Multiplicity results for a semilinear elliptic
equation involving sign-changing weight function
∗
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 nonlinearities on the number of solutions for a semilinear elliptic equation with sign-changing weight functions. With the help of the Nehari manifold, we prove that there are at least two solutions for the equation (Ea,b).
1
Introduction
1In this paper, we consider the multiplicity results of solutions of the following
semilinear elliptic equation: −∆u = λa (x) uq+ b (x) up in Ω, u ≥ 0, u 6≡ 0 in Ω, u = 0 in ∂Ω, (Ea,b)
where Ω is a bounded domain in RN, 0 ≤ q < 1 < p < 2∗ − 1 (2∗ = 2N N −2
if N ≥ 3, 2∗ = ∞ if N = 2), λ > 0 and the weight functions a, b are satisfying
the following conditions:
(A) a+ = max {a, 0} 6≡ 0 and a ∈ Lrq(Ω) where r
q = r−(q+1)r for some r ∈
(q + 1, 2∗) , with in addition a (x) ≥ 0 a.e. in Ω in case q = 0;
(B) b+ = max {b, 0} 6≡ 0 and b ∈ Lsp(Ω) where s
p = s−(p+1)s for some s ∈
(p + 1, 2∗) .
The fact that the number of positive solutions of equation (Ea,b) is affected by
the concave and convex nonlinearities has been the focus of a great deal of research ∗2000 Mathematics Subject Classification. 35J20, 35J65
in recent years. If the weight functions a ≡ b ≡ 1, the authors Ambrosetti-Brezis-Cerami [1] have investigated equation (E1,1) . They found that there exists
λ0 > 0 such that equation (E1,1) 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. Wu [8] proved that equation (Ea,b) has at least two positive solutions
under the assumptions the weight functions a change sign in Ω, b ≡ 1 and λ is sufficiently small. For more general result, de Figueiredo-Grossez-Ubilla [4] proved the following result:
Theorem 1.1 Assume that the conditions (A) and (B) hold, and in addition (C1) there exists a nonempty open subset Ω1 ⊂ Ω such that, on Ω1, a (x) ≥ ε1
for some ε1 > 0 and b (x) is bounded from below;
(C2) there exists a nonempty open subset Ω2 ⊂ Ω such that, on Ω2, b (x) ≥ ε2 for
some ε2 > 0 and a (x) is bounded from below.
Then there exists λ > 0 such that if λ ∈ ¡0, λ¢, then equation (Ea,b) has at least
two solutions.
The main purpose of this paper is using a new method to improve Theorem 1.1. In particular, we do this without assuming the conditions (C1) and (C2) . Our main result is the following.
Theorem 1.2 Assume that the conditions (A) and (B) hold. Then there exists Λ0 > 0 such that for λ ∈ (0, Λ0) , equation (Ea,b) has at least two solutions.
Among the other interesting problems which are similar of equation (Ea,b) for
q = 1, Brown-Zhang [2] have investigated the following equation:
½
−∆u = λa (x) u + b (x) |u|p−1u in Ω, u ∈ H1
0(Ω) ,
(1) where Ω is a bounded domain in RN and a, b : Ω → R are smooth functions which
change sign in Ω. They found existence and non-existence results for positive solutions of equation (1) as λ changes.
This paper is organized as follows. In section 2, we give some notations and preliminaries. In section 3, we prove that equation (Ea,b) has at least two solutions
for λ sufficiently small.
2
Notations and Preliminaries
Throughout this section, we denote by Sl the best Sobolev constant for the
em-bedding of H1
0(Ω) in Ll(Ω) , where 1 < l ≤ 2∗. Associated with equation (Ea,b) ,
we consider the energy functional Jλ, for each u ∈ H01(Ω) ,
Jλ(u) = 1 2 Z Ω |∇u|2dx − λ q + 1 Z Ω a |u|q+1dx − 1 p + 1 Z Ω b |u|p+1dx.
It is well-known that the solutions of equation (Ea,b) are the critical points of
the energy functional Jλ (see Rabinowitz [6]). Moreover, we consider the Nehari
minimization problem: for λ > 0,
αλ(Ω) = inf {Jλ(u) | u ∈ Mλ(Ω)} ,
where Mλ(Ω) = {u ∈ H01(Ω) \ {0} | hJλ0 (u) , ui = 0} .
