Existence and multiplicity of nodal solutions
for Dirichlet problems in upper half strip with
holes
Tsung-fang Wu
Department of Applied Mathematics,
National University of Kaohsiung, Kaohsiung 811, Taiwan
Abstract
In this paper, we consider the existence and multiplicity of nodal solutions of semilin-ear elliptic boundary value problems. We prove that the semilinsemilin-ear elliptic equations in large domains does not admit any least energy nodal (sign-changing) solution and in an upper half strip with m–holes has at least m2 2–nodal solutions.
Key words: Semilinear elliptic equations; Nodal solutions; Nehari manifold
1 Introduction
In this paper, we study the existence and multiplicity of nodal solutions of semilinear elliptic problems of the form
−∆u + u = |u|p−2u++ |u|q−2u− in Ω,
u = 0 on ∂Ω, (Ep,q)
where Ω is a domain in RN, 2 < p, q < ∞ (N = 2) , 2 < p, q < 2N
N −2(N ≥ 3) , u+=
max {0, u} and u− = min {u, 0} . Associated with equation (E
p,q) , we consider
the energy functional J in the Sobolev space H1 0 (Ω) , J(u) = 1 2kuk 2− 1 p Z Ω ¯ ¯ ¯u+ ¯ ¯ ¯p −1 q Z Ω ¯ ¯ ¯u− ¯ ¯ ¯q,
where kuk =³RΩ|∇u|2+ u2´1/2 is a standard norm in H1
0 (Ω) . It is well known
that the functonal J ∈ C2(H1
0 (Ω) , R) and the solutions of equation (Ep,q) in
Ω are the critical points of the energy functional J in H1 0 (Ω).
Generally, a standard technique to find the one sign solutions of equation (Ep,q) in Ω is using the Nehari minimization problems:
α±(Ω) = inf
v∈M±(Ω)J(v),
where M±(Ω) = {u ∈ H1
0(Ω) \{0} | hJ0(u) , ui = 0, ±u ≥ 0}. Note that α±(Ω)
are positive numbers and α±(Ω
1) ≥ α±(Ω2) if Ω1 ⊂ Ω2 (see Willem [18]).
Furthermore, we called a nonzero critical point u0 of J is a least energy
pos-itive (or negative) solution of equation (Ep,q) in Ω if u0 > 0 (or < 0) and
J (u0) = α+(Ω) (or α−(Ω)).
By the Rellich compactness theorem, it is easy to obtain a least energy pos-itive (or negative) solution of equation (Ep,q) in bounded domains. For general
unbounded domains Ω, because the lack of compactness, the existence of one sign solutions of equation (Ep,q) in Ω is very difficult and unclear. Indeed, a
by now classical result of Esteban-Lions [11] states that for unbounded do-mains satisfying the condition: there exists χ ∈ RN, ||χ|| = 1 such that
n(x) · χ ≥ 0 and n(x) · χ 6≡ 0 on ∂Ω, where n(x) is the unit outward normal
vector to ∂Ω at the point x, equation (Ep,p) does not admit any nontrivial
solution. Recently, there have been some progresses for the existence of least energy positive (or negative) solutions of equation (Ep,q) in unbounded
do-mains as follows: Berestycki-Lions [5] for Ω = RN, Lien-Tzeng-Wang [15] for
Ω is a periodic domain, Del Pino-Felmer [9,10] for Ω is a quasicylindrical do-main, Wu [20] for Ω is a multi-bump domain. On the other hand, when Ω is an exterior domain in RN, it is well known that equation (E
p,q) in exterior
domain does not admit any least energy positive (or negative) solution (see Benci-Cerami [4]). However, Benci-Cerami [4] proved that equation (Ep,q) in
exterior domain has a higher energy positive solution.
In the aforementioned works, the authors considered one sign solutions. For other situations, Bartsch [2] obtained infinite nodal (sign-changing) solutions for equation (Ep,p) in bounded domains. Furtado [12,13], showed that the
domain topology is related with the number of 2–nodal solutions of equation (Ep,p) , where the definition of 2–nodal solution is: for a nontrivial solution u is
such that the set {x ∈ Ω | u (x) 6= 0} has exactly two connected components,
u is positive in one of them and negative in the other (see Castro-Clapp [7]
or Bartsch-Weth [3]). Huang-Wu [14] proved that equation (Ep,p) in a finite
strip with a hole has at least four 2–nodal solutions. Bartsch-Weth [3], proved that the equation (Ep,q) in a bounded domain Ω that contains a large ball has
three nodal solutions in which two 2–nodal solutions. Wu [20], proved that the equation (Ep,q) in an m−bump domain has at least m2 2–nodal solutions.
Motivated by the above results, we are interested the relation between the topology of domain and the existence of nodal solutions of equation (Ep,q).
Before stating our main results, we need the following definitions and nota-tions. Denote the N−ball BN(z
0; r) in RN, the infinite strip A, the upper half
strip A+ and the finite strip A (s, l) as follows:
BN(z
0; r) = {z ∈ RN | |z − z0| < r};
A =n(x, y) ∈ RN −1× R | x ∈ ω, ω is a bounded domain in RN −1o;
A+ = {(x, y) ∈ A | y > 0} ;
A (s, l) = {(x, y) ∈ A | s < y < l} .
Definition 1.1 (i) The domain Ω is called large domain in A if Ω ⊂ A and
for any n > 0 there exist s < l such that l − s = n and A (s, l) ⊂ Ω;
(ii) The domain Ω is called strictly large domain in A if Ω is a large domain
in A and Ω 6= A.
