonlinearities). Thus, the existence of more than one nontrivial solution for problem (Eλ,µ) is expected. In this paper, we give
a very simple variational method which is similar to the ”fibering method” of Pohozaev’s (see [3] or [4]) to prove the existence of at least two nontriv-ial nonnegative solutions of problem (Eλ,µ). In particular we do this without
the extraction of Palais–Smale sequences in the Nehari manifold. Throughout this section, we let S and S be the best Sobolev and the best Sobolev trace constants for the embedding of H1
0 (Ω) in Lα+β(Ω) and H01(Ω) in Lq(∂Ω) ,
re-spectively. And let C0 =
q
2
2/(2−q)
Cα, β, q, S, Sbe a positive number where Cα, β, q, S, S = α+β−q2−q Sα+β2/(2−α−β)α+β−2
α+β−qS −q 2
2−q
. Then we have the following result.
Theorem 1.1 If the parameters λ, µ satisfy
0 < |λ|2−q2 + |µ| 2 2−q < C
0,
then problem (Eλ,µ) has at least two solutions
u+0, v+0 and u−0, v0− such that u±0 ≥ 0, v0± ≥ 0 in Ω and u±0 6= 0, v0± 6= 0. Furthermore, if f ≥ 0, then u±0 > 0, v0± > 0 in Ω.
This paper is organized as follows. In section 2, we give some the properties of the Nehari manifold. In section 3, we prove Theorem 1.1.
2 Nehari manifold
Problem (Eλ,µ) is posed in the framework of the Sobolev space H = H1(Ω)×
H1(Ω) with the standard norm
k(u, v)kH = Z Ω |∇u|2+ u2dx + Z Ω |∇v|2+ v2dx 12 .
Moreover, a pair of functions (u, v) ∈ H is said to be a weak solution of problem (Eλ,µ) if Z Ω (∇u∇ϕ1+ uϕ1) dx + Z Ω (∇v∇ϕ2+ vϕ2) dx − α α + β Z Ω f |u|α−2u |v|βϕ1dx − β α + β Z Ω f |u|α|v|β−2vϕ2dx − λ Z ∂Ω g |u|q−2uϕ1ds − µ Z ∂Ω h |v|q−2vϕ2ds = 0
for all (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 α + β Z Ω f |u|α|v|βdx − 1 qKλ,µ(u, v)
for (u, v) ∈ H, where Kλ,µ(u, v) = λ
R ∂Ωg |u| q ds + µR ∂Ωh |v| q ds.
As the energy functional Jλ,µ is not bounded below on H, it is useful to
consider the functional on the Nehari manifold Nλ,µ=
n
(u, v) ∈ H\ {(0, 0)} | DJλ,µ0 (u, v) , (u, v)E= 0o.
Thus, (u, v) ∈ Nλ,µ if and only if
D
Jλ,µ0 (u, v) , (u, v)E= k(u, v)k2H −
Z
Ω
f |u|α|v|βdx − Kλ,µ(u, v) = 0. (2.1)
Note that Nλ,µ contains every nonzero solution of problem (Eλ,µ) . Moreover,
we have the following results.
Lemma 2.1 The energy functional Jλ,µ is coercive and bounded below on
Nλ,µ.
Jλ,µ(u, v) = α + β − 2 2 (α + β) k(u, v)k 2 H − α + β − q q (α + β) ! Kλ,µ(u, v) ≥α + β − 2 2 (α + β) k(u, v)k 2 H −Sq α + β − q q (α + β) ! |λ|2−q2 + |µ| 2 2−q 2−q2 k(u, v)kqH. (2.2)
Thus, Jλ is coercive and bounded below on Nλ,µ.
Define
Φλ,µ(u, v) =
D
Jλ,µ0 (u, v) , (u, v)E. Then for (u, v) ∈ Nλ,µ,
D Φ0λ,µ(u, v) , (u, v)E = 2 k(u, v)k2H − (α + β) Z Ω f |u|α|v|βdx − qKλ,µ(u, v) (2.3) = (2 − α − β) Z Ω f |u|α|v|βdx − (q − 2) Kλ,µ(u, v) . (2.4)
Now, we split Nλ,µ into three parts:
N+λ,µ=n(u, v) ∈ Nλ,µ | D Φ0λ,µ(u, v) , (u, v)E> 0o; N0λ,µ=n(u, v) ∈ Nλ,µ | D Φ0λ,µ(u, v) , (u, v)E= 0o; N−λ,µ=n(u, v) ∈ Nλ,µ | D Φ0λ,µ(u, v) , (u, v)E< 0o.
