Three Positive Solutions of Nonhomogeneous
Semilinear Elliptic Equations
Huei-li Lin
Center for General Education Chang Gung University, Tao-Yuan, Taiwan
Hwai-chiuan Wang
Department of Applied Mathematics Hsuan Chuang University, Hsinchu, TaiwanTsung-fang Wu
Department of Applied Mathematics,
National University of Kaohsiung, Kaohsiung 811, Taiwan
Abstract
In this paper, we use new analyses to assert that there are three positive solutions of equation (1.1) in infinite cylinder domain with hole A\D.
Key words: Semilinear elliptic equations, Positive solutions, Palais–Smale
1 Introduction
Throughout this paper, let z = (x, y) ∈ RN −m× Rm= RN, where 1 ≤ m ≤ N − 1 and we denote ω ⊂ RN −m is a bounded smooth domain, the infinite
cylinder domain A = ω × Rm; the infinite cylinder domain with hole A\D,
where D $ A is a bounded domain in RN which contained in BN(0; r
0) ∩ A
for some r0 > 0.
In this paper, we consider the multiplicity of positive solutions for semilinear elliptic equation −∆u + u = |u|p−2u + h (z) in Ω; u ∈ H1 0(Ω) , (1.1) where 2 < p < 2N
N −2(N ≥ 3), Ω = A\D and h (x, y) ∈ L2(Ω) \ {0}. Associated
with equation (1.1), we consider the functionals a, b, and Jh, for u ∈ H01(Ω),
a(u) = Z Ω ³ |∇u|2+ u2´, b(u) = Z Ω|u| p, J h(u) = 1 2a(u) − 1 pb (u) − Z Ωhu.
By Rabinowitz [15, Proposition B.10.], a, b, and Jh are of C1. It is
well-known that the solutions of equation (1.1) and the critical points of the energy functional Jh are the same. For the limiting case of equation (1.1) : h = 0, we
consider the semilinear elliptic equation
−∆u + u = |u|p−2u in Ω; u ∈ H1 0(Ω) . (1.2)
It is known that the existence of positive solutions of the homogeneous equa-tion (1.2) is affected by the shape of the domain. By the Rellich compactness theorem and the minimax method, it is easy to obtain a positive solution of equation (1.2) in bounded domains (see Ambrosetti-Rabinowitz [3]). For gen-eral unbounded domains Ω, because the luck of compactness, the existence of positive solutions of equation (1.2) in Ω is very difficult and unclear. The breakthrough was made by Esteban-Lions [11]. They asserted that the equa-tion (1.2) in Esteban–Lions domain does not admit any nontrivial soluequa-tion. Recently, there have been some progresses for the existence of positive solu-tions of equation (1.2) in unbounded domains as follows: Benci-Cerami [4] for Ω is an exterior domain, Berestycki-Lions [5] for Ω = RN, Lien-Tzeng-Wang
[14] for Ω is an infinite cylinder domain A, Chen-Wang [6] for Ω is an inte-rior flask domain, Del Pino-Felmer [9,10] for Ω is a quasicylindrical domain, Wang [17] for Ω is a Esteban–Lions domain with holes, Wu [19] for Ω is a multi–bump domain.
In this paper, we are interesting the multiplicity of positive solutions for equation (1.1) in A\D. Before stating our main results, we need the some notations: let λ1 be the first eigenvalue of −∆ in ω with the Dirichlet
prob-lem and φ1 the corresponding positive eigenfunction to λ1. Then we have the
Theorem 1.1 There exist positive numbers d0, δ such that if khkL2 < d0 and 0 ≤ h (x, y) ≤ c exp µ − q 1 + λ1+ δ |y| ¶
for all (x, y) ∈ A\D,
for some c > 0, then equation (1.1) in A\D has at least three positive solutions.
Our result generalizes previous results in two folds.
(1) Hirano [12], Zhu [20], and Cao-Zhu [8] proved in RN and Hsu-Wang [13]
in an exterior domain, equation (1.1) admits two positive solutions. We generalize their results to obtain three positive solutions.
