Multiple positive solutions of a nonlinear
boundary value problem involving a
sign-changing weight
Tsung-fang Wu
Department of Applied Mathematics,
National University of Kaohsiung, Kaohsiung 811, Taiwan
Abstract
In this paper, we study the multiplicity of positive solutions for the following elliptic
equation: ∆u − u = 0 in RN +, ∂u
∂ν = λa (x) |u|q−2u + b (x) |u|p−2u on ∂RN+,
where 1 < q < 2 < p < 2∗ (2∗ = 2(N −1)N −2 if N ≥ 3, 2∗ = ∞ if N = 2), RN + = © (x0, x N) ∈ RN −1× R | xN > 0 ª
is an upper half space in RN, λ > 0 and the functions a and b are satisfying some suitable conditions. Using the decompo-sition of the Nehari manifold, we prove that the equation has at least two positive solutions provided λ is sufficiently small.
Key words: Nonlinear boundary conditions, Sign-changing weight, Multiple
positive solutions
1 Introduction
In this paper, we study the multiplicity of positive solutions for the following elliptic equation: ∆u − u = 0 in RN +, ∂u ∂ν = λa (x) |u| q−2u + b (x) |u|p−2u on ∂RN +, (Eλ)
where 1 < q < 2 < p < 2∗ (2∗ = 2(N −1)N −2 if N ≥ 3, 2∗ = ∞ if N = 2), RN+ =
n
(x0, x
N) ∈ RN −1× R | xN > 0
o
is an upper half space in RN and λ > 0. We
assume that the functions a and b satisfy the following conditions: (D1) a ∈ Lp−qp
³ ∂RN
+
´
\ {0} with a±(x) = ± max {±a (x) , 0} 6≡ 0;
(D2) b ∈ C³∂RN
+
´
and there is a positive number rb < q such that
b (x) ≥ 1 + c0exp (−rb|x|) for some c0 < 1 and for all x ∈ ∂RN+ and
b (x) → 1 as |x| → ∞.
Associated with equation (Eλ) , we consider the energy functional Jλ in
H1³RN + ´ , Jλ(u) = 1 2kuk 2 H1 − 1 q Z ∂RN + λa |u|qdσ −1 p Z ∂RN + b |u|pdσ,
where dσ is the measure on the boundary and kukH1 =
³R RN + ³ |∇u|2+ u2´dx´1/2 is the standard in H1³RN + ´
. It is well known that Jλ ∈ C1
³
H1³RN
+
´´
and the solutions of equation (Eλ) are the critical points of the energy functional
Jλ.
The study of existence and multiplicity when the nonlinear terms are placed in the equation, that is if one considers a problem of the form −∆u = g (x, u) with Dirichlet boundary conditions, has received considerable attention, see for example [1,2,5,10,14,17,20,21], etc. However, nonlinear boundary condi-tions have only been considered in recent years. For the semilinear (or linear) elliptic equations with a nonlinear term boundary condition, see for example [8,9,11,15,16,18,19], etc.
The purpose of this paper, we are facing two nonlinear terms are placed in the boundary condition. Using the decomposition of the Nehari manifold as λ varies to prove that the following result.
Theorem 1.1 There exists λ0 > 0 such that for λ ∈ (0, λ0) , equation (Eλ)
has at least two positive solutions.
This paper is organized as follows. In section 2, we give some notations and preliminaries. In section 3, we establish the existence of a local minimum for
2 Notations and Preliminaries
Throughout this paper, we denote by Cp the best Sobolev trace constant
for the imbedding of H1³RN
+ ´ into Lp³∂RN + ´ : Cp = inf u∈H1(RN +)\{0} R RN + ³ |∇u|2 + u2´dx ³R ∂RN + |u| pdσ´2/p > 0. (2.1) In particular, ³R∂RN + |u| pdσ´1/p≤ C−1/2
p kukH1 for all u ∈ H1
³
RN
+
´ .