Then, we have the following results.
Lemma 2.2 Suppose that (u0, v0) is a local minimizer for Jλ,µ on Nλ,µ and
that (u0, v0) /∈ N0λ,µ. Then Jλ,µ0 (u0, v0) = 0 in H−1 (the dual space of the
Sobolev space H).
Proof. Our proof is almost the same as that in Brown-Zhang [3, Theorem
2.3] (or see Binding-Drabek-Huang [2]).
Lemma 2.3 We have
(i) if (u, v) ∈ N+λ,µ, then Kλ,µ(u, v) > 0;
(ii) if (u, v) ∈ N0λ,µ, then Kλ,µ(u, v) > 0 and
R
Ωf |u| α
|v|βdx > 0; (iii) if (u, v) ∈ N−λ,µ, then R
Ωf |u| α|v|β
dx > 0.
Proof. The proof is immediate from (2.1) and (2.4) .
Lemma 2.4 If 0 < |λ|2−q2 + |µ| 2 2−q < C α, β, q, S, S, then N0λ,µ= ∅.
Proof. Suppose otherwise, that is there exists (λ, µ) ∈ R2\ {(0, 0)} with
0 < |λ|2−q2 + |µ| 2
2−q < Cα, β, q, S, S
such that N0
λ,µ 6= ∅. Then for (u, v) ∈ N0λ,µ we have
0 =DΦ0λ,µ(u, v) , (u, v)E= (2 − q) k(u, v)k2H − (α + β − q)
Z
Ω
f |u|α|v|βdx = (2 − α − β) k(u, v)k2H − (q − α − β) Kλ,µ(u, v) .
By the H¨older inequality and the Sobolev imbedding theorem, k(u, v)kH ≥ α + β − q 2 − q S α+β !1/(2−α−β) and k(u, v)kH ≤ α + β − q α + β − 2 !2−q1 S q 2−q |λ|2−q2 + |µ| 2 2−q 12 . This implies |λ|2−q2 + |µ| 2 2−q ≥ C α, β, q, S, S which is a contradiction. Thus, we can conclude that if
0 < |λ|2−q2 + |µ| 2 2−q < C α, β, q, S, S, we have N0λ,µ= ∅. By Lemma 2.4, we write Nλ,µ= N+λ,µ∪ N − λ,µ and define θλ,µ+ = inf (u,v)∈N+λ,µ Jλ,µ(u, v) ; θλ,µ− = inf (u,v)∈N−λ,µ Jλ,µ(u, v) .
Then we have the following result. Theorem 2.5 If 0 < |λ|2−q2 + |µ|
2 2−q < C
0, then we have
(i) θ+λ,µ< 0;
(ii) θ−λ,µ> d0 for some d0 = d0
α, β, q, S, S, λ, µ> 0.
Proof. (i) Let (u, v) ∈ N+λ,µ. By (2.3)
2 − q α + β − qk(u, v)k 2 H > Z Ω f |u|α|v|βdx
and so Jλ,µ(u, v) = 1 2− 1 q ! k(u, v)k2H + 1 q − 1 α + β ! Z Ω f |u|α|v|βdx < " 1 2− 1 q ! + 1 q − 1 α + β ! 2 − q α + β − q # k(u, v)k2H = −(2 − q) (α + β − 2) 2q (α + β) k(u, v)k 2 H < 0. Thus, θλ,µ+ < 0.