Note that the infinite strip A is a large domain in itself and the upper half strip with m–holes
Ω (t) = A+\h∪mi=1BN ((0, it) ; r
0)
i
is a strictly large domain in A, where t > 2r0 > 0 and BN −1(0; r0) $ ω.
Furthermore, equation (Ep,q) in A has a ground state solution and in Ω (t)
does not admit any least energy positive (or negative) solution for all t > 0 (see Wu [19, Lemma 11]). Thus, equation (Ep,q) in Ω (t) only has higher energy
solution. However, Wu [19] proved that equation (Ep,q) in Ω (t) has at least m
higher energy positive solutions for t sufficiently large.
In this paper, we can show that equation (Ep,q) in large domains does not
admit any least energy nodal solution. Here all nodal solutions of equation (Ep,q) lie in the set
N (Ω) =nu ∈ H01(Ω) | u±∈ M±(Ω)o.
Let θ (Ω) = infu∈N(Ω)J (u). Then we have the following result.
Theorem 1.2 If Ω is a large domain in A, then equation (Ep,q) in Ω does not
admit any nodal solution v0 such that J (v0) = θ (Ω) , that is equation (Ep,q)
in Ω does not admit any least energy nodal solution. Furthermore, θ (Ω) = α+(A) + α−(A).
By Theorem 1.2, equation (Ep,q) in the upper half strip with m–holes Ω (t)
does not admit any least energy nodal solution for all t > 0. Next, we will use a method of Clapp-Weth [8] to construct the local minimizers of the restriction of
the energy functional J on the set N(Ω (t)). Then proved that the multiplicity of higher energy nodal solutions of equation (Ep,q) in Ω (t). Furthermore, we
have the following result.
Theorem 1.3 For each 0 < ε ≤ minn p
p−2α+(A) , q
q−2α−(A)
o
, there exists t0 > 2r0 such that for t > t0, equation (Ep,q) in Ω (t) has m2 2–nodal solutions
n
ui,j0 o
i,j∈{1,2,...,m} such that J
³ ui,j0 ´> θ (Ω (t)) , Z [A((i−1)t,it)]c ¯ ¯ ¯ ¯ ³ ui,j0 ´+ ¯ ¯ ¯ ¯ p < ε and Z [A((j−1)t,jt)]c ¯ ¯ ¯ ¯ ³ ui,j0 ´− ¯ ¯ ¯ ¯ q < ε for all i, j ∈ {1, 2 . . . , m} .
This paper is organized as follow. In section 2, we prove Theorem 1.2. In section 3, we prove Theorem 1.3.
2 Existence of Nodal Solutions
First, we define the Palais–Smale (denoted by (PS)) sequences in H1 0(Ω)
for J as follows.
Definition 2.1 For β ∈ R, a sequence {un} is a (PS)β–sequence in H01(Ω)
for J if J(un) = β + o(1) and J0(un) = o(1) strongly in H−1(Ω) as n → ∞.
We need the following result.
Lemma 2.2 If Ω is a large domain in A, then α±(Ω) = α±(A). Furthermore,
if Ω is a strictly large domain in A, then equation (Ep,q) in Ω does not admits
any solution u0 such that J (u0) = α±(Ω).
Proof. See Lien-Tzeng-Wang [15, Lemma 2.5] and Wu [19, Lemma 11]. ¤ Proposition 2.3 If Ω is a large domain in A, then θ (Ω) = α+(Ω) + α−(Ω).
Furthermore, every minimizing sequence {vn} in N (Ω) for J is a (PS)θ(Ω)–
sequence in H1
0 (Ω) for J.
Proof. First, by Lien-Tzeng-Wang [15], we have the equation (Ep,q) there
exist a positive solution u(1)0 and a negative solution u(2)0 such that J³u(1)0 ´=
α+(A) and J³u(2) 0
´
= α−(A) . Since Ω is a large domain in A. For n =
1, 2, · · · , there exist two sequencesnt(1)
n o ,nt(2) n o ⊂ R such that t(2) n > t(1)n , t(2)n − t(1) n = 2n and A ³ t(1) n , t(2)n ´ ⊂ Ω. Let t(3) n = ³ t(1) n + t(2)n ´ /2. Then t(2) n − t(3)n = t(3) n − t(1)n = n, A ³ t(1) n , t(3)n ´ ⊂ Ω and A³t(3) n , t(2)n ´ ⊂ Ω. Set l(1) n = ³ t(1) n + t(3)n ´ /2
and l(2) n = ³ t(2) n + t(3)n ´