Define
ψλ(u) = hJλ0 (u) , ui = kuk 2 H1 − λ Z Ω a |u|q+1dx − Z Ω b |u|p+1dx. Then for u ∈ Mλ(Ω) , hψλ0 (u) , ui = 2 kuk2H1 − (q + 1) λ Z Ω a |u|q+1dx − (p + 1) Z Ω b |u|p+1dx.
Similarly to the method used in Tarantello[7], we split Mλ(Ω) into three parts:
M+λ (Ω) = {u ∈ Mλ(Ω) | hψ0λ(u) , ui > 0} ,
M0
λ(Ω) = {u ∈ Mλ(Ω) | hψ0λ(u) , ui = 0} ,
M−
λ (Ω) = {u ∈ Mλ(Ω) | hψ0λ(u) , ui < 0} .
Then, we have the following results.
Lemma 2.1 There exists λ1 > 0 such that for each λ ∈ (0, λ1) we have M0λ(Ω) =
∅.
Proof. Suppose otherwise, that is M0
λ(Ω) 6= ∅ for all λ > 0. Then for u0 ∈
M0 λ(Ω) we have 0 = hψ0λ(u) , ui = (1 − q) kuk2H1 − (p − q) Z Ω b |u|p+1dx (3) = (1 − p) kuk2H1 − λ (q − p) Z Ω a |u|q+1dx. (4)
By the Sobolev imbedding theorem, there exist positive numbers C1, C2 such that
kuk2H1 ≤ C1kukp+1H1 and kuk2H1 ≤ λC2kukq+1H1
or
kukH1 ≥ C11/1−p and kukH1 ≤ (λC2)1/1−q.
If λ is sufficiently small, this is impossible. Thus, we can conclude that there exists λ1 > 0 such that for λ ∈ (0, λ1), we have M0λ(Ω) = ∅. ¤
Lemma 2.2 (i) If u ∈ M+ λ (Ω) , then R Ωa |u| q+1dx > 0; (ii) If u ∈ M− λ (Ω) , then R Ωb |u| p+1dx > 0
Proof. The proof is immediate from (3) and (4) . ¤ By Lemma 2.1, for λ ∈ (0, λ1) we write Mλ(Ω) = M+λ (Ω) ∪ M−λ (Ω) and
define α+ λ (Ω) = inf u∈M+λ(Ω) Jλ(u) ; α−λ (Ω) = inf u∈M−λ(Ω) Jλ(u) .
The following lemma shows that the minimizers on Mλ(Ω) are ”usually” critical
points for Jλ.
Lemma 2.3 For λ ∈ (0, λ1). If u0 is a local minimizer for Jλ on Mλ(Ω) , then
J0
λ(u0) = 0 in H−1(Ω) .
Proof. Our proof is almost the same as that in Brown-Zhang [2, Theorem 2.3]. ¤ For each u ∈ M− λ (Ω) , we write tmax = Ã (1 − q) kuk2H1 (p − q)RΩb |u|p+1dx ! 1 p−1 > 0.
Then, we have the following lemmas. Lemma 2.4 Let λ2 = ³ p−1 p−q ´ ³ 1−q p−q ´1−q p−1³ 1 kakLrqSrq+1 ´ ³ 1 Sp+1s kbkLsp ´1−q p−1 . Then for each u ∈ M− λ (Ω) and λ ∈ (0, λ2), we have
(i) if RΩa |u|q+1dx ≤ 0, then Jλ(u) = supt≥0Jλ(tu) > 0;
(ii) if RΩa |u|q+1dx > 0, then there is a unique 0 < t+ = t+(u) < t
max such that
t+u ∈ M+ λ and Jλ ¡ t+u¢= inf 0≤t≤tmax
Jλ(tu) , Jλ(u) = sup t≥tmax
Jλ(tu) .
Proof. Fix u ∈ M−λ (Ω). Let
h (t) = t1−qkuk2
H1 − tp−q
Z
Ω
b |u|p+1dx for t ≥ 0.