(ii) Let (u, v) ∈ N−λ,µ. By (2.3)
2 − q α + β − qk(u, v)k 2 H < Z Ω f |u|α|v|βdx. Moreover, by the Sobolev imbedding theorem
Z Ω f |u|α|v|βdx ≤ Sα+βk(u, v)kα+βH . This implies k(u, v)kH > 2 − q (α + β − q) Sα+β ! 1 α+β−2
for all (u, v) ∈ N−λ,µ. (2.5)
By (2.2) in the proof of Lemma 2.1
Jλ,µ(u, v) ≥ k(u, v)kqH " α + β − 2 2 (α + β) k(u, v)k 2−q H − S q α + β − q q (α + β) ! |λ|2−q2 + |µ| 2 2−q 2−q2 # > 2 − q (α + β − q) Sα+β !α+β−2q α + β − 2 2 (α + β) 2 − q (α + β − q) Sα+β !α+β−22−q −Sq α + β − q q (α + β) ! |λ|2−q2 + |µ| 2 2−q 2−q2 # . Thus, if 0 < |λ|2−q2 + |µ| 2 2−q < C 0, then
Jλ,µ(u, v) > d0 for all (u, v) ∈ N−λ,µ,
for some d0 = d0
For each (u, v) ∈ H with R Ωf |u| α |v|βdx > 0, we write tmax = (2 − q) k(u, v)k2H (α + β − q)R Ωf |u| α |v|βdx !α+β−21 > 0. Then the following lemma hold.
Lemma 2.6 For each (u, v) ∈ H with R
Ωf |u| α|v|β
dx > 0, we have
(i) if Kλ,µ(u, v) ≤ 0, then there is unique t−> tmaxsuch that (t−u, t−v) ∈ N−λ,µ
and Jλ,µ t−u, t−v= sup t≥0 Jλ,µ(tu, tv) ;
(ii) if Kλ,µ(u, v) > 0, then there are unique 0 < t+ < tmax < t− such that
(t+u, t+v) ∈ N+ λ,µ, (t −u, t−v) ∈ N− λ,µand Jλ,µ t+u, t+v= inf 0≤t≤tmax Jλ,µ(tu, tv) ; Jλ,µ t−u, t−v= sup t≥0 Jλ,µ(tu, tv) .
Proof. Fix (u, v) ∈ H with R
Ωf |u| α |v|βdx > 0. Let m (t) = t2−qk(u, v)k2H − tα+β−qZ Ω f |u|α|v|βdx for t ≥ 0. (2.6) Clearly, m(0) = 0, m(t) → −∞ as t → ∞. Since m0(t) = (2 − q) t1−qk(u, v)k2H − (α + β − q) tα+β−q−1Z Ω f |u|α|v|βdx we have m0(t) = 0 at t = tmax, m0(t) > 0 for t ∈ [0, tmax) and m0(t) < 0
for t ∈ (tmax, ∞) . Then m (t) achieves its maximum at tmax, is increasing for
t ∈ [0, tmax) and decreasing for t ∈ (tmax, ∞) . Moreover,
m (tmax) = k(u, v)kqH 2 − q α + β − q !α+β−22−q − 2 − q α + β − q !α+β−qα+β−2 k(u, v)kα+βH R Ωf |u| α |v|βdx ! 2−q α+β−2 ≥ k(u, v)kqH α + β − 2 α + β − q ! α + β − q 2 − q S α+β !2−α−β2−q . (2.7)
(i) Kλ,µ(u, v) ≤ 0 : There is a unique t− > tmaxsuch that m (t−) = Kλ,µ(u, v)
and m0(t−) < 0. Now, (2 − q)t−2k(u, v)k2H − (α + β − q)t−α+β Z Ω f |u|α|v|βdx =t−1+qm0t−< 0,
and D Jλ,µ0 t−u, t−v,t−u, t−vE =t−qhmt−− Kλ,µ(u, v) i = 0.
Thus, (t−u, t−v) ∈ N−λ,µ. Since for t > tmax, we have
(2 − q) k(tu, tv)k2H − (α + β − q) Z Ω f |tu|α|tv|βdx < 0, d 2 dt2Jλ,µ(tu, tv) < 0 and d dtJλ,µ(tu, tv) = t k(u, v)k 2 H − t qK λ,µ(u, v) − tα+β Z Ω f |u|α|v|βdx = 0 for t = t−.
Thus, Jλ,µ(t−u, t−v) = supt≥0Jλ,µ(tu, tv) .