We define the Palais–Smale (simply by (PS)) sequences, (PS)–values, and (PS)–conditions in H1³RN
+
´
for Jλ as follows.
Definition 2.1 (i) For β ∈ R, a sequence {un} is a (PS)β–sequence in H1
³
RN
+
´ for Jλ if Jλ(un) = β + o(1) and Jλ0(un) = o(1) strongly in H∗
³
RN
+
´
as n → ∞;
(ii) Jλ satisfies the (PS)β–condition in H1
³ RN + ´ if every (PS)β–sequence in H1³RN + ´
for Jλ contains a convergent subsequence.
As the energy functional Jλ is not bounded below on H1
³
RN
+
´
, it is useful
to consider the functional on the Nehari manifold Mλ = n u ∈ H1³RN + ´ \ {0} | hJ0 λ(u) , ui = 0 o .
Thus, u ∈ Mλ if and only if
kuk2H1 − Z ∂RN + λa |u|qdσ − Z ∂RN + b |u|pdσ = 0.
Furthermore, we have the following results.
Lemma 2.2 The energy functional Jλ is coercive and bounded below on Mλ.
Proof. If u ∈ Mλ, then by the H¨older and Sobolev trace inequalities,
Jλ(u) = Ã 1 2 − 1 p ! kuk2H1 − Ã 1 q − 1 p ! Z ∂RN + λa |u|qdσ ≥ Ã 1 2 − 1 p ! kuk2H1 − λ Ã p − q pq ! kak L p p−q C −q2 p kukqH1. (2.2)
Thus, Jλ is coercive and bounded below on Mλ. ¤
The Nehari manifold Mλ is closed linked to the behavior of the function of
and were introduced by Dradek-Pohozaev in [12] and are also discussed in Brown-Zhang [7] and Brown-Wu [5,6]. If u ∈ H1³RN
+ ´ , we have gu(t) =t 2 2 kuk 2 H1 − tq q Z ∂RN + λa |u|qdσ −t p p Z ∂RN + b |u|pdσ; g0 u(t) = t kuk2H1 − tq−1 Z ∂RN + λa |u|qdσ − tp−1 Z ∂RN + b |u|pdσ; g00 u(t) = kuk 2 H1 − (q − 1) tq−2 Z ∂RN + λa |u|qdσ − (p − 1) tp−2Z ∂RN + b |u|pdσ.
It is easy to see that
tg0u(t) = ktuk2H1 − Z ∂RN + λa |tu|qdσ − Z ∂RN + b |tu|pdσ
and so, for u ∈ H1³RN
+
´
\ {0} and t > 0, g0
u(t) = 0 if and only if tu ∈ Mλ,
i.e., positive critical points of gu correspond to points on the Nehari manifold.
In particular, g0
u(1) = 0 if and only if u ∈ Mλ. Thus it is natural split Mλ
into three parts corresponding to local minima, local maxima and points of inflection. Accordingly, we define
M+
λ = {u ∈ Mλ | g00u(1) > 0} ;
M0λ= {u ∈ Mλ | g00u(1) = 0} ;
M−λ = {u ∈ Mλ | g00u(1) < 0} .
We now derive some basic properties of M+λ, M0
λ and M−λ.
Lemma 2.3 Suppose that u0 is a local minimizer for Jλ on Mλ and that
u0 ∈ M/ 0λ. Then Jλ0 (u0) = 0 in H∗ ³ RN + ´ .
Proof. Our proof is almost the same as that in Brown-Zhang [7, Theorem
2.3] (or see Binding-Drabek-Huang [3]). ¤
For each u ∈ Mλ we have
g00 u(1) = kuk2H1 − (q − 1) Z ∂RN + λa |u|qdσ − (p − 1) Z ∂RN + b |u|pdσ = (2 − p) kuk2H1 − (q − p) Z ∂RN + λa |u|qdσ (2.3) = (2 − q) kuk2H1 − (p − q) Z ∂RN + b |u|pdσ. (2.4)
Lemma 2.4 (i) For any u ∈ M+λ ∪ M0 λ, we have R ∂RN + a |u| qdσ > 0;
(ii) for any u ∈ M−λ, we have R∂RN
+ b |u|
pdσ > 0.