(2) Consider the generalized equation of Equation (1.1)
−∆u + u = p(z) |u|p−2u + h (z) in Ω; u ∈ H1 0 (Ω) , (1.3)
where 0 < p(z) 1. Adachi-Tanaka [1] asserted that there are three positive solutions of equation (1.3) in RN. RN is contractible and there
is a ground state solution in it. Our domain A\D is not contractible and there is no any ground state solution in it. So we need more analyses to work for it. We generalize the result of Adachi-Tanaka [1] to that p(x) = 1 and Ω = A\D.
Main tools are from Adachi-Tanaka [1], Tarantello [16], and Cao-Zhou [8]. We then develop analyses which includes some lemmas for the limiting case
h = 0 to complete our theory.
2 (PS)–Theory
We define the Palais–Smale (denoted by (PS)) sequences and (PS)–conditions in H1
0(Ω) for Jh as follows.
Definition 2.1 (i) For β ∈ R, a sequence {un} is a (PS)β–sequence in H01(Ω)
for Jh if Jh(un) = β + o(1) and Jh0(un) = o(1) strongly in H−1(Ω) as n → ∞;
(ii) Jh satisfies the (PS)β–condition in H01(Ω) if every (PS)β–sequence in H1
0 (Ω) for Jh contains a convergent subsequence.
For the limiting case h = 0, we consider the Nehari minimazation problem:
α0(Ω) = inf
u∈M0
where M0 = {u ∈ H01(Ω) \ {0} | hJ00(u) , ui = 0} . Note that a nonzero critical
point u ∈ H1
0(Ω) of J0 is a ground state solution of equation (1.1) in Ω if
J0(u) = α0(Ω). Then we have the following results.
Lemma 2.2 There is a bijective C1,1 map m from the unit sphere Σ in H1 0(Ω)
to M0. Moreover, M0 is path-connected and there exists a constant c > 0 such
that for u ∈ M0, kukH1 ≥ c and J0(u) ≥ c.
Proof. See Chen-Wang [6]. ¤
Lemma 2.3 Let β > 0 and {un} be a sequence in H01(Ω)\{0} for J0 such that
J0(un) = β + o(1) and a(un) = b(un) + o(1). Then there is a sequence {sn} in
R+ such that s
n= 1 + o(1), {snun} in M0 and J0(snun) = β + o(1).
Proof. See Chen-Wang [6]. ¤
Lemma 2.4 If u ∈ H1 0(Ω) \ {0} , then a (u) p 2 b (u) 1 p−2 ≥ Ã 2p p − 2 !1 2 α0(Ω) 1 2.
Proof. See Chen-Wang [6]. ¤
Lemma 2.5 {un} is a (PS)α0(Ω)–sequence in H
1
0(Ω) for J0 if and only if
J0(un) = α0(Ω) + o (1) and a (un) = b (un) + o (1) . In particular, every min-imizing sequence {un} in M0 of α0(Ω) is a (PS)α0(Ω)–sequence in H
1
0(Ω) for
J0.
Proof. See Wang-Wu [18]. ¤
Associated with equation (1.2) in A, we consider the functional J∞, for u ∈ H1 0(A), J∞(u) = 1 2 Z A ³ |∇u|2+ u2´−1 p Z A|u| p
By Lien-Tzeng-Wang [14], the equation (1.2) in A has a positive solution
w (x) such that J∞(w) = α
0(A). Moreover, the positive solution w (x) of
equation (1.2) plays an important role in describing the asymptotic behavior of a (PS)–sequence for Jh.
Proposition 2.6 Let {un} be a (PS)–sequence in H01(Ω) for Jh. Then there exist a subsequence {un} , an integer k ∈ N∪{0} , k sequences {z1n} , {zn2} , . . . ,
n
zk n
o
of equation (1.2) in A such that |zi n| → ∞ for 1 ≤ i ≤ k, un* u0 weakly in H01(Ω), un= u0+ w1(z − z1n) + w2(z − z1n− zn2) + wk(z − z1n− . . . − zkn) + o(1) in H01(A), Jh(un) = Jh(u0) + Pk i=1J∞(wi) + o(1).
Proof. This is a standard result. See Lien-Tzeng-Wang [14] for analogous
arguments. ¤
Next, we give some properties of the functional J0.
Lemma 2.7 We have
(i) inf {J0(u) | u ∈ M0} = α0(Ω) = α0(A) .