(ii) Kλ,µ(u, v) > 0 : By (2.7) and
m (0) = 0 < Kλ,µ(u, v) ≤ Sq|λ|2−q2 + |µ| 2 2−q 2−q2 k(u, v)kqH < k(u, v)kqH α + β − 2 α + β − q ! α + β − q 2 − q S α+β !2−α−β2−q ≤ m (tmax) , for 0 < |λ|2−q2 + |µ| 2
2−q < Cα, β, q, S, S, there are unique t+ and t− such
that 0 < t+< t max < t−, mt+= Kλ,µ(u, v) = m t− and m0t+> 0 > m0t−.
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, Jλ,µ t+u, t+v= inf 0≤t≤tmax Jλ,µ(tu, tv) ; Jλ,µ t−u, t−v= sup t≥0 Jλ,µ(tu, tv) .
For each (u, v) ∈ H with Kλ,µ(u, v) > 0, we write tmax = (α + β − q) Kλ,µ(u, v) (α + β − 2) k(u, v)k2H !2−q1 > 0. (2.8)
Then we have the following lemma.
Lemma 2.7 For each (u, v) ∈ H with Kλ,µ(u, v) > 0, we have
(i) if R
Ωf |u| α
|v|βdx ≤ 0, then there is unique 0 < t+ < t
max such that
(t+u, t+v) ∈ N+λ,µ and Jλ,µ t+u, t+v= inf t≥0Jλ,µ(tu, tv) ; (ii) if R Ωf |u| α|v|β
dx > 0, then there are unique 0 < t+ < t
max< t− such that
(t+u, t+v) ∈ N+ λ,µ, (t −u, t−v) ∈ N− λ,µand Jλ,µ t+u, t+v= inf 0≤t≤tmax Jλ,µ(tu, tv) ; Jλ,µ t−u, t−v= sup t≥0 Jλ,µ(tu, tv) .
Proof. Fix (u, v) ∈ H with Kλ,µ(u, v) > 0. Let
m (t) = t2−α−βk(u, v)k2H − tq−α−βKλ,µ(u, v) for t > 0. (2.9)
Clearly, m(t) → −∞ as t → 0+, m(t) → 0 as t → ∞. Since
m0(t) = (2 − α − β) t1−α−βk(u, v)k2H − (q − α − β) tq−α−β−1K
λ,µ(u, v)
we have m0(t) = 0 at t = tmax, m0(t) > 0 for t ∈ [0, tmax) and m0(t) < 0
for t ∈ (tmax, ∞) . Then m (t) achieves its maximum at tmax, is increasing for
t ∈ (0, tmax) and decreasing for t ∈ (tmax, ∞) . Similar to the argument in
Lemma 2.6, we can obtain the results of Lemma 2.7.
3 Proof of Theorem 1.1
First, we establish the existence of a local minimum for Jλ,µ on N+λ,µ.
Theorem 3.1 If 0 < |λ|2−q2 +|µ| 2 2−q < C
0, then Jλ,µhas a minimizer
u+0, v0+ in N+λ,µ and it satisfies (i) Jλ,µ u+0, v+0= θ+λ,µ;
(ii) 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.
Proof. Let {(un, vn)} be a minimizing sequence for Jλ,µ on N+λ,µ. Then by
Lemma 2.1 and the compact imbedding theorem, there exist a subsequence {(un, vn)} and
u+0, v+0∈ H such thatu+0, v0+is a solution of problem (Eλ,µ)
and un* u+0 weakly in H01(Ω), un→ u+0 strongly in L q(∂Ω) and in Lα+β(Ω), vn* v+0 weakly in H 1 0(Ω), vn→ v+0 strongly in Lq(∂Ω) and in Lα+β(Ω). This implies Kλ,µ(un, vn) → Kλ,µ u+0, v+0 as n → ∞, Z Ω f |un|α|vn|β→ Z Ω f u + 0 α v + 0 β as n → ∞. Since Jλ,µ(un, vn) = α + β − 2 2 (α + β) k(un, vn)k 2 H − α + β − q q (α + β)Kλ,µ(un, vn)
and by Theorem 2.5 (i)
Jλ,µ(un, vn) → θλ,µ+ < 0 as n → ∞.
Letting n → ∞, we see that Kλ,µ
u+0, v0+> 0. Now we prove that
un→ u+0 strongly in H 1
(Ω), vn→ v+0 strongly in H
1(Ω).