Proof. The results now follows immediately from (2.3) and (2.4) . ¤ Let Λ0 = ³ 2−q kbk∞ ´2−q p−2 p−2 kak L p p−q ³ Cp p−q ´p−q p−2
. Then we have the following results.
Lemma 2.5 For each λ ∈ (0, Λ0) , we have M0λ = ∅.
Proof. Suppose otherwise, then there exists λ ∈ (0, Λ0) such that M0λ 6= ∅.
Then for u ∈ M0
λ, by (2.3) and the H¨older and Sobolev trace inequalities we
have kuk2H1 = p − q p − 2 Z ∂RN + λa |u|qdσ ≤ λC−q2 p p − q p − 2kakL p p−q kuk q H1 and so kuk2H1 ≤ C q q−2 p " λ kak Lp−qp p − q p − 2 # 2 2−q .
Similarly using (2.4) and the Sobolev trace inequality we have 2 − q p − qkuk 2 H1 = Z ∂RN + b |u|pdσ ≤ C−p2 p kbk∞kukpH1, this implies kuk2H1 ≥ C p p−2 p " 2 − q (p − q) kbk∞ # 2 p−2 .
Hence we must have
λ ≥ à 2 − q kbk∞ !2−q p−2 p − 2 kak L p p−q à Cp p − q !p−q p−2 = Λ0,
a contradiction. This completes the proof. ¤
Denote Φ³RN + ´ = nu ∈ H1³RN + ´ \ {0} | R∂RN + b |u| pdσ > 0o is a subset of H1³RN + ´
. In order to get a better understanding of the Nehari manifold and
fibering maps, we consider the function hu : R+→ R defined by
hu(t) = t2−qkuk2H1 − tp−q
Z
∂RN
+
b |u|pdσ for t > 0. (2.5) Clearly tu ∈ Mλ if and only if hu(t) =
R ∂RN + b |u| pdσ. Moreover, h0 u(t) = (2 − q)t1−qkuk 2 H1 − (p − q)tp−q−1 Z ∂RN + b |u|pdσ (2.6)
and so it is easy to see that, if tu ∈ Mλ, then tq−1h0u(t) = g00u(t). Hence tu ∈ M+ λ( or M−λ) if and only if h0u(t) > 0( or < 0). Suppose u ∈ Φ³RN + ´
. Then by (2.6), hu has a unique critical point at
t = tmax(u) where
tmax(u) = (2 − q) kuk 2 H1 (p − q)R∂RN + b |u| pdσ 1 p−2 > 0 (2.7)
and clearly hu is strictly increasing on (0, tmax(u)) and strictly decreasing on (tmax(u) , ∞) with limt→∞hu(t) = −∞. Moreover, if λ ∈ (0, Λ0) , then
hu(tmax(u)) > Z ∂RN + λa |u|qdσ. For each u ∈ Φ³RN + ´ , we write tmax = (2 − q) kuk 2 H1 (p − q)R∂RN + b |u| pdσ 1 p−2 .
Thus, we have the following lemma. Lemma 2.6 For each u ∈ Φ³RN
+
´
we have
(i) if R∂RN
+ a |u|
qdσ ≤ 0, then there is a unique t− = t−(u) > t
max(u) such
that t−u ∈ M−
λ and hu is increasing on (0, t−) and decreasing on (t−, ∞).