(ii) inf {J0(u) | u ∈ M0} is not achieved.
Proof. See Wang [17]. ¤
Lemma 2.8 There exists a δ0 > 0 such that if u ∈ M0 and J0(u) ≤ α0(A) +
δ0, then Z A y |y| ³ |∇u|2+ u2´dydx 6= 0.
Proof. On the contrary, there exists a sequence {un} in M0 such that
J0(un) = α0(A) + o (1) and Z A y |y| ³ |∇u|2+ u2´dydx = 0.
By Lemmas 2.5, 2.7, {un} is a (PS)α0(A)–sequence in H01(Ω) for J0. By
Proposi-tion 2.6 and Lemma 2.7, there exists a sequence {yn} in Rmsuch that |yn| → ∞
as n → ∞ and
un(x, y) = w(x, y − yn) + o (1) strongly in H01(A) .
Assume yn
|yn| → y0 as n → ∞, where y0 is a unit vector in R
m. Then by the
Lebesgue dominated theorem, we have 0 = Z A y |y| ³ |∇un|2+ u2n ´ dydx = Z A y + yn |y + yn| ³ |∇w|2+ w2´dydx + o (1) = Ã 2p p − 2 ! y0α0(A) + o (1) , which is a contradiction. ¤
3 Existence of a Local Minimum
In this section, we prove that there exists a positive solution of equation (1.1). First, we consider the Nehari manifold Mh, where
Mh = n u ∈ H1 0 (Ω) \ {0} | hJh0 (u) , ui = 0 o . Define ψ (u) = hJ0
h(u) , ui = a (u) − b (u) −
R
Ωhu. Then we have
Lemma 3.1 Suppose that h (z) ≥ 0 satisfies 0 < khkL2 < (p − 2) Ã 1 p − 1 !p−1 p−2 Ã 2p p − 2 !1 2 α (Ω)12 .
Then for each u ∈ Mh, we have hψ0(u) , ui = a (u) − (p − 1) b (u) 6= 0.
Proof. Our proof is almost the same as that in Tarantello [16]. ¤ By Lemma 3.1, we write Mh = M+h ∪ M−h, where
M+ h = {u ∈ Mh | a (u) − (p − 1) b (u) > 0} , M−h = {u ∈ Mh | a (u) − (p − 1) b (u) < 0} , and define αh(Ω) = inf u∈Mh Jh(u) ; α+h (Ω) = inf u∈M+h Jh(u) ; α − h (Ω) = inf u∈M−h Jh(u) . For each u ∈ H1 0 (Ω) \ {0} , we write
tmax = tmax(u) = Ã a (u) (p − 1) b (u) ! 1 p−2 > 0.
By elementary calculus, we have the following two lemmas. Lemma 3.2 For each u ∈ H1
0(Ω) \ {0} , we have the following results.
(i) There is a unique t− = t−(u) > t
max > 0 such that t−u ∈ M−h and Jh(t−u) = maxt≥tmaxJh(tu) ;
(ii) t−(u) is a continuous function for nonzero u;
(iii) M− h = n u ∈ H1 0(Ω) \ {0} | kuk1H1t− ³ u kukH1 ´ = 1o;
(iv) If RΩhu > 0, then there is a unique 0 < t+ = t+(u) < t
max such that
t+u ∈ M+
h and Jh(t+u) = min0≤t≤t−Jh(tu) .
Lemma 3.3 (i) For each u ∈ M+h, we have RΩhu > 0 and Jh(u) < 0. In particular, αh(Ω) ≤ α+h (Ω) < 0;
(ii) Jh is coercive and bounded below on Mh.
Proof. Similar to the proof of theorem 1 in Tarantello [16, p.288]. ¤ Proposition 3.4 Jh satisfies the (PS)β–condition for β < αh(Ω) + α0(A).
Proof. Similar to the proof of corollary 1.10 in Adachi-Tanaka [1]. ¤ Furthermore, we have the following theorem.