We have h(0) = 0, h(t) → −∞ as t → ∞, h (t) is achieves its maximum at tmax,
increasing for t ∈ [0, tmax) and decreasing for t ∈ (tmax, ∞) . Moreover,
h (tmax) = Ã (1 − q) kuk2H1 (p − q)RΩb |u|p+1dx !1−q p−1 kuk2H1 − Ã (1 − q) kuk2H1 (p − q)RΩb |u|p+1dx !p−q p−1 Z Ω b |u|p+1dx = kukq+1H1 "µ 1 − q p − q ¶1−q p−1 − µ 1 − q p − q ¶p−q p−1 # Ã kukp+1H1 R Ωb |u| p+1dx !1−q p−1 ≥ kukq+1H1 µ p − 1 p − q ¶ µ 1 − q p − q ¶1−q p−1 Ã 1 Ssp+1kbkLsp !1−q p−1 ,
or h (tmax) ≥ kukq+1H1 µ p − 1 p − q ¶ µ 1 − q p − q ¶1−q p−1 Ã 1 Ssp+1kbkLsp !1−q p−1 . (5) (i)RΩa |u|q+1dx ≤ 0. There is a unique t− > t
max such that h (t−) =
R Ωa |u| q+1dx and h0(t−) < 0. Now, (1 − q)°°t−u°°2 H1 − (p − q) Z Ω b¯¯t−u¯¯p+1dx = ¡t−¢2+q · (1 − q)¡t−¢−qkuk2 H1 − (p − q) ¡ t−¢p−q−1 Z Ω b |u|p+1dx ¸ = ¡t−¢2+qh0¡t−¢< 0, and J0 λ ¡ t−u¢, t−u® = ¡t−¢2kuk2H1 − ¡ t−¢q+1λ Z Ω a |u|q+1dx −¡t−¢p+1 Z Ω b |u|p+1dx = ¡t−¢q+1 · h¡t−¢− λ Z Ω a |u|q+1dx ¸ = 0. Thus, t−u ∈ M−
λ (Ω) or t− = 1. Since for t > tmax, we have
(1 − q) ktuk2H1 − (p − q) Z Ω b |tu|p+1ds < 0, d 2 dt2Jλ(tu) < 0 and d dtJλ(tu) = t kuk 2 H1 − λtq Z Ω a |u|q+1dx − tp Z Ω b |tu|p+1dx = 0 for t = t−.
Thus, Jλ(u) = supt≥0Jλ(tu) . Moreover,
Jλ(u) ≥ Jλ(tu) ≥ t 2 2 kuk 2 H1 − tp+1 p + 1 Z Ω
b |u|p+1dx for all t ≥ 0
Similar to the argument in the function h (t), we obtain
Jλ(u) ≥ p − 1 2 (p + 1) Ã kukp+1H1 R Ωb |u| p+1dx ! 2 p−1 > 0
(ii)RΩa |u|qdx > 0. By (5) and h (0) = 0 < λ
Z
Ω
a |u|q+1dx ≤ λ kakLrq Srq+1kukq+1H1
< kukq+1H1 µ p − 1 p − q ¶ µ 1 − q p − q ¶1−q p−1 Ã 1 Ssp+1kbkLsp !1−q p−1 ≤ h (tmax) for λ ∈ (0, λ2)
there are unique t+ and t− such that 0 < t+< t max < t−, h¡t+¢= λ Z Ω a |u|q+1dx = h¡t−¢ and h0¡t+¢ > 0 > h0¡t−¢. We have t+u ∈ M+
λ (Ω) , t−u ∈ M−λ (Ω), and Jλ(t−u) ≥ Jλ(tu) ≥ Jλ(t+u) for
each t ∈ [t+, t−] and J
λ(t+u) ≤ Jλ(tu) for each t ∈ [0, t+] . Thus, t− = 1 and
Jλ(u) = sup t≥0 Jλ(tu) , Jλ ¡ t+u¢= inf 0≤t≤tmax Jλ(tu) .
This completes the proof. ¤
Lemma 2.5 (i) αλ(Ω) ≤ α+λ (Ω) < 0;
(ii) Jλ is coercive and bounded below on Mλ(Ω) .
Proof. (i) Given u ∈ M+
λ (Ω) , we have Jλ(u) = p − 1 2 (p + 1)kuk 2 H1 + µ q − p (p + 1) (q + 1) ¶ λ Z Ω a |u|q+1dx < · 1 2− 1 q + 1 ¸ p − 1 p + 1kuk 2 H1 < 0 This yields αλ(Ω) ≤ αλ+(Ω) < 0.