/2. Consider the cut-off function ζ ∈ C∞([0, ∞)) such
that 0 ≤ ζ ≤ 1 and ζ (t) = 1 for t ∈ [0, 1] 0 for t ∈ [2, ∞). Let ζ(i) n (x, y) = ζ 2 ¯ ¯ ¯ ³ x,y−l(i)n ´¯¯ ¯ n and u(i) n (x, y) = ζn(i)(x, y) u(i)0 ³ x, y − l(i) n ´ for i ∈ {1, 2} . Then u(i) n (x, y) ∈ H01(Ω) , ° ° °u(i) n ° ° °2 =RΩ¯¯¯u(i) n ¯ ¯ ¯p+ o (1) and J³u(i) n ´ =
J³u(i)0 ´+ o (1) . By Wang-Wu [17, Lemma 7], u(1)
n is a (PS)α+(A)–sequence
in H1
0(Ω) for J and u(2)n is a (PS)α−(A)–sequence in H01(Ω) for J. Moreover,
by the routine computations, there exists a sequence {s(i)
n } ⊂ R+ such that
s(i)
n = 1 + o(1), {s(1)n u(1)n } ⊂ M+(Ω) , {s(2)n u(2)n } ⊂ M−(Ω) and J(s(i)n u(i)n ) =
J³u(i)0 ´ + o(1). Let vn = s(1)n un(1) + s(2)n u(2)n , then by suppu(1)n ∩ suppu(2)n =
∅ we have vn ∈ N (Ω) and J(vn) ≥ θ (Ω) ≥ α+(Ω) + α−(Ω) . Moreover,
suppu(1)
n ∩ suppu(2)n = ∅ and so
J(vn) = J(s(1)n u(1)n ) + J(s(2)n u(2)n ) = α+(A) + α−(A) + o(1)
= α+(Ω) + α−(Ω) + o (1) . Thus, θ (Ω) = α+(Ω) + α−(Ω) . Let {u n} be a minimizing sequence in N (Ω) for J. Then J (un) = θ (Ω) + o (1) = α+(Ω) + α−(Ω) + o (1) and J (un) = J ³ u+n´+ J³u−n´≥ α+(Ω) + α−(Ω) . Thus, J (u±
n) → α±(Ω) . Since u±n ∈ M±(Ω) , again use the result of Wang-Wu
[17, Lemma 7], {u±
n} are (PS)α±(Ω)–sequences in H01(Ω) for J, that is
J³u± n ´ = α±(Ω) + o (1) and J0(u± n) = o(1) in H−1(Ω) . Moreover, for ϕ ∈ C∞ c (Ω), we have hJ0(un) , ϕi = D J0³u+n + u−n´, ϕE = Z Ω ³ ∇³u+n´∇ϕ + u+nϕ´+ Z Ω ³ ∇³u−n´∇ϕ + u−nϕ´ − Z Ω ¯ ¯ ¯u+n¯¯¯p−2u+nϕ − Z Ω ¯ ¯ ¯u−n¯¯¯q−2u−nϕ =DJ0(u+ n), ϕ E +DJ0(u− n), ϕ E . Therefore, J0(u
n) = o(1) in H−1(Ω) as n → ∞. We conclude that J (un) =
θ (Ω) + o (1) and J0(u
Now we proof Theorem 1.2: Suppose otherwise. Then equation (Ep,q) there
exists a nodal solution v0 such that J (v0) = θ (Ω) . Clearly, v0± ∈ M±(Ω) ⊂
M±(A) and J³v± 0 ´ ≥ α±(A) . Since θ (Ω) = α+(Ω) + α−(Ω) = α+(A) + α−(A). Thus, J³v± 0 ´ = α±(Ω) . Then v±
0 are ground state solutions of equation (Ep,q) .
By the strong maximum principle, we obtain v+
0 > 0 and v0− < 0 in Ω, which
contradicts to nx ∈ Ω | v0±(x) 6= 0o$ Ω. ¤
3 Multiple 2–Nodal Solutions
For each i, j ∈ {1, 2, . . . , m} and 0 < ε ≤ minnp−2p α+(A) , q
q−2α−(A) o , we denote Mi+(ε, t) = ( u ∈ M+(Ω (t)) | Z [A((i−1)t,it)]c ¯ ¯ ¯u+¯¯¯p < ε ) ; Oi+(ε, t) = ( u ∈ M+(Ω (t)) | Z [A((i−1)t,it)]c ¯ ¯ ¯u+¯¯¯p = ε ) ; M− i (ε, t) = ( u ∈ M−(Ω (t)) | Z [A((i−1)t,it)]c ¯ ¯ ¯u− ¯ ¯ ¯q < ε ) ; O− i (ε, t) = ( u ∈ M−(Ω (t)) | Z [A((i−1)t,it)]c ¯ ¯ ¯u− ¯ ¯ ¯q = ε ) ; Ni,j(ε, t) = n u ∈ H1 0(Ω (t)) | u+ ∈ Mi+(ε, t) and u− ∈ Mj−(ε, t) o ; Oi,j(ε, t) = n u ∈ H1 0(Ω (t)) | u+ ∈ Mi+(ε, t), u− ∈ Mj−(ε, t) and either u+∈ O+i (ε, t) or u− ∈ O−j (ε, t)o, where M±
i (ε, t) = Mi±(ε, t) ∪ O±i (ε, t) for all i ∈ {1, 2, . . . , m} . If Ni,j(ε, t)
are denoted the closure of Ni,j(ε, t) , then we have Ni,j(ε, t) = Ni,j(ε, t) ∪
Oi,j(ε, t) and Oi,j(ε, t) is the boundary of Ni,j(ε, t) for all i, j ∈ {1, 2, . . . , m} .
Furthermore, we have the following results.
Lemma 3.1 For each t > 2r0, we have Ni,j(ε, t) are disjoint.
Proof. Since the proof of all cases are similar, we only need to prove the case ”1, 1” and ”1, 2”. Assume the contrary, there exist v0 ∈ H01(Ω (t)) such that
v0 ∈ N1,1(ε, t) ∩ N1,2(ε, t). Then Z [A(0,t)]c ¯ ¯ ¯v0− ¯ ¯ ¯q < ε and Z [A(t,2t)]c ¯ ¯ ¯v−0 ¯ ¯ ¯q ≤ ε.