Supposing the contrary, then either
u + 0 H1 < lim infn→∞ kunkH1 or v + 0 H1 < lim infn→∞ kvnkH1. (3.1)
Fix (u, v) ∈ H with Kλ,µ(u, v) > 0. Let
φ(u,v)(t) = m (t) −
Z
Ω
f |u|α|v|βdx
where m (t) is as in (2.9) . Clearly, φ(u,v)(t) → −∞ as t → 0+ and
φ(u,v)(t) → −
Z
Ω
Since φ0(u,v)(t) = m0(t) , similar argument as in the proof of Lemma 2.7 we have φ(u,v)(t) achieves its maximum at tmax(u, v) , is increasing for t ∈
(0, tmax(u, v)) and decreasing for t ∈ (tmax(u, v) , ∞) , where
tmax(u, v) = (α + β − q) Kλ,µ(u, v) (α + β − 2) k(u, v)k2H !2−q1 is as in (2.8) . Since Kλ,µ
u+0, v+0 > 0, by Lemma 2.7, there is a unique 0 < t+0 < tmax u+0, v0+ such thatt+0u+0, t+0v0+∈ N+ λ,µ and Jλ,µ t+0u+0, t+0v0+= inf 0≤t≤tmax(u+0,v + 0) Jλ,µ tu+0, tv0+. Then φ(u+0,v+0) t+0 =t+0−(α+β) t+0u+0, t+0v0+ 2 H − Kλ,µ t+0u+0, t+0v0+− Z Ω f t + 0u + 0 α t + 0v + 0 β dx = 0 (3.2) By (3.1) and (3.2) we obtain φ(un,vn)
t+0> 0 for n sufficiently large. (3.3)
Since (un, vn) ∈ N+λ,µ, we have tmax(un, vn) > 1. Moreover,
φ(un,vn)(1) = k(un, vn)k 2 H − Kλ,µ(un, vn) − Z Ω f |un|α|vn|βdx = 0
and φ(un,vn)(t) is increasing for t ∈ (0, tmax(un, vn)) . This implies φ(un,vn)(t) ≤
0 for all t ∈ (0, 1] and n sufficiently large. We obtain 1 < t+0 ≤ tmax
u+0, v+0. But t+0u+0, t+0v+0∈ N+ λ,µ and Jλ,µ t+0u+0, t+0v0+= inf 0≤t≤tmax(u+0,v + 0) Jλ,µ tu+0, tv0+. This implies Jλ,µ t+0u+0, t+0v0+< Jλ,µ u+0, v0+< lim n→∞Jλ,µ(un, vn) = θ + λ,µ,
which is a contradiction. Hence
un→ u+0 strongly in H 1(Ω),
vn→ v+0 strongly in H 1(Ω).
This implies
Jλ,µ(un, vn) → Jλ,µ
u+0, v0+= θλ,µ+ as n → ∞.
Thus,u+0, v0+is a minimizer for Jλ,µon N+λ,µ. 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 non-negative solution of problem (Eλ,µ). Finally, we prove that u+0 6= 0, v0+ 6= 0.
We assume that, without loss of generality, v0+ ≡ 0. Then as u+
0 is a nonzero solution of −∆u + u = 0 in Ω, ∂u ∂n = λg (x) |u| q−2 u on ∂Ω, we have u + 0 2 H1 = λ Z ∂Ω g u + 0 q ds > 0
Moreover, by the conditions (A) , (B) we may choose w ∈ H1(Ω) \ {0} such
that kwk2H1 = µ Z ∂Ω h |w|qds > 0 and Z Ω f u + 0 α |w|βdx ≥ 0. Now Kλ,µ u+0, w= λ Z ∂Ω g u + 0 q ds + µ Z ∂Ω h |w|qds > 0
and so by Lemma 2.7 there is a unique 0 < t+< t
max such that
t+u+ 0, t+w ∈ N+λ,µ. Moreover, tmax= (α + β − q) Kλ,µ u+0, w (α + β − 2) u+0, w 2 H 1 2−q = α + β − q α + β − 2 !2−q1 > 1 and Jλ,µ t+u+0, t+w= inf 0≤t≤tmax Jλ,µ tu+0, tw. This implies Jλ,µ t+u+0, t+w≤ Jλ,µ u+0, w< Jλ,µ u+0, 0= θ+λ,µ which is a contradiction.