(ii) For u ∈ Mλ(Ω) , we have kuk2H1 = λ
R
Ωa |u|
q+1dx +R Ωb |u|
p+1dx. Then by
the H¨older and Young inequality,
Jλ(u) = p − 1 2 (p + 1)kuk 2 H1 − λ µ p − q (p + 1) (q + 1) ¶ Z Ω a |u|q+1dx ≥ p − 1 2 (p + 1)kuk 2 H1 − λ µ p − q (p + 1) (q + 1) ¶ kakLrq Srq+1kukq+1H1
Thus, Jλ is coercive and bounded below on Mλ(Ω). ¤
3
Proof of Theorem 1
First, we will use the idea of Ni–Takagi [5] to get the following results.
Lemma 3.1 For each u ∈ Mλ(Ω) , there exist ² > 0 and a differentiable function
ξ : B (0; ²) ⊂ H1
0(Ω) → R+ such that ξ (0) = 1, the function ξ (v) (u − v) ∈
Mλ(Ω) and hξ0(0) , vi = 2 R Ω∇u∇vdx − (q + 1) λ R Ωa |u| q−1uvdx − (p + 1)R Ωb |u| p−1uvdx (1 − q)RΩ|∇u|2dx − (p − q)RΩb |u|p+1dx (6) for all v ∈ H1 0(Ω) .
Proof. For u ∈ Mλ(Ω), define a function F : R × H01(Ω) → R by Fu(ξ, w) = hJλ0 (ξ (u − w)) , ξ (u − w)i = ξ2 Z Ω |∇ (u − w)|2dx − ξq+1λ Z Ω a |u − w|q+1dx −ξp+1 Z Ω b |u − w|p+1dx.
Then Fu(1, 0) = hJλ0 (u) , ui = 0 and
d dξFu(1, 0) = 2 Z Ω |∇u|2dx − (q + 1) λ Z Ω a |u|q+1dx − (p + 1) Z Ω b |u|p+1dx = (1 − q) Z Ω |∇u|2dx − (p − q) Z Ω b |u|p+1dx 6= 0.
According to the implicit function theorem, there exist ² > 0 and a differentiable function ξ : B (0; ²) ⊂ H1¡RN¢→ R such that ξ (0) = 1,
hξ0(0) , vi = 2 R Ω∇u∇vdx − (q + 1) λ R Ωa |u| q−1uvdx − (p + 1)R Ωb |u| p−1uvdx (1 − q)RΩ|∇u|2dx − (p − q)RΩb |u|p+1dx and Fu(ξ (v) , v) = 0 for all v ∈ B (0; ²) which is equivalent to hJ0
λ(ξ (v) (u − v)) , ξ (v) (u − v)i = 0 for all v ∈ B (0; ²) ,
that is ξ (v) (u − v) ∈ Mλ(Ω) . ¤
Lemma 3.2 For each u ∈ M−
λ (Ω) , there exist ² > 0 and a differentiable function
ξ− : B (0; ²) ⊂ H1
0 (Ω) → R+ such that ξ−(0) = 1, the function ξ−(v) (u − v) ∈
M−λ (Ω) and D¡ ξ−¢0(0) , v E = 2 R Ω∇u∇vdx − (q + 1) λ R Ωa |u| q−1uvdx − (p + 1)R Ωb |u| p−1uvdx (1 − q)RΩ|∇u|2dx − (p − q)RΩb |u|p+1dx (7) for all v ∈ H1 0(Ω) .
Proof. Similar to the argument in Lemma 3.1, there exist ² > 0 and a dif-ferentiable function ξ− : B (0; ²) ⊂ H1¡RN¢ → R such that ξ−(0) = 1 and
ξ−(v) (u − v) ∈ M
λ(Ω) for all v ∈ B (0; ²) . Since
hψ0
λ(u) , ui = (1 − q) kuk2H1 − (p − q)
Z
Ω
Thus, by the continuity of the function ξ−, we have ψ0 λ ¡ ξ−(v) (u − v)¢, ξ−(v) (u − v)® = (1 − q)°°ξ−(v) (u − v)°°2H1 − (p − q) Z Ω b¯¯ξ−(v) (u − v)¯¯p+1dx < 0
if ² sufficiently small, this implies that ξ−(v) (u − v) ∈ M−
λ (Ω) . ¤
Proposition 3.3 Let Λ0 = min {λ1, λ2}, then for λ ∈ (0, Λ0)
(i) there exists a minimizing sequence {un} ⊂ Mλ(Ω) such that
Jλ(un) = αλ(Ω) + o (1) ,
Jλ0 (un) = o (1) in H−1(Ω) ;
(ii) there exists a minimizing sequence {un} ⊂ M−λ (Ω) such that
Jλ(un) = α−λ (Ω) + o (1) ,
J0
λ(un) = o (1) in H−1(Ω) .