Since v0−∈ M−(Ω (t)) and A (0, t) ∩ A (t, 2t) = ∅ for all t > 2r 0, we have Z Ω(t) ¯ ¯ ¯v− 0 ¯ ¯ ¯q ≤ Z [A(0,t)]c ¯ ¯ ¯v− 0 ¯ ¯ ¯q+ Z [A(t,2t)]c ¯ ¯ ¯v− 0 ¯ ¯ ¯q < 2q q − 2α −(A) . Therefore, α−(A) = α−(Ω (t)) ≤ J³v− 0 ´ = q − 2 2q Z Ω(t) ¯ ¯ ¯v−0 ¯ ¯ ¯q < α−(A) , which is a contradiction. ¤
Define the minimization problems in Ni,j
³ ε 2, t ´ and Oi,j ³ ε 2, t ´ for J, γi,j(t) = inf v∈Ni,j(ε 2,t) J (v) , γei,j(t) = inf v∈Oi,j(ε 2,t) J (v) .
Clearly, γi,j(t) ≥ θ (Ω (t)) for all t > 0. Furthermore, we have the following
results.
Lemma 3.2 For each positive number σ < min {α+(A) , α−(A)} there exists
t1 > 2r0 such that
γi,j(t) < α+(A) + α−(A) + σ
for all i, j ∈ {1, 2, . . . , m} and t > t1.
Proof. By Lien-Tzeng-Wang [15, Lemma 2.2], we have
α± µ A µt 2, t ¶¶ = α± µ A µ 0, t 2 ¶¶ & α±(A) as t % ∞.
Thus, there exists t1 > 2r0 such that
α± µ A µt 2, t ¶¶ = α µ A± µ 0, t 2 ¶¶ < α±(A) + σ 2 (3.1)
for all t > t1. By Ambrosetti-Rabinowitz [1] (or see Willem [18]),
equa-tion (Ep,q) in A ³ r0,2t ´ and in A³t 2, t − r0 ´
has a positive solution v+ ∈
M+³A³r 0,2t
´´
and a negative solution v− ∈ M−³A³t
2, t − r0 ´´ such that J (v+) = α+³A³r 0,2t ´´ and J (v−) = α−³A³t 2, t − r0 ´´ . Set vi(x, y) = v+(x, y − (i − 1) t) and v j(x, y) = v−(x, y − (j − 1) t) . Clearly, vi ∈ M+(Ω (t)) , vj ∈ M−(Ω (t)) and Z [A((i−1)t,it)]c ¯ ¯ ¯v+i ¯ ¯ ¯p = Z [A((j−1)t,jt)]c ¯ ¯ ¯vj− ¯ ¯ ¯q= 0. Thus, vi ∈ Mi+ ³ ε 2, t ´ and vj ∈ Mj− ³ ε 2, t ´
. Set vi,j = vi+ vj, we obtain v0 ∈
Ni,j ³ ε 2, t ´ and
γi,j(t) ≤ J (v) < α+(A) + α−(A) + σ
Lemma 3.3 There exist positive numbers δ and t2 > 2r0 such that for each
i, j ∈ {1, 2, . . . , m} we have
e
γi,j(t) > α+(A) + α−(A) + δ for all t ≥ t2.
Proof. Fix i, j ∈ {1, 2, . . . , m} . Assume the contrary, there exist tn→ ∞ as
n → ∞ and {un} ⊂ Oi,j ³ ε 2, tn ´ such that J (un) → α+(A) + α−(A) , (3.2) Z Ω(tn) ¯ ¯ ¯∇u+ n ¯ ¯ ¯2+³u+ n ´2 = Z Ω(tn) ¯ ¯ ¯u+ n ¯ ¯ ¯p, (3.3) Z Ω(tn) ¯ ¯ ¯∇u−n ¯ ¯ ¯2+ ³ u− n ´2 = Z Ω(tn) ¯ ¯ ¯u−n ¯ ¯ ¯q (3.4) and either u+ n ∈ Oi+ ³ ε 2, tn ´ or u− n ∈ Oj− ³ ε 2, tn ´ . Since u± n ∈ M±(Ω (tn)) ⊂ M±(A+) ⊂ M±(A) and J (u n) = J (u+n) + J (u−n) . By (3.2) and Wang-Wu [17, Lemma 7] {u±
n} are (PS)α±(A)–sequences in H01(A) for J. Moreover, y (3.4)
and the Sobolev imbedding theorem, there exists c > 0 such thatRA+|∇u±n|2+
(u±
n)2 > c for all n. From the concentration compactness principle of Lions [16]
(or see [15, Theorem 4.1]), there exist R > 0, d > 0 and {(0, y±
n)} ∈ RN −1×R+ such that Z A(−R,R)+(0,y+ n) ¯ ¯ ¯u+n ¯ ¯ ¯p ≥ d and Z A(−R,R)+(0,y− n) ¯ ¯ ¯u−n ¯ ¯ ¯q ≥ d for all n.
Without loss of generality, we may assume that u+
n ∈ O+i ³ ε 2, tn ´ that is Z [A((i−1)tn,itn)]c ¯ ¯ ¯u+n ¯ ¯ ¯p = ε.