Theorem 3.5 Let r0 = ³ 1 p−1 ´ 1 p−2³ 2p p−2 ´1 2 α (Ω)12 . Then (i) M+h ⊂ B (r0) = {u ∈ H01(Ω) | kukH1 < r0} ;
(ii) There is a unique local minimum u0 ∈ M+h of Jh such that Jh(u0) =
α+
h(Ω) = αh(Ω);
(iii) u0 is a positive solution of equation (1.1)
Proof. Similar to the proof of same results in Adachi-Tanaka [1] and
Cao-Zhou [8]. ¤
Remark 3.1 Throughout this paper, let u0 be the positive solution of Equation
(1.1) in Theorem 3.5.
4 Existence of Three Solutions
In this section, we assert that there are three positive solutions of equation (1.1) in Ω = A\D. By Lien-Tzeng-Wang [14], there is a positive ground state solution of Equation (1.2) in A. Let λ1 be the first eigenvalue of −∆ in ω with
the Dirichlet problem, and φ1 the corresponding positive eigenfunction to λ1.
Then we have the following results.
Theorem 4.1 Let w be a positive ground state solution of equation (1.2) in A. Then for each 0 < δ < 1 + λ1 there exist γ > 0 and β > 0 such that
γφ1(x) exp µ − q 1 + λ1+ δ|y| ¶ ≤ u(z) ≤ βφ1(x) exp µ − q 1 + λ1− δ|y| ¶ for z = (x, y) ∈ A.
Proof. See Chen-Chen-Wang [7]. ¤
Lemma 4.2 Let u be a positive solution of equation (1.1) in Ω. Then for any 0 < δ < 1 + λ1, there exist positive constants γ1, γ2 and R0 > r0 such that for
|y| ≥ R0 γ1φ1(x) exp µ − q 1 + λ1+ δ |y| ¶ ≤ u (x, y) ≤ γ2φ1(x) exp µ − q 1 + λ1− δ |y| ¶ .
Proof. By the regularity results, we have u ∈ W2,s(Ω) ∩ C1,θ(Ω) for some
0 < θ < 1 and u (z) → 0 as |y| → ∞.
(i) Take R1 ≥ r0 such that D ⊂ BN(0; R1) ∩ A. For 0 < δ < min {ε, 1} , we
choose R2 > R1 such that
δ − √
1 + λ1 + δ (m − 1)
|y| ≥ 0 for |y| ≥ R2. (4.1)
Define v1(x, y) = φ1(x) exp ³ −√1 + λ1+ δ (|y| − R2) ´ . Let γ1 = inf z∈Ω |y|=R2 u (x, y) v1(x, y) .
Similarly to the proof in Chen-Chen-Wang [7], γ1 > 0. Then min
|y|=R2 (u − γ1v1) (x, y) ≥ 0. By (4.1) , for |y| > R2 4 (u − γ1v1) (x, y) = u − |u|p−2u − h (x, y) − Ã β2 −β (m − 1) |y| ! γ1v1 ≤ u − Ã β2− β (m − 1) |y| ! γ1v1 ≤ (u − γ1v1) (x, y) .
By the maximum principle, for |y| > R2
u (x, y) − γ1v1(x, y) ≥ min
|y|=R2
(u − γ1v1) (x, y) ≥ 0.
Thus, we have
u (x, y) ≥ γ1v1(x, y) = γ1exp (−β (|y| − R2))
= γ1exp (R2β) exp (−β |y|)
≥ γ1exp µ − q 1 + λ1+ δ |y| ¶ for |z| ≥ R2. (4.2)
(ii) We know that positive numbers ε, c exist such that 0 ≤ h (x, y) ≤ c exp µ − q 1 + λ1+ δ |y| ¶ for any (x, y) ∈ Ω. For 0 < δ < 1 + λ1, by (4.2) , there is R3 > R2 > 0 such that
δ
Since lim
|y|→∞u (x, y) = 0, there exists R0 > R3 > 0 such that
1 − up−2(x, y) ≥ 1 − δ
2 for |y| ≥ R0. (4.4)
Let γ = √1 + λ1− δ and v2(x, y) = νφ1(x) exp (−γ (|y| − R)) , where ν =
max
|y|=Ru (x, y) > 0. Thus min|y|=R(v2− u) (x, y) ≥ 0. By (4.3) and (4.4) , for |y| > R0 4 (v2− u) (x, y) = Ã γ2− γ (m − 1) |y| ! v2(x, y) − u + |u|p−2u + h (x, y) ≤ γ2v2(x, y) − Ã 1 −δ 2 ! u (x, y) + h (x, y) = (1 + λ1− δ) (v2(x, y) − u (x, y)) − δ 2u + h (x, y) ≤ (1 + λ1− δ) (v2(x, y) − u (x, y)) .