Proof. (i) By Lemma 2.5 (ii) and the Ekeland variational principle [3], there exists a minimizing sequence {un} ⊂ Mλ(Ω) such that
Jλ(un) < αλ(Ω) + 1 n (8) and Jλ(un) < Jλ(w) + 1 nkw − unkH1 for each w ∈ Mλ(Ω) . (9)
By taking n large, from Lemma 2.5 (i) we have
Jλ(un) = µ 1 2 − 1 p + 1 ¶ kunk2H1 − µ 1 q + 1 − 1 p + 1 ¶ λ Z Ω a |un|q+1dx (10) < αλ(Ω) + 1 n < αλ(Ω) 2 . This implies kakLrq Srq+1kunkq+1H1 ≥ Z Ω a |un|q+1dx > − (q + 1) (p + 1) λ (p − q) αλ(Ω) 2 > 0. (11) Consequently un 6= 0 and putting together (10), (11) and the H¨older inequality,
we obtain kunkH1 > · (q + 1) (p + 1) λ (p − q) αλ(Ω) 2 S −(q+1) r kak−1Lrq ¸ 1 q+1 (12)
and kunkH1 < · 2λ (p − q) (p − 1) (q + 1)kakLrq Srq+1 ¸ 1 1−q (13) Now, we will show that
kJ0
λ(un)kH−1 → 0 as n → ∞.
Applying Lemma 3.1 with unto obtain the functions ξn: B (0; ²n) → R+for some
²n > 0, such that ξn(w) (un− w) ∈ Mλ(Ω) . Fixed n ∈ N, we choose 0 < ρ < ²n.
Let u ∈ H1
0(Ω) with u 6≡ 0 and let wρ = kukρu
H1. We set ηρ = ξn(wρ) (un− wρ) .
Since ηρ∈ Mλ(Ω) , we deduce from (9) that
Jλ(ηρ) − Jλ(un) ≥ −1
nkηρ− unkH1
and by the mean value theorem, we have
hJ0 λ(un) , ηρ− uni + o ¡ kηρ− unkH1 ¢ ≥ −1 nkηρ− unkH1. Thus, hJ0 λ(un) , −wρi + (ξn(wρ) − 1) hJλ0 (un) , (un− wρ)i (14) ≥ −1 n kηρ− unkH1 + o ¡ kηρ− unkH1 ¢ .
From ξn(wρ) (un− wρ) ∈ Mλ(Ω) and (14) it follows that
−ρ ¿ Jλ0 (un) , u kukH1 À + (ξn(wρ) − 1) hJλ0 (un) − Jλ0 (ηρ) , (un− wρ)i ≥ −1 nkηρ− unkH1+ o ¡ kηρ− unkH1 ¢ . Thus, ¿ J0 λ(un) , u kukH1 À ≤ kηρ− unkH1 nρ + o¡kηρ− unkH1 ¢ ρ +(ξn(wρ) − 1) ρ hJ 0 λ(un) − Jλ0 (ηρ) , (un− wρ)i . (15) Since kηρ− unkH1 ≤ ρ |ξn(wρ)| + |ξn(wρ) − 1| kunkH1 and lim n→∞ |ξn(wρ) − 1| ρ ≤ kξ 0 n(0)k .
If we let ρ → 0 in (15) , then by (13) we can find a constant C > 0, independent of ρ, such that ¿ J0 λ(un) , u kukH1 À ≤ C n (1 + kξ 0 n(0)k) .
We are done once we show that kξ0
n(0)k is uniformly bounded in n. By (6) , (13)
and the H¨older inequality, we have
hξ0 n(0) , vi ≤ d kvkH1 ¯ ¯(1 − q)RΩ|∇un|2dx − (p − q) R Ωb |un| p+1dx¯¯ for some d > 0.