Set ¯un(x, y) = u+n(x, y + yn+). From the translation invariance of the
func-tional in y–axis, we get that also {un} is satisfying
¯ un ∈ M+ ³ A+−³0, y+ n ´´ ⊂ M+(A) (3.5)
and is (PS)α+(A)–sequences in H01(A) for J. Then there exist a subsequence
{¯un} and a nonnegative function u0 ∈ H01(A) such that
¯ un* u0 weakly in H01(A) as n → ∞, ¯ un→ u0 a.e. in A as n → ∞ and Z A(−R,R)|¯un| p →Z A(−R,R)|u0| p ≥ d as n → ∞. (3.6)
This implies u0 is a nonzero nonnegative solution of equation (Ep,q) in A. By
α+(A) ≤ J (u0) = à 1 2 − 1 p ! Z A|u0| p ≤ lim inf à 1 2 − 1 p ! Z A|un| p = α+(A) ,
and so J (u0) = α+(A) . Moreover, by the strong maximum principle, u0 is
a positive solution of equation (Ep,q) in A. Set wn = un− u0, Since {un} is
uniformly bounded, by Br´ezis-Lieb lemma [6] we obtain Z A|wn| p =Z A|un| p−Z A|u0| p+ o (1) . (3.7)
Moreover, un* u0 weakly in H01(A) we have
kwnk2 = kunk2− ku0k2+ o (1) . (3.8) Then kwnk2 = Z A|wn| p+ o (1) and so à 1 2 − 1 p ! kwnk2 = J (wn) = J (un) − J (u0) + o (1) = o (1) .
This implies ¯un → u0 strongly in H01(A) as n → ∞. We will show that
y+
n → ∞ as n → ∞. Suppose otherwise, that is {y+n} is a bounded sequence
in R or there exists a subsequence {y+
n} such that yn+ → −∞ as n → ∞.
If {y+
n} is a bounded sequence in R, we may assume yn+ → y0. Since u+n ∈
M+(Ω (t
n)) ⊂ M+(A+) and ¯un(x, y) = u+n (x, y + yn+) . Then by (3.5) and
lim n→∞ h A+−³0, y+ n ´i → A+ y0 = {(x, y) ∈ A | y > −y0} we have u0 ∈ M+ ³ A+ y0 ´
which contradicts to u0 is a positive solution of
equation (Ep,q) in A. If yn+ → −∞ as n → ∞, then there exists n0 ∈ N such
that h
A+−³0, y+
n
´i
∩ A (−R, R) = ∅ for all n ≥ n0,
this implies ¯un ≡ 0 on A (−R, R) for all n ≥ n0 which contradicts (3.6) . Thus
y+
n → ∞ as n → ∞. By passing to a subsequence, we may assume that one
of the following cases occurs:
(a) {itn− yn+} is bounded for any subsequence;
(b) dist ((0, y+
n) , A ((i − 1) tn, itn)) → ∞ for a sebsequence;
(c) (0, y+
n) ∈ A ((i − 1) tn, itn) for a subsequence.
Since ¯un → u0 strongly in H01(A) as n → ∞. By the Sobolev imbedding
0 such that Z A(−R(ε),R(ε))|¯un| p > 2p p − 2α +(A) − ε 2 for all n ≥ n0. (3.9) In case (a) we may assume itn− y+n → y0. Since y+n → ∞ as n → ∞ and
¯ un∈ H01 ³ Ω (tn) − ³ 0, y+ n ´´ . Then lim n→∞ h Ω (tn) − ³ 0, y+ n ´i → A\BN((0, y 0) ; r0) and u0 ∈ M+ ³ A\BN((0, y 0) ; r0) ´
which contradicts to u0 is a positive
solu-tion of equasolu-tion (Ep,q) in A.
In case (b) : Since dist (y+
n, A ((i − 1) tn, itn)) → ∞, there exists n1 ≥ n0 such
that h
A (−R (ε) , R (ε)) +³0, y+n´i∩ A ((i − 1) tn, itn) = ∅ for all n ≥ n1.
Thus, Z A(−R(ε),R(ε))|¯un| p=Z A(−R(ε),R(ε))+(0,y+n) ¯ ¯ ¯u+ n ¯ ¯ ¯p ≤ Z [A((i−1)tn,itn)]c ¯ ¯ ¯u+n¯¯¯p = ε which contradicts (3.9) . In case (c) : Since (0, y+ n) ∈ A ((i − 1) tn, itn) and yn+ → ∞ as n → ∞. By
case (a) itn− yn+ → ∞ as n → ∞. We will show that (i − 1) tn− yn+ → −∞
as n → ∞. Clearly
(i − 1) tn− y+n → −∞ as n → ∞ for i = 1.
For i ≥ 2. Suppose otherwise, by (i − 1) tn< y+n we may assume {(i − 1) tn− yn+}
is a bounded sequence in R. Then there exists a subsequence {(i − 1) tn− yn+}
and ye0 ≤ 0 such that
(i − 1) tn− y+n →ye0 as n → ∞ and so lim n→∞ h Ω (tn) − ³ 0, y+ n ´i = A\BN((0,ye 0) ; r0) .
Similar to the argument in case (a) , u0 ∈ M+
³
A\BN((0,ye
0) ; r0)
´
which con-tradicts to u0 is a positive solution of equation (Ep,q) in A. Thus, (i − 1) tn−
y+
n → −∞ as n → ∞, this mean there exists n2 ≥ n0 such that
A (−R (ε) , R (ε)) ⊂hA ((i − 1) tn, itn) − ³ 0, y+ n ´i for all n ≥ n2.