Moreover, Jλ ³ t−u´= sup t≥0 Jλ(tu) ; (2.8) (ii) ifR∂RN + a |u|
qdσ > 0, then there are unique 0 < t+ = t+(u) < t
max(u) < t−
such that t+u ∈ M+
λ, t−u ∈ M−λ, hu is decreasing on (0, t+), increasing on
(t+, t−) and decreasing on (t−, ∞). Moreover,
Jλ ³ t+u´= inf 0≤t≤tmax(u) Jλ(tu) ; Jλ ³ t−u´= sup t≥t+Jλ(tu) ; (2.9)
(iii) t−(u) is a continuous function on Φ³RN
+ ´ ; (iv) M−λ =nu ∈ Φ³RN + ´ | 1 kukH1t− ³ u kukH1 ´ = 1o. Proof. Fix u ∈ Φ³RN + ´ . (i) Suppose R∂RN + a |u| qdσ ≤ 0. Then h u(t) = R ∂RN + λa |u|
qdσ has a unique
so-lution t− > t
max(u) and h0u(t−) < 0. Hence hu has a unique turning point at
t = t− and g00
u(t−) < 0. Thus t−u ∈ M−λ and (2.8) holds.
(ii) SupposeR∂RN + a |u| qdσ > 0. Since h u(tmax(u)) > R ∂RN + λa |u| qdσ, the equa-tion hu(t) = R ∂RN + λa |u|
qdσ has exactly two solutions t+ < t
that h0
u(t+) > 0 and h0u(t−) < 0. Hence there are exactly two multiples of u
lying in Mλ, viz, t+u ∈ M+λ and t−u ∈ M−λ. Thus, gu has turning points at
t = t+ and t = t− with h00
u(t+) > 0 and h00u(t−) < 0. Thus, gu is decreasing
on (0, t+) , increasing on (t+, t−) and decreasing on (t+, ∞) . Hence (2.9) must hold.
(iii) By the uniqueness of t−(u) and the external property of t−(u) , we have
t−(u) is a continuous function on Φ³RN
+
´ .
(iv) For u ∈ M−
λ. Let v = kukuH1. By parts (i), (ii) , there is a unique t−(v) > 0
such that t−(v) v ∈ M− λ or t− ³ u kukH1 ´ 1
kukH1u ∈ M−λ. Since u ∈ M−λ, we have
t−³ u kukH1
´
1
kukH1 = 1, this implies
M−λ ⊂ ( u ∈ H1³RN´ | 1 kukH1 t− Ã u kukH1 ! = 1 ) . Conversely, let u ∈ Φ³RN + ´ such that 1 kukH1t− ³ u kukH1 ´ = 1. Then t− Ã u kukH1 ! u kukH1 ∈ M− λ. Thus, M− λ = ( u ∈ Φ³RN + ´ | 1 kukH1 t− Ã u kukH1 ! = 1 ) .
This completes the proof. ¤
3 Existence of a first solution
Firstly, we remark that it follows from Lemma 2.5, we may write Mλ =
M+λ ∪ M−λ for all λ ∈ (0, Λ0). Furthermore, by Lemma 2.6 that M+λ and M−λ
are nonempty and by Lemma 2.2 may define
α+
λ = inf
u∈M+λJλ(u) and α −
λ = inf
u∈M−λ Jλ(u) .
Then we have the following results. Theorem 3.1 We have
(i) α+λ < 0 for all λ ∈ (0, Λ0) ; (ii) if λ ∈³0,qΛ0
2
´
, then α−λ > c0 for some c0 > 0.
In particular, for each λ ∈³0,qΛ0
2
´
, we have α+
Proof. (i) For u ∈ M+λ, we have by (2.3) kuk2H1 < p − q p − 2 Z ∂RN + λa |u|qdσ.
Hence by (2.2) and Lemma 2.4
Jλ(u) = p − 2 2p kuk 2 H1− p − q pq Z ∂RN + λa |u|qdσ < −(p − q) (2 − q) 2pq Z ∂RN + λa |u|qdσ < 0 and so α+λ < 0. (ii) For u ∈ M− λ, we have 2 pkuk 2 H1 < Z ∂RN + b |u|pdσ, if Z ∂RN + a |u|qdσ ≤ 0 and 2 − q p − qkuk 2 H1 < Z ∂RN + b |u|pdσ, if Z ∂RN + a |u|qdσ > 0, this implies 2 − q p − qkuk 2 H1 < Z ∂RN +
b |u|pdσ, for all u ∈ M−λ.