We only need to show that ¯ ¯ ¯ ¯(1 − q) Z Ω |∇un|2dx − (p − q) Z Ω b |un|p+1dx ¯ ¯ ¯ ¯ > c (16) for some c > 0 and n large enough. We argue by contradiction. Assume that there exists a subsequence {un} , we have
(1 − q) Z Ω |∇un|2dx − (p − q) Z Ω b |un|p+1dx = o (1) . (17)
Combining (17) with (12) , we can find a suitable constant k > 0 such that Z
Ω
b |un|p+1dx ≥ k for n sufficiently large. (18)
In addition (17) , and the fact that un ∈ Mλ(Ω) also give
λ Z Ω a |un|q+1dx = kunk2H1 − Z Ω b |un|p+1dx = p − 1 1 − q Z Ω b |un|p+1dx + o (1) and kunkH1 ≤ · λ µ p − q p − 1 ¶ kakLp∗ Srq+1 ¸ 1 1−q + o (1) . (19) This implies Iλ(un) = K (p, q) Ã kunk2pH1 R Ωb |un| p+1dx ! 1 p−1 − λ Z Ω a |un|q+1dx = µ 1 − q p − q ¶ p p−1µp − 1 1 − q ¶ ³ p−q 1−q ´p¡R Ωb |un| p+1dx¢p R Ωb |un| p+1dx 1 p−1 −p − 1 1 − q Z Ω b |un|p+1dx = o (1) . (20)
However, by (18) , (19) and λ ∈ (0, Λ0) Iλ(un) ≥ K (p, q) Ã kunk2pH1 Ssp+1kbkLspkunkp+1H1 ! 1 p−1 − λSq+1 r kakLrqkunkq+1H1 = kunkq+1H1 µ K (p, q) S p+1 1−p s kbk 1 1−p Lsp kunk−qH1 − λSrq+1kakLrq ¶ ≥ kunkq+1H1 ( K (p, q) S p+1 1−p s kbk 1 1−p Lsp λ −q 1−q ·µ p − q p − 1 ¶ kakLrq Ssq+1 ¸−q 1−q −λSq+1 r kakLrq ª .
this contradicts (20). We get ¿ J0 λ(un) , u kukH1 À ≤ C n.
This completes the proof of (i).
(ii) Similarly, by using Lemma 3.2, we can prove (ii) . We will omit detailed proof
here. ¤
Now, we establish the existence of a local minimum for Jλ on M+λ (Ω).
Theorem 3.4 Let Λ0 > 0 as in Proposition 3.3, then for λ ∈ (0, Λ0) the
func-tional Jλ has a minimizer u+0 in M+λ (Ω) and it satisfies
(i) Jλ ¡ u+ 0 ¢ = αλ(Ω) = α+λ (Ω) ;
(ii) u+0 is a solution of equation (Ea,b) .
Proof. Let {un} ⊂ Mλ(Ω) be a minimizing sequence for Jλ on Mλ(Ω) such that
Jλ(un) = αλ(Ω) + o (1) and Jλ0 (un) = o (1) in H−1(Ω) .
Then by Lemma 2.5 and the compact imbedding theorem, there exist a subse-quence {un} and u+0 ∈ H01(Ω) such that
un * u+0 weakly in H01(Ω)
and
un → u+0 strongly in Lr(Ω) for 1 < r < 2∗. (21)
First, we claim that RΩa¯¯u+ 0
¯ ¯q+1
dx 6= 0. If not, by (21) and the H¨older inequality
we can conclude that Z Ω a |un|q+1dx → Z Ω a¯¯u+ 0 ¯ ¯q+1 dx = 0 as n → ∞. Thus, Z Ω |∇un|2dx = Z Ω p |un|p+1dx + o (1)
and Jλ(un) = µ 1 2 − 1 p + 1 ¶ Z Ω |∇un|2dx + o (1) ,
this contradicts Jλ(un) → αλ(Ω) < 0 as n → ∞. Moreover,
o (1) = hJ0
λ(un) , φi = hJλ0 (u0) , φi + o (1) for all φ ∈ H01(Ω) .
Thus, u+
0 ∈ Mλ is a nonzero solution of equation (Ea,b) and Jλ
¡
u+ 0
¢
≥ αλ(Ω) .