Thus, by (3.9) andRA|¯un|p → p−22p α+(A) as n → ∞, there exists n3 ≥ n2 such
that Z
[A((i−1)tn,itn)−(0,y+n)] c|¯un| p ≤Z [A(−R(ε),R(ε))]c|¯un| p < ε for all n ≥ n 3. This implies, Z [A((i−1)tn,itn)]c ¯ ¯ ¯u+n ¯ ¯ ¯p < ε for all n ≥ n3,
which contradicts R[A((i−1)tn,itn)]c|u+n|p = ε. Therefore, we have completed our
proof. ¤
By Lemmas 3.2, 3.3, there exists t0 > 2r0 such that for t > t0
γi,j(t) < min
n
α+(A) + α−(A) + minnα+(A) , α−(A)o,γe i,j(t)
o
for all i, j ∈ {1, 2, . . . , m} . Furthermore, we will use the idea of Bartsch-Weth [3] and Clapp-Weth [8] to get the following results.
Lemma 3.4 There exists µ0 > 0 such that, for each v ∈ Ni,j
³ ε 2, t ´ and u ∈ H1
0 (Ω (t)) such that kv − uk < µ0, there holds
¯ ¯ ¯ ¯ ¯ Z [A((i−1)t,it)]c ¯ ¯ ¯u+ ¯ ¯ ¯p− Z [A((i−1)t,it)]c ¯ ¯ ¯v+ ¯ ¯ ¯p ¯ ¯ ¯ ¯ ¯< ε 2 and ¯ ¯ ¯ ¯ ¯ Z [A((j−1)t,jt)]c ¯ ¯ ¯u− ¯ ¯ ¯q− Z [A((j−1)t,jt)]c ¯ ¯ ¯v− ¯ ¯ ¯q ¯ ¯ ¯ ¯ ¯< ε 2.
Proof. If not, there exist {vn} ⊂ Ni,j
³
ε
2, r
´
and {un} ⊂ H01(Ω (t)) such that
kvn− unk → 0, but ¯ ¯ ¯ ¯ ¯ Z [A((i−1)t,it)]c ¯ ¯ ¯u+n ¯ ¯ ¯p− Z [A((i−1)t,it)]c ¯ ¯ ¯v+n ¯ ¯ ¯p ¯ ¯ ¯ ¯ ¯≥ ε 2 (3.10) or ¯ ¯ ¯ ¯ ¯ Z [A((j−1)t,jt)]c ¯ ¯ ¯u−n¯¯¯q− Z [A((j−1)t,jt)]c ¯ ¯ ¯vn−¯¯¯q ¯ ¯ ¯ ¯ ¯≥ ε 2. (3.11) Since Z [A((i−1)t,it)]c ¯ ¯ ¯u+n − vn+ ¯ ¯ ¯p ≤ Z Ω(t) ¯ ¯ ¯u+n − v+n ¯ ¯ ¯p ≤ Z Ω(t)|un− vn| p → 0 and Z [A((j−1)t,jt)]c ¯ ¯ ¯u−n − vn− ¯ ¯ ¯q≤ Z Ω(t) ¯ ¯ ¯u−n − vn− ¯ ¯ ¯q ≤ Z Ω(t)|un− vn| q → 0.
Thus, by the Minkowski inequality ¯ ¯ ¯ ¯ ¯ Z [A((i−1)t,it)]c ¯ ¯ ¯u+ n ¯ ¯ ¯p− Z [A((i−1)t,it)]c ¯ ¯ ¯v+ n ¯ ¯ ¯p ¯ ¯ ¯ ¯ ¯→ 0 and ¯ ¯ ¯ ¯ ¯ Z [A((j−1)t,jt)]c ¯ ¯ ¯u−n¯¯¯q− Z [A((j−1)t,jt)]c ¯ ¯ ¯v−n¯¯¯q ¯ ¯ ¯ ¯ ¯→ 0
which contradicts (3.10) and (3.11) . ¤
Lemma 3.5 For each v0 ∈ Ni,j
³
ε
2, t
´
there exists a map h : H1
0 (Ω (t)) → R2
such that
(i) h³l1v+0 + l2v0−
´
= (l1, l2) for l1, l2 ≥ 0;
(ii) h (u) = (1, 1) if and only if u ∈ N (Ω (t)) .
Proof. Similarly to the method used in Clapp-Weth [8, Lemma 13]. ¤ The next result is a variant of Proposition 14 in Clapp-Weth [8], and that the proof follows the arguments applied the Leray-Schauder continuation prin-ciple. Let
b = minnα+(A) + α−(A) + minnα+(A) , α−(A)o,γei,j(t)
o
.
Proposition 3.6 Let λ0 = b − γi,j(t) . Then for each λ ∈ (0, λ0) and µ ∈
(0, µ0) there exists u0 ∈ H01(Ω (t)) such that
(i) dist³u0, Ni,j
³
ε
2, t
´´
≤ µ;
(ii) J (u0) ∈ [γi,j(t) , γi,j(t) + λ);
(iii) k∇J (u0)k ≤ max n√ λ,λ µ o ; (iv) R[A((i−1)t,it)]c ¯ ¯ ¯u+0¯¯¯p < ε and R[A((j−1)t,jt)]c ¯ ¯ ¯u−0¯¯¯q < ε.
Proof. Fix v0 ∈ Ni,j
³
ε
2, t
´
such that J (v0) < γi,j(t) + λ, and fix l0 > 1
such that J³l0v0± ´ ≤ 0. Let h : H1 0 (Ω (t)) → R2 as in Lemma 3.5. We put K = [0, l0] × [0, l0] and define η : K → H1 0(Ω (t)) , η (s1, s2) = s1v+0 + s2v0−. Then h ◦ η = id : K → K, in particular deg (h ◦ η, K, (1, 1)) = 1. (3.12)
Notice also that
J (η (s1, s2)) ≤ J (v0) < γi,j(t) + λ for all (s1, s2) ∈ K. (3.13)
We now choose a Lipschitz continuous function χ : R → R such that 0 ≤
C2(H1 0 (Ω (t)) , R) , there is a semiflow ϕ : [0, ∞) × H01(Ω (t)) → H01(Ω (t)) satisfying ∂ ∂sϕ (s, u) = −χ (J (ϕ (s, u))) ∇J (ϕ (s, u)) , ϕ (0, u) = u.