Moreover, by (2.1) 2 − q p − qkuk 2 H1 < Z ∂RN + b |u|pdσ ≤ C−p2 p kbk∞kukpH1. Thus, kukH1 > Cpp/2(p−2) à 2 − q kbk∞(p − q) !1/(p−2) for all u ∈ M−λ. By (2.2) , Jλ(u) ≥ kukqH1 à p − 2 2p kuk 2−q H1 − λ kak L p p−q C −q2 p à p − q pq !! > C pq 2(p−2) p à 2 − q kbk∞(p − q) ! q p−2 · p − 2 2p C p(2−q) 2(p−2) p à 2 − q kbk∞(p − q) !2−q p−2 − λ kak L p p−q C −q2 p à p − q pq ! . Thus, if λ ∈³0,qΛ0 2 ´ , then α− λ > c0 for some c0 > 0.
This completes the proof. ¤ Next, we consider the following elliptic equation:
∆u − u = 0 in RN +; ∂u ∂ν = |u| p−2u on ∂RN +. (E∞)
Associated with the equation (E∞) , we consider the energy functional J∞ in
H1³RN + ´ J∞(u) = 1 2kuk 2 H1 − 1 p Z RN|u| pdx.
Consider minimizing problems inf
u∈M∞J
∞(u) = α∞
where
M∞=nu ∈ H1³RN´\ {0} | D(J∞)0(u) , uE = 0o.
By del Pino-Flores [11], equation (E∞) has a positive ground state solution
w0 such that J∞(w0) = α∞ > 0. Furthermore, if λ ∈
³
0,qΛ0
2
´
, then we have
the following proposition provides a precise description for the (PS)–sequence of Jλ. Proposition 3.2 If {un} is a (PS)β–sequence in H1 ³ RN + ´ for Jλ with β <
α+λ + α∞, then there exists a subsequence {u
n} and a nonzero u0 in H1
³
RN
+
´ such that un → u0 strongly in H1
³ RN + ´ and Jλ(u0) = β. Furthermore, u0 is a solution of equation (Eλ) .
Proof. By Lemma 2.2 and the Rellich–Kondrachov theorem there exist a subsequence {un} and u0 ∈ H1(RN+) is a solution of equation (Eλ) such that
un* u0 weakly in H1(RN+); un→ u0 strongly in Lploc ³ ∂RN + ´ ; un→ u0 a.e. in RN+.
First, we claim that u0 6≡ 0. If not, then by a ∈ L
p p−q ³ ∂RN + ´ and (D2) we have Z ∂RN + a |un|qdσ → 0 as n → ∞ and Z ∂RN + b |un|pdσ = Z ∂RN + |un|pdσ + o (1) .
Thus, kunk2H1 = Z ∂RN + |un|pdσ + o (1) (3.1) and à 1 2 − 1 p ! Z ∂RN + b |un|pdσ =1 2kunk 2 H1 − 1 q Z ∂RN + λa |un|qdσ − 1 p Z ∂RN + b |un|pdσ + o (1) = β + o (1) .
From {un} ⊂ M−λ and the Sobolev trace embedding we obtain
kunkH1 > c for some c > 0
and β ≥ α∞, this contradicts to β < α+
λ + α∞ < α∞. Thus, u0 is a nontrivial solution of equation (Eλ) and Jλ(u0) ≥ α+λ. Write un = u0+ vn with vn *
0 weakly in H1(RN
+). By the Brezis-Lieb lemma [4] and vn * 0 weakly in
H1(RN +), we have Z ∂RN + b |un|pdσ = Z ∂RN + b |u0+ vn|pdσ = Z ∂RN + b |u0|pdσ + Z ∂RN + |vn|pdσ + o (1) .