By the maximum principle, for |y| > R0
v2(x, y) − u (x, y) ≥ min
|y|=R0
(v2 − u) (x, y) ≥ 0.
Thus, we have
u (x, y) ≤ v2(x, y) = ν exp (−γ (|y| − R))
= ν exp (Rγ) exp (−γ |y|)
≤ γ2exp µ − q 1 + λ1− δ |y| ¶ for |y| ≥ R.
This complete the proof. ¤
By Lemma 4.2, there is a R > 0 such that ω ⊂ BN(0; R) ∩ A. For such R,
let ψR: A → [0, 1] be a C∞–function on A such that 0 ≤ ψR≤ 1,
ψR(x, y) = 1 for |y| ≥ R + 1; 0 for |y| ≤ R. For y0 is a unit vector in Rl, we define
vl(x, y) = ψR(x, y) w (x, y − ly0) .
Clearly, vl(x, y) ∈ H01(Ω) .
Lemma 4.3 (i) a (vl) = b (vl) + o (1) as l → ∞;
(ii) J (vl) = α0(A) + o (1) as l → ∞;
(iii) vl * 0 weakly in H01(Ω) as l → ∞.
Proof. See Wang [17, Lemma 30]. ¤
Since Ω is non contraction and in which there is no any ground state solu-tion, we need more analyses.
Lemma 4.4 There exists l0 > 0 such that for l ≥ l0
sup
t≥0Jh(u0+ tvl) < Jh(u0) + α0(A) uniformly in unit vector y0.
Proof. Our proof is almost the same as that in Hsu-Wang [13]. ¤ For the Lusternik-Schnirelman category theory, see Ambrosetti [2] and Adachi-Tanaka [1, Lemma 2.5]. In the following, we take the idea of Adachi-Adachi-Tanaka [1]. For c ∈ R, we denote [Jh ≤ c] = n u ∈ M−h | Jh(u) ≤ c and u ≥ 0 o .
we show for a sufficiently small σ > 0
cat ([Jh ≤ αh(Ω) + α0(A) − σ]) ≥ 2. (4.5) Let A1 = n u ∈ H1 0(Ω) \ {0} ¯ ¯ ¯ kuk1 H1t −³ u kukH1 ´ > 1o∪ {0} A2 = n u ∈ H1 0(Ω) \ {0} ¯ ¯ ¯ kuk1 H1t −³ u kukH1 ´ < 1o.
Lemma 4.5 We have the following results: (i) H1
0 (Ω) \M−h = A1∪ A2;
(ii) M+h ⊂ A1;
(iii) there exist t0 > 1 and l1 ≥ l0 such that u0+ t0vm ∈ A2 for each l ≥ l1,
where l0 is defined as in Lemma 4.4;
(iv) there exist sl∈ (0, 1) such that u0 + slt0vl ∈ M−h for each l ≥ l1;
(v) α−
h (Ω) < αh(Ω) + α0(A) .
Proof. Our proof is almost the same as that in Tarantello [16]. ¤ By Lemma 4.5 (iv) , there exist sl ∈ (0, 1) such that u0 + slt0vl ∈ M−h for
each l ≥ l1. For l ≥ l1, we define a map Fl: Sm−1 → H01(Ω) by
Fl(y0) (z) = u0(z) + slt0vl(z) for y0 ∈ Sm−1.
Lemma 4.6 There exists a sequence {σl} in R+ such that Fl
³
Sm−1´⊂ [Jh ≤ αh(Ω) + α0(A) − σl] .
Proof. By Lemma 4.5 (iv) and Lemma 4.4, we have that for each l ≥ l1,
Fl(z) = u0+slt0vl∈ M−h and Jh(Fl(z)) = Jh(u0+ slt0vl) ≤ αh(Ω)+α0(A)−
σl, the conclusion holds. ¤
For c > 0, we define
bc(u) =
R
Ωc |u|p; Ic(u) = 12a (u) −1pbc(u) ;
MIc = {u ∈ H01(Ω) \ {0} | hIc0(u) , ui = 0} .