Next, we establish the existence of a local minimum for Jλ,µ on N−λ,µ
Theorem 3.2 If 0 < |λ|2−q2 +|µ| 2 2−q < C
0, then Jλ,µhas a minimizer
u−0, v0− in N−λ,µ and it satisfies (i) Jλ,µ u−0, v−0= 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 u−0 6= 0, v0− 6= 0.
Proof. Let {(un, vn)} be a minimizing sequence for Jλ,µ on N−λ,µ. Then by
Lemma 2.1 and the compact imbedding theorem there exist a subsequence {(un, vn)} and u−0, v−0∈ H such that un* u−0 weakly in H 1 0(Ω), un→ u−0 strongly in L q(∂Ω) and in Lα+β(Ω), vn* v−0 weakly in H 1 0(Ω), vn→ v−0 strongly in Lq(∂Ω) and in Lα+β(Ω). This implies Kλ,µ(un, vn) → Kλ,µ u−0, v−0 as n → ∞, Z Ω f |un|α|vn|β→ Z Ω f u − 0 α v − 0 β as n → ∞. Moreover, by (2.3) we obtain Z Ω f |un|α|vn|βdx > 2 − q α + β − qk(un, vn)k 2 H; (3.4)
By (2.5) and (3.4) there exists a positive number C such that
Z Ω f |un| α |vn| β dx > C. This implies Z Ω f u − 0 α v − 0 β dx ≥ C. (3.5)
Now we prove that
un→ u−0 strongly in H 1 0(Ω), vn→ v−0 strongly in H 1 0(Ω).
Suppose otherwise, then either u − 0 H1 < lim infn→∞ kunkH1 or v − 0 H1 < lim infn→∞ kvnkH1.
By Lemma 2.6, there is a unique t−0 such that t−0u−0, t−0v−0 ∈ N−λ,µ. Since (un, vn) ∈ N−λ,µ, Jλ,µ(un, vn) ≥ Jλ,µ(tun, tvn) for all t ≥ 0, we have
Jλ,µ t−0u−0, t−0v0−< lim n→∞Jλ,µ t−0un, t−0vn ≤ lim n→∞Jλ,µ(un, vn) = θ − λ,µ
un→ u−0 strongly in H 1 0(Ω), vn→ v−0 strongly in H 1 0(Ω). This implies Jλ,µ(un, vn) → Jλ,µ u−0, v0−= θλ,µ− as n → ∞. Since Jλ,µ u−0, v0−= Jλ,µ u − 0 , v − 0 and u − 0 , v − 0 ∈ N−λ,µ, by Lemma 2.2 and (3.5) 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.
Now, we complete the proof of Theorem 1.1: By Theorems 3.1, 3.2 problem (Eλ,µ) has two solutions
u+0, v+0 ∈ N+ λ,µ and u−0, v−0 ∈ N−λ,µ such that u±0 ≥ 0, v± 0 ≥ 0 in Ω and u ± 0 6= 0, v ± 0 6= 0. Since N + λ,µ∩ N − λ,µ = ∅, this implies
that u+0, v0+ and u−0, v−0 are distinct. Moreover, if f ≥ 0, then by the maximum principle we obtain u±0 > 0, v0±> 0 in Ω.
References
[1] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994) 519–543. [2] P. A. Binding, P. Drabek and Y. X. Huang, On Neumann boundary value
problems for some quasilinear elliptic equations, Electr. J. Diff. Eqns. 5 (1997) 1–11.
[3] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Eqns 193 (2003) 481–499.
[4] P. Drabek and S. I. Pohozaev, Positive solutions for the p–Laplacian: application of the fibering method, Proc. Royal Soc. Edinburgh Sect. A 127 (1997) 703–727. [5] J. Garcia-Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a
nonlinear boundary condition, J. Diff. Eqns. 198 (2004) 91–128.
[6] T. F. Wu, A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential, Electr. J. Diff. Eqns. 131 (2006) 1–15.
[7] T. F. Wu, Multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition, Appl. Math. Comput. (2007), doi:10.1016/j.amc. 2006.12.052.