We now prove that Jλ
¡ u+0¢= αλ(Ω) . Since Jλ ¡ u+ 0 ¢ = 1 2 ° °u+ 0 ° °2 H1 − λ q + 1 Z Ω a¯¯u+ 0 ¯ ¯q+1 dx − 1 p + 1 Z Ω b¯¯u+ 0 ¯ ¯p+1 dx = µ 1 2− 1 p + 1 ¶° °u+ 0 ° °2 H1 + µ λ p + 1 − λ q + 1 ¶ Z Ω a¯¯u+ 0 ¯ ¯q+1 dx ≤ lim inf n→∞ µµ 1 2− 1 p + 1 ¶ kunk2H1 + µ λ p + 1 − λ q + 1 ¶ Z Ω a |un|q+1dx ¶ = lim inf n→∞ Jλ(un) = αλ(Ω) . Thus, Jλ ¡
u+0¢ = αλ(Ω) . Moreover, we have u+0 ∈ M+λ (Ω) . In fact, if u+0 ∈
M−
λ (Ω) , by Lemma 2.4, there are unique t+0 and t−0 such that t+0u+0 ∈ M+λ (Ω)
and t− 0u+0 ∈ M−λ (Ω) , we have t+0 < t−0 = 1. Since d dtJλ ¡ t+0u+0¢= 0 and d 2 dt2Jλ ¡ t+0u+0¢ > 0,
there exists t+0 < ¯t ≤ t−0 such that Jλ
¡ t+0u+0¢< Jλ ¡ ¯tu+ 0 ¢ . By Lemma 2.4, Jλ ¡ t+ 0u+0 ¢ < Jλ ¡ ¯tu+ 0 ¢ ≤ Jλ ¡ t− 0u+0 ¢ = Jλ ¡ u+ 0 ¢ ,
which is a contradiction. Since Jλ
¡ u+ 0 ¢ = Jλ ¡¯ ¯u+ 0 ¯ ¯¢ and ¯¯u+ 0 ¯ ¯ ∈ M+ λ (Ω) , by
Lemma 2.3 we may assume that u+0 is a solution of equation (Ea,b). ¤
Next, we establish the existence of a local minimum for Jλ on M−λ (Ω)
Theorem 3.5 Let Λ0 > 0 as in Proposition 3.3, then for λ ∈ (0, Λ0) the
func-tional Jλ has a minimizer u−0 in M−λ (Ω) and it satisfies
(i) Jλ
¡
u−0¢= α−λ (Ω) ; (ii) u−
0 is a solution of equation (Ea,b) .
Proof. By Proposition 3.3 (ii), there exists a minimizing sequence {un} for Jλ
on M−
λ (Ω) such that
By Lemma 2.5 and the compact imbedding theorem, there exist a subsequence
{un} and u−0 ∈ M−λ (Ω) such that
un * u−0 weakly in H01(Ω), un → u−0 strongly in Ls(Ω) and un → u−0 strongly in Lr(Ω) for 1 ≤ r < 2∗. Since o (1) = hJ0
λ(un) , φi = hJλ0 (u0) , φi + o (1) for all φ ∈ H01(Ω)
and 0 > hψ0 λ(un) , uni = (2 − q) kunk2H1 − (p − q) Z ∂Ω g |un|pds ≥ (2 − q) ku0k2H1 − (p − q) Z ∂Ω g |u0|pds. Thus, u−
0 ∈ M−λ (Ω) is a nonzero solution of equation (Ea,b) . We now prove that
un → u−0 strongly in H01(Ω). Suppose otherwise, then
° °u− 0 ° ° H1 < lim inf n→∞ kunkH1 and so ° °u− 0 ° °2 H1 − λ Z Ω a¯¯u− 0 ¯ ¯q+1 dx − Z Ω b¯¯u− 0 ¯ ¯p+1 dx < lim inf n→∞ µ kunk2H1 − λ Z Ω a |un|q+1dx − Z Ω b |un|p+1dx ¶ = 0.
this contradicts u−0 ∈ M−λ (Ω). Hence un → u−0 strongly in H01(Ω). This implies
Jλ(un) → Jλ ¡ u− 0 ¢ = α− λ (Ω) as n → ∞. Since Jλ ¡ u−0¢= Jλ ¡¯ ¯u− 0 ¯ ¯¢ and¯¯u− 0 ¯ ¯ ∈ M−
λ(Ω) by Lemma 2.3 we may assume that
u−
0 is a solution of equation (Ea,b). ¤
Now, we complete the proof of Theorem 1.2: By Theorems 3.4, 3.5 equation (Ea,b) there exist two solutions u+0 and u−0 such that u0+ ∈ M+λ (Ω) , u−0 ∈ M−λ (Ω).
Since M+
λ (Ω) ∩ M−λ (Ω) = ∅, this implies that u+0 and u−0 are different.
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