We will frequently write ϕs in place of ϕ (s, ·) . Since
J³v±
0
´
< α+(A) + α−(A) and J³l
0v±0
´
≤ 0,
it follows that
sup J (η (∂K)) < α+(A) + α−(A) . Hence
(ϕs◦ η) (∂K) ∩ N (Ω (t)) = ∅ for all s ≥ 0
and, by Lemma 3.5, this implies
(h ◦ ϕs◦ η) (y) 6= (1, 1) for all y ∈ ∂K, s ≥ 0.
Equality (3.12) and the global continuation principle of Leray-Schauder (see e.g. Zeider [21, p.629]) imply that there exists a connected subset Z ⊂ K×[0, 1] such that (1, 1, 0) ∈ Z; ϕs(η (l 1, l2)) ∈ N (Ω (t)) for all (l1, l2, s) ∈ Z; Z ∩ (K × {1}) 6= ∅. We put e Z = {ϕs(η (l 1, l2)) ∈ N (Ω (t)) | (l1, l2, s) ∈ Z} . By inequality (3.13) , sup u∈Ze J (u) < γi,j(t) + λ < b and e Z ∩ Oi,j µ ε 2, t ¶ = ∅. So, since Z is connected, we obtain thatZ ⊂ Ne i,j
³ ε 2, t ´ . We now pick³¯l1, ¯l2, 1 ´ ∈ Z ∩ (K × {1}) and write v1 := η ³ ¯l1, ¯l2 ´ , v2 := ϕ1(v1) . Then v2 ∈Z ⊂ Ne i,j ³ ε 2, t ´
. We distinguish two case.
Case 1. kϕs(v
1) − v2k ≤ µ for all s ∈ [0, 1] . Then
Z [A((i−1)t,it)]c ¯ ¯ ¯(ϕs(v1))+ ¯ ¯ ¯p < ε and Z [A((j−1)t,jt)]c ¯ ¯ ¯(ϕs(v1))− ¯ ¯ ¯q < ε
for all s ∈ [0, 1] by Lemma 3.4. We choose s0 ∈ [0, 1] with k∇J (ϕs0(v 1))k = min 0≤s≤1k∇J (ϕ s(v 1))k
and put u0 = ϕs0(v1) . Then
λ ≥ J (v1) − J (v2) = − Z 1 0 ∂ ∂sJ (ϕ s(v 1)) ds = Z 1 0 k∇J (ϕ s(v 1))k2ds ≥ k∇J (u0)k2.
Hence u0 has the desired properties.
Case 2. There exists ¯s ∈ [0, 1] such that kϕ¯s(v
1) − v2k > µ. Then let s1 = sup {s ≥ ¯s | kϕs(v1) − v2k > µ} . By Lemma 3.4, Z [A((i−1)t,it)]c ¯ ¯ ¯(ϕs(v1))+ ¯ ¯ ¯p < ε and Z [A((j−1)t,jt)]c ¯ ¯ ¯(ϕs(v1))− ¯ ¯ ¯q < ε
for all s ∈ [s1, 1] . We choose s0 ∈ [s1, 1] with
k∇J (ϕs0(v
1))k = mins
1≤s≤1
k∇J (ϕs(v1))k
and put u0 = ϕs0(v1) . Then
µ ≤ Z 1 s1 ° ° ° ° ° ∂ ∂sϕ s(v 1) ° ° ° ° °ds ≤ Z 1 s1 k∇J (ϕs(v 1))k ds and λ ≥ J (ϕs1(v 1)) − J (v2) = Z 1 s1 k∇J (ϕs(v 1))k2ds ≥ k∇J (u0)k Z 1 s1 k∇J (ϕs(v 1))k ds.
We conclude that k∇J (u0)k ≤ λµ. Thus, u0 has the desired properties. ¤
Corollary 3.7 For each t > t0 there exists a sequence {ui,jn } ⊂ H01(Ω (t))
such that
(i) dist³ui,j n , Ni,j ³ ε 2, t ´´ → 0; (ii) J (ui,j
n ) → γi,j(t) < min {α+(A) + α−(A) + min {α+(A) , α−(A)} ,γei,j(t)} ;
(iii) J0(ui,j n ) = o(1) strongly in H−1(Ω (t)) ; (iv) R[A((i−1)t,it)]c ¯ ¯ ¯(ui,jn )+ ¯ ¯ ¯p < ε and R[A((j−1)t,jt)]c ¯ ¯ ¯(ui,jn )− ¯ ¯ ¯q< ε.
Now, we give an outline of the proof of Theorem 1.3. First, we consider a function ξ ∈ C∞([0, ∞)) such that 0 ≤ ξ ≤ 1 and
ξ(t) = 0, for t ∈ [0, 1] 1, for t ∈ [2, ∞). (3.14) Let ξn(z) = ξ( 2|z| n ). (3.15)
Then we have the following results.