Since {un} is a bounded sequence in H1
³
RN
+
´
and so {vn} is also bounded
sequence in H1³RN + ´ . Moreover, by a ∈ Lp−qp ³ ∂RN + ´
, the Egorov theorem and
the H¨older inequality we have
Z ∂RN + a |vn|qdσ = Z ∂RN + a |un|qdσ − Z ∂RN + a |u0|qdσ + o (1) = o (1) .
Hence for n large enough, we can conclude that
α+λ + α∞> Jλ(u0+ vn) = Jλ(u0) + 1 2kvnk 2 H1 − 1 p Z ∂RN + |vn|pdσ + o (1) ≥ α+λ + 1 2kvnk 2 H1 − 1 p Z ∂RN + |vn|pdσ + o (1) or 1 2kvnk 2 H1 − 1 p Z ∂RN + |vn|pdσ < α∞+ o (1) . (3.2)
Also from J0 λ(un) = o (1) in H∗ ³ RN + ´
, {un} is uniformly bounded and u0 is a solution of equation (Eλ) follows
o (1) = hJ0 λ(un) , uni = kvnk2H1 − Z ∂RN + |vn|pdσ + o (1) . (3.3)
We claim that (3.2) and (3.3) can be hold simultaneously only if {vn} admits
a subsequence {vni} which converges strongly to zero. If not, the kvnkH1 is
bounded away from zero, that is
kvnkH1 ≥ c for some c > 0.
From (3.3) then it follows
Z ∂RN + |vn|pdσ ≥ Ã 2p p − 2 ! α∞+ o (1) .
By (3.2) and (3.3) for n large enough
α∞≤ Ã 1 2 − 1 p ! Z ∂RN + |vn|pdσ + o (1) =1 2kvnk 2 H1 − 1 p Z ∂RN + |vn|pdσ + o (1) < α∞,
a contradiction. Consequently, un → u0 strongly in H1(RN+) and Jλ(u0) = β. ¤
Theorem 3.3 For each λ ∈³0,qΛ0
2
´
, the functional Jλ has a minimizer umin
in M+
λ and it satisfies
(i) Jλ(umin) = α+λ;
(ii) umin is a positive solution of equation (Eλ) ;
(iii) kuminkH1 → 0 as λ → 0.
Proof. By the Ekeland variational principle [13] (or see Wu [21]), there exists
{un} ⊂ M+λ such that Jλ(un) = α+λ + o (1) and Jλ0 (un) = o (1) in H∗ ³ RN + ´ .
Then by Proposition 3.2, there exist a subsequence {un} and umin ∈ M+λ,
a solution of equation (Eλ) such that un → umin strongly in H1(RN+) and
Jλ(umin) = α+λ. Since Jλ(umin) = Jλ(|umin|) and |umin| ∈ M+λ, by Lemma 2.3,
we may assume that umin ≥ 0. Moreover, by the maximum principle and the Hopf lemma, we obtain umin > 0 in RN+. Finally, by (2.3)
kumink2−qH1 < λ p − q p − 2kakLp−qp C −q 2 p .
This implies kuminkH1 → 0 as λ → 0. ¤
4 Proof of Theorem 1
First, let w0(x) be a positive solution of equation (E∞) such that J∞(w0) =
α∞. Then by the proof of lemma 3.3 in [11], there exists a positive number C
0 such that
|w0(x)| ≤ C0exp (− |x|) for all x ∈ RN+ (4.1) Let
wl(x) = w0(x + le) , for l ∈ R and e ∈ S, (4.2) where S =nx ∈ ∂RN + | |x| = 1 o . Clearly, R∂RN + a |wl| qdσ = 0 as l → ∞. Then
we have the following results.