Therefore, if u ∈ MIc, then a (u) = bc(u), and Ic(u) =
³ 1 2 − 1p
´
bc(u). Recall
that there exist unique t− = t−(u) > 0 and t0 = t0(u) > 0 such that t−u ∈
M− h, t0u ∈ M0, and t0(u) = ³ 1 b(u) ´1/p−2 . Similarly, we have
Lemma 4.7 For each u ∈ Σ, we have the following results: (i) there exists a unique tc(u) > 0 such that tc(u) u ∈ M
Ic and max t≥0 Ic(tu) = Ic(t c(u) u) = Ã 1 2 − 1 p ! bc(u)− 2 p−2 ;
(ii) for 0 < µ < 1, there exists a d1(µ) > 0 such that for khkL2 < d1(µ)
Jh ³ t−u´≥ (1 − µ)p−2p J 0 ³ t0u´− 1 2µkhk 2 L2.
Proof. (i) By elementary calculus. (ii) For 0 < µ < 0, let c = 1
1−µ, tc> 0 and t0 = t0(u) > 0 such that tcu ∈ MIc
and t0u ∈ M 0. We have ¯ ¯ ¯ ¯ Z Ωht cudz ¯ ¯ ¯ ¯≤ ktcukH1khkL2 ≤ µ 2kt cuk2 H1 + 1 2µkhk 2 L2.
Then by part (i) , sup t≥0Jh(tu) ≥ Jh(t cu) ≥ 1 − µ 2 kt cuk2 H1− 1 pb (t cu) − 1 2µkhk 2 L2 = (1 − µ) " 1 2kt cuk2 H1 − 1 (1 − µ) p Z Ω|t cu|p # − 1 2µkhk 2 L2 = (1 − µ) Ic(tcu) − 1 2µ khk 2 L2 = (1 − µ)p−2p à 1 2 − 1 p ! b (u)−p−22 − 1 2µkhk 2 L2 = (1 − µ)p−2p J 0 ³ t0u´− 1 2µkhk 2 L2 ≥ (1 − µ)p−2p α 0(Ω) − 1 2µkhk 2 L2.
For µ ∈ (0, 1), there exists a d1(µ) > 0 such that for khkL2 < d1(µ)
sup
t≥0
Jh(tu) > 0.
By Lemma 3.2, there exists a t−= t−(u) > 0 such that t−u ∈ M− h and sup t≥0Jh(tu) = Jh ³ t−u´. Thus, for khkL2 < d1(µ) Jh ³ t−u´≥ (1 − µ)p−2p J 0 ³ t1u´− 1 2µkhk 2 L2.
This complete the proof. ¤
Lemma 4.8 There exists a positive number d0 < d1(µ) such that for khkL2 <
d0, we have Z A y |y| ³ |∇u|2+ u2´dydx 6= 0, for u ∈ [Jh < αh(Ω) + α0(A)] .
Proof. For u ∈ [Jh < αh(Ω) + α0(A)]. There exists a t0 > 0 such that
t0u/ kuk
H1 ∈ M0. By Lemma 4.7 (ii), we have for each µ ∈ (0, 1) , there is
d1(µ) > 0 such that khkL2 < d1(µ) implies
J à t1u kukH1 ! ≤ (1 − µ)−p−2p à Jh(u) + 1 2µkhk 2 L2 ! , (4.6)
where t−u
kukH1 = u ∈ M −
h. Since αh(Ω) < 0, we have [Jh < αh(Ω) + α0(A)] ⊂
[Jh < α0(A)] . Thus by (4.6), we have, for u ∈ [Jh < αh(Ω) + α0(A)] ,
J³ t1u kukH1 ´ ≤ (1 − µ)−p−2p ³ α0(A) + 2µ1 khk2L2 ´ = α0(A) + ²(µ) + (1 − µ)− p p−2 1 2µkhk 2 L2,
where ²(µ) → 0 as µ → 1. Thus for δ0 > 0 in Lemma 2.8, there exist µ ∈ (0, 1)
and d0 > 0 such that for khkL2 < d0, we have
J Ã t1u kukH1 ! ≤ α0(A) + δ0. (4.7) Since t0u/ kuk
H1 ∈ M0, by Lemma 2.8 and (4.7) we have Z A y |y| ¯ ¯ ¯ ¯ ¯∇ à t1u kukH1 !¯¯ ¯ ¯ ¯ 2 + à t1u kukH1 !2 dydx 6= 0, or, Z A z |z| ³ |∇u|2+ (u)2´dz 6= 0.