Lemma 3.8 Suppose that {un} is a Palais-Smale sequence in H01(Ω (t)) for
J. Then we have
(i) if u+
n * 0 weakly in H01(Ω (t)) , then there exists a subsequence {un} such
that ku+
n − ξnu+nk = o(1) as n → ∞ and kξnu+nk
2 =R
Ω(t)|ξnu+n|p+ o (1) ;
(ii) if u−
n * 0 weakly in H01(Ω (t)) , then there exists a subsequence {un} such
that ku−
n − ξnu−nk = o(1) as n → ∞ and kξnu−nk2 =
R
Ω(t)|ξnu−n|p+ o (1)
Proof. Our proof is almost the same as that in Wu [20, Lemma 3.1] and is
omitted here. ¤
Lemma 3.9 If u is a nodal solution of equation (Ep,q) in Ω (t) and J (u) ≤
α+(Ω (t))+α−(Ω (t))+min {α+(Ω (t)), α−(Ω (t))} , then u is a 2–nodal solution
of equation (Ep,q) in Ω (t).
Proof. The proof is similar to that of Proposition 3.1 in Furtado [12] (or see
Bartsch-Weth [3]. ¤
Now we prove Theorem 1.3: It follows from Corollary 3.7 that there exists
t0 > 2r0 such that for each t > t0 and i, j ∈ {1, 2, . . . , m} we can find a
sequence {ui,j
n } ⊂ H01(Ω (t)) such that (i) − (iv) are hold in Corollary 3.7.
Since {ui,j n } is bounded in H01(Ω (t)). we have n (ui,j n ) +o and n(ui,j n ) −o are also bounded in H1 0(Ω (t)) and ° ° ° ° ³ ui,j n ´+°° ° ° 2 = Z Ω(t) ¯ ¯ ¯ ¯ ³ ui,j n ´+¯¯ ¯ ¯ p + o (1) and ° ° ° ° ³ ui,j n ´−°° ° ° 2 = Z Ω(t) ¯ ¯ ¯ ¯ ³ ui,j n ´−¯¯ ¯ ¯ q + o (1) . Thus, there exist a subsequence {ui,j
n } and ui,j0 in H01(Ω (t)) such that
ui,j n * ui,j0 ; ³ ui,j n ´± *³ui,j0 ´± weakly in H1 0(Ω (t)) and ui,j n → ui,j0 ; ³ ui,j n ´± →³ui,j0 ´± a.e. in Ω (t) .
Moreover, ui,j0 is a solution of equation (Ep,q) in Ω (t) . We will show that
³
ui,j0 ´± 6≡ 0. If not, then we may assume that³ui,j0 ´+≡ 0. Thus,
³
ui,jn ´+* 0 weakly in H01(Ω (t)) as n → ∞.
By Lemma 3.8, there exists ¯n0 ∈ N such that for n ≥ ¯n0,
2p p − 2α (Ω (t)) ≤ Z Ω(t)| ³ ui,j n ´+ |p =Z Ω(t)|ξn ³ ui,j n ´+ |p+ o(1) ≤ Z [A((i−1)t,it)]c ¯ ¯ ¯ ¯ ³ ui,j n ´+¯¯ ¯ ¯ p + o (1) < p p − 2α (A) + o (1) ,
which is a contradiction. Therefore,³ui,j0 ´±6≡ 0 and so ui,j0 is a nodal solution of equation (Ep,q) in Ω (t) . By the Fatou lemma, we have
Z [A((i−1)t,it)]c ¯ ¯ ¯ ¯ ³ ui,j0 ´+ ¯ ¯ ¯ ¯ p ≤ lim inf Z [A((i−1)t,it)]c ¯ ¯ ¯ ¯ ³ ui,j n ´+¯¯ ¯ ¯ p ≤ ε, Z [A((j−1)t,jt)]c ¯ ¯ ¯ ¯ ³ ui,j0 ´− ¯ ¯ ¯ ¯ q ≤ lim inf Z [A((j−1)t,jt)]c ¯ ¯ ¯ ¯ ³ ui,j n ´−¯¯ ¯ ¯ q ≤ ε, and
J³ui,j0 ´≤ lim inf J³ui,j n ´ = γi,j(t) <γei,j(t) . (3.16) Thus,³ui,j0 ´+∈ M+ i (ε, t) and ³ ui,j0 ´−∈ M− j (ε, t). If Z [A((j−1)t,jt)]c ¯ ¯ ¯ ¯ ³ ui,j0 ´+ ¯ ¯ ¯ ¯ p = ε or Z [A((j−1)t,jt)]c ¯ ¯ ¯ ¯ ³ ui,j0 ´− ¯ ¯ ¯ ¯ q = ε. Then ui,j0 ∈ Oi,j(ε, t) and from (3.16) that
e
γi,j(t) ≤ J
³
ui,j0 ´<γei,j(t) ,
which is a contradiction. Thus, Z [A((j−1)t,jt)]c ¯ ¯ ¯ ¯ ³ ui,j0 ´+ ¯ ¯ ¯ ¯ p < ε and Z [A((j−1)t,jt)]c ¯ ¯ ¯ ¯ ³ ui,j0 ´− ¯ ¯ ¯ ¯ q < ε.
This implies ui,j0 ∈ Ni,j(ε, t) and J
³
ui,j0 ´= γi,j(t) . Therefore, by Lemma 3.9
(ii), ui,j0 are 2–nodal solutions of equation (Ep,q) in Ω (t). Furthermore, by
Lemma 3.1 and Theorem 1.2, we obtain ui,j0 6= ui,k0 for j 6= k and J³ui,j0 ´ > θ (Ω (t)) for all i, j ∈ {1, 2, . . . , m} . ¤
Acknowledgment. The author is grateful for the referee’s valuable sug-gestions.
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