Lemma 4.1 Let umin be a positive solution of (Eλ) as in Theorem 3.3. Then
for each λ ∈ (0,qΛ0
2 ) there exists l0 > 0 such that for l ≥ l0 we have sup t≥0Jλ(umin+ twl) < α + λ + α∞. Proof. Since Jλ(umin+ twl) =1 2kumin+ twlk 2 H1 − λ q Z ∂RN + a (umin+ twl)qdσ − 1 p Z ∂RN + b (umin+ twl)pdσ = Jλ(umin) + J∞(twl) + t ÃZ ∂RN + bup−1minwldσ + Z ∂RN + λauq−1minwldσ ! −λ q "Z ∂RN + a (umin+ twl)qdσ − Z ∂RN + auqmindσ # −1 p "Z ∂RN + b (umin+ twl)pdσ − Z ∂RN + bupmindσ − Z ∂RN + b (twl)pdσ # ≤ α+λ + J∞(twl) + Λ0tq 2 Z ∂RN + |a| wlqdσ − tp Z ∂RN + (b − 1) wpldσ − Z ∂RN +
bh(umin+ twl)p− uminp − tpwpl − pup−1mintwl
i
dσ. (4.3)
By Brown-Zhang [7], we know that
We also remark that (u + v)p− up− vp− pup−1v ≥ 0 for all (u, v) ∈ [0, ∞) ×
[0, ∞). Thus,
Z
∂RN
+
bh(umin+ twl)p− uminp − tpwlp− pup−1mintwl
i dσ ≥ 0. Thus, Jλ(umin+ twl) ≤ α+λ + α0+ Λ0tq 2 Z ∂RN + |a| wqldσ − tpZ ∂RN + (b − 1) wpldσ. (4.4) Since Jλ(umin+ twl) → Jλ(umin) = α+λ < 0 as t → 0 and Jλ(umin+ twl) → −∞ as t → ∞,
we can easily find 0 < t1 < t2 such that
Jλ(umin+ twl) < αλ++ α∞ for all t ∈ [0, t1] ∪ [t2, ∞). (4.5) Thus, we only need to show that there exists l0 > 0 such that for l > l0,
sup
t1≤t≤t2
Jλ(umin+ twl) < αλ++ α∞.
From the condition (D2) and del Pino-Flores [11, Lemma 2.2]
Z ∂RN + (b − 1) wpldσ = Z ∂RN + (b (x − le) − 1) w0p(x) dσ ≥ d0 Z BN +(1) (b (x − le) − 1) dσ ≥ce0exp (−rbl) , (4.6) where d0 = minx∈BN +(1)w p 0(x) > 0 and B+N(1) = n x ∈ ∂RN + | |x| < 1 o . From
the condition (D1) and (4.1) , we also have
Z ∂RN + |a| wlqdσ ≤ Z ∂RN +
C0q|a| exp (−q |x + le|) dσ
≤ C exp (−ql) . (4.7)
Since rb < q and t1 ≤ t ≤ t2, by (4.4) − (4.7) we can find l0 > 0 such that sup
t≥0Jλ(umin+ twl) < α
+
λ + α∞ for all l ≥ l0.
This completes the proof. ¤
Theorem 4.2 For each λ ∈ ³0,qΛ0
2
´
, the functional Jλ has a minimizer u−0
in M−λ and it satisfies
(i) Jλ
³
u−0´= α−λ; (ii) u−
0 is a positive solution of equation (Eλ) .
Proof. First, we claim that α−
λ < α+λ + α∞. Let U1= ( u ∈ Φ³RN + ´ ¯¯ ¯ ¯ ¯ 1 kukH1 t− à u kukH1 ! > 1 ) , U2= ( u ∈ Φ³RN + ´ ¯¯ ¯ ¯ ¯ 1 kukH1 t− à u kukH1 ! < 1 ) . By Lemma 2.6, M−λ disconnects Φ³RN + ´
in two components U1 and U2 and Φ³RN
+
´ \M−
λ = U1∪ U2. Moreover, for each u ∈ M+λ, we have
1 < tmax(u) < t−(u) and
Z ∂RN + b |u|pdσ > 0. Since t−(u) = 1 kukH1t− ³ u kukH1 ´ , then M+ λ ⊂ U1. In particular, umin ∈ U1. We claim that for each l ≥ l0 there exists s0 > 0 such that umin+ s0wl ∈ U2. Since
Z
∂RN
+
b |umin+ swl|pdσ > 0 for all s ≥ 0.