This complete the proof. ¤
We hence define G : [Jh < αh(Ω) + α0(A)] → Sm−1 by G (u) = Z A y |y| ³
|∇u|2+ |u|2´dydx
¯ ¯ ¯ ¯ ¯ Z A y |y| ³
|∇u|2+ |u|2´dydx
¯ ¯ ¯ ¯ ¯. Then we have
Lemma 4.9 For l ≥ l1 and khkL2 < d0, the map
G ◦ Fl : Sm−1 → Sm−1 is homotopic to the identity.
Proof. Since 0 ≤ h(x, y) ≤ c exp³−√1 + λ1+ δ |y| ´
for any (x, y) ∈ Ω, then by regularities, we have u0, w ∈ C1,θ
³
Ω´∩ L∞(Ω).We define ζl(θ, y0) : [0, 1] × Sm−1 → Sm−1
by ζl(θ, y0) = G ((1 − 2θ) Fl(y0) + 2θψw (x, y − ly0)) for θ ∈ [0, 1/2) ; G³ψw³x, y − l 2(1−θ)y0 ´´ for θ ∈ [1/2, 1) ; y0 for θ = 1. We have (a) lim θ→1−ζl(θ, y0) = y0 : since Z A y |y| ¯ ¯ ¯ ¯ ¯∇ " ψw à x, y − l 2 (1 − θ)y0 !#¯ ¯ ¯ ¯ ¯ 2 + " ψw à x, y − l 2 (1 − θ)y0 !#2 dydx = Z A y + l 2(1−θ)y0 ¯ ¯ ¯y + l 2(1−θ)y0 ¯ ¯ ¯ ³ |∇w|2 + w2´dydx = à 2p p − 2 ! α0(A) y0+ o(1) as θ → 1−. (b) lim θ→1 2 −ζl(θ, y0) = G (ψw (z − ly0)) : we have k(1 − 2θ) Fl(y0) + 2θψw (x, y − ly0)kH1 = kw (x, y − ly0)kH1+o(1) as θ → 1 2 − .
By the continuity of G, we obtain lim
θ→12− ζl(θ, y0) = G (w (x, y − ly0)) . Thus, ζl(θ, y0) ∈ C ([0, 1] × Sm−1, Sm−1) and ζl(0, y0) = G (Fl(y0)) for all y0 ∈ Sm−1, ζl(1, y0) = y0 for all y0 ∈ Sm−1, provided l ≥ l1 and khkL2 < d0. ¤ Thus we have
Lemma 4.10 Jh(u) has at least two critical points in
[Jh < αh(Ω) + α0(A)] .
Proof. Applying Adachi-Tanaka [1, Lemma 2.5], Proposition 3.4 and Lemma 4.9, we have for sufficiently large l ≥ l1 and khkL2 < d0,
cat ([Jh ≤ αh(Ω) + α0(A) − σl]) ≥ 2.
By Ambrosetti [2, Theorem 2.3] and Lemma 4.5 (v) , Jh(u) has at least two
We can now complete the proof of Theorem 1.1:we need to show that
Jh(u0) = αh(Ω) < αh−(Ω) = Jh(u0),
for each critical point u0 in [J
h < αh(Ω) + α0(A)] ⊂ M−h. Otherwise, assume
that Jh(u0) = α−h (Ω) = Jh(u0) = αh(Ω). Since
R
Ωhu0 > 0, by Lemma 3.2,
there exists t+(u0) > 0 such that t+(u0)u0 ∈ M+
h and α+
h(Ω) ≤ Jh(t+(u0)u0) < Jh(u0) = αh(Ω) = α−h (Ω) ,
which contradicts to α+h(Ω) = αh(Ω) in Lemma 3.5. Therefore, we have that
the equation (1.1) in A\D has at least three nonnegative solutions. Moreover, since h 0, then by maximum principle, Equation (1.1) in A\D has at least
three positive solutions. ¤
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