By Lemma 2.6, for each s ≥ 0 there is a unique
t− Ã umin+ swl kumin+ swlkH1 ! > 0 such that t− Ã umin+ swl kumin+ swlkH1 ! umin+ swl kumin+ swlkH1 ∈ M−λ.
First, we find a constant d > 0 such that 0 < t−³ umin+swl
kumin+swlkH1
´
< d for
each l ≥ 0. Otherwise, there exists a sequence {sn} such that sn → ∞ and
t−³ umin+snwl
kumin+snwlkH1
´
→ ∞ as n → ∞. Let vn = kuminumin+s+snwnwlkl
H1. Since t
−(v
n) vn ∈
M−λ ⊂ Mλ and by the Lebesgue dominated convergence theorem,
Z ∂RN + bvp ndσ = 1 kumin+ snwlkpH1 Z ∂RN + b (umin+ snwl)pdσ = ° 1 ° °umin sn + wl ° ° °p H1 Z ∂RN + b µu min sn + wl ¶p dσ → R ∂RN + bw p ldσ kwlkpH1 as n → ∞.
We have Jλ ³ t−(vn) vn ´ = 1 2 h t−(vn) i2 − [t −(v n)]q q Z ∂RN + λavnqdσ −[t −(v n)]p p Z ∂RN + bvpndσ → −∞ as n → ∞,
this contradicts Jλ is bounded below on Mλ. Let
s0 = ¯ ¯ ¯d2− ku mink2H1 ¯ ¯ ¯ 1 2 kw0kH1 + 1. Then kumin+ s0wlk2H1 = kumink2H1 + s20kw0k2H1 + 2s0humin, wliH1 > kumink2H1 + ¯ ¯ ¯d2− ku mink2H1 ¯ ¯ ¯+ 2l0 ÃZ ∂RN + λauq−1minwldσ + Z ∂RN + bup−1minwldσ ! > d2 > " t− Ã umin+ s0wl kumin+ s0wlkH1 !#2 ,
that is umin+ s0wl ∈ U2. Let
g (s) = 1 kumin+ swlkH1 t− Ã umin+ swl kumin+ swlkH1 ! .
By Lemma 2.6 (ii) , g (s) is a continuous function on [0, ∞). Since g (0) > 1 and g (s0) < 1. By the intermediate value theorem, there exists s1 ∈ (0, s0) such that g (s1) = 1 kumin+ s1wlkH1 t− Ã umin+ s1wl kumin+ s1wlkH1 ! = 1.
Thus, umin+ s1wl ∈ M−λ and Jλ(umin+ s1wl) ≥ α−λ. Moreover, by Lemma 4.1
α−λ ≤ Jλ(umin+ s1wl) < αλ++ α∞.
By the Ekeland variational principle [13] (or see Wu [21]), there exists {un} ⊂
M−
λ such that
Jλ(un) = α−λ + o (1) and Jλ0 (un) = o (1) in H∗
³
RN+´.
Since α−λ < αλ++ α0 by Proposition 3.2, there exist a subsequence {un} and
u−0 ∈ H1(RN+) such that
This implies u−0 ∈ M−λ and Jλ(un) → Jλ ³ u− 0 ´ = α− λ as n → ∞. Since Jλ ³ u−0´ = Jλ ³¯¯ ¯u−0¯¯¯ ´
and ¯¯¯u−0¯¯¯ ∈ M−λ. By Lemma 2.3 we may assume
that u−0 ≥ 0. Moreover, by the maximum principle and the Hopf lemma, we
obtain u−
0 is a positive solution of equation (Eλ) . ¤
Now, we begin to show the proof of Theorem 1.1: By Theorems 3.3, 4.2, for λ ∈ ³0,qΛ0
2
´
, equation (Eλ) has two positive solutions umin and u−0 such that umin ∈ M+λ and u−0 ∈ M−λ. Since M+λ ∩ M−λ = ∅, this implies that umin and u−0 are different.
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