The effect of domain shape on the number of
positive and nodal solutions for semilinear elliptic
equations
∗
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 effect of domain shape on the number of positive and nodal (sign-changing) solutions for semilinear elliptic equa-tions. We prove a semilinear elliptic equation in a domain Ω that contains
m disjoint large balls has m2 2–nodal solutions in addition to m positive
solutions.
1
Introduction
In this paper, we study the multiplicity of positive and nodal solutions for the following semilinear elliptic equation:
½
−∆u + u = |u|p−2u++ |u|q−2u− in Ω,
u ∈ H1 0(Ω) , (Ep,q) where Ω is a domain in RN, 2 < p, q < 2∗(2∗ = 2N N −2 if N ≥ 3, 2∗ = ∞ if N = 2), u+ = max {0, u} , u−= u − u+ and H1
0 (Ω) is the Sobolev space in Ω with
dual space H−1(Ω). Associated with equation (E
p,q) , we consider the energy
functional J in H1 0(Ω) J(u) = 1 2kuk 2− 1 p Z Ω |u+|p− 1 q Z Ω |u−|q
where kuk = ¡RΩ|∇u|2+ u2¢1/2 is a standard norm in H1
0 (Ω) . It is well known
that the functional J ∈ C2(H1
0 (Ω) , R) and the solutions of equation (Ep,q) are the
critical points of the energy functional J in H1
0(Ω). (see Ambrosetti-Rabinowitz
[1] and Willem [23]).
Generally, a standard technique to find the one sign solutions of equation (Ep,q) in Ω is using the Nehari minimization problem:
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 (see Willem [23]). The existence of positive solutions of equation (Ep,q) is affected by the shape of the domain Ω has been the focus of
a great deal of research in recent years. By the Rellich compactness theorem, it is easy to obtain a positive solution of equation (Ep,q) in bounded domains. For
general unbounded domains Ω, because of the lack of compactness, the existence of positive solutions of equation (Ep,q) in Ω is very difficult and unclear. Indeed
a classical, by now, result Esteban-Lions [17] states that, for a very large class of unbounded domains, those 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. They asserted that equation (Ep,q) does
not admit any nontrivial solution. Recently, there have been some progresses for the existence of positive solutions of equation (Ep,q) in unbounded domains as
follows: Benci-Cerami [5] for Ω is an exterior domain, Berestycki-Lions [6] for Ω = RN, Lien-Tzeng-Wang [18] for Ω is a periodic domain, Chen-Wang [9] for
Ω is an interior flask domain, Del Pino-Felmer [13, 14] for Ω is a quasicylindrical domain, Wu [24] for Ω is a multi-bump domain and Wu [25] for Ω is an unbounded dumbbell type domain.
In the aforementioned works, the authors considered positive solutions. For other situations, Bartsch [4] obtained infinite nodal solutions for equation (Ep,q) in
bounded domains. Furtado [11, 12] showed that the domain topology is related with the number of 2–nodal solutions of equation (Ep,q). A 2–nodal solution
is a nontrivial solution u 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]). Bartsch-Weth [3], proved that equation (Ep,q) in a bounded domain Ω that contains a large ball has three nodal solutions
in which two 2–nodal solutions.
Motivated by the above results, we are interested in relation between the shape of domain and the number of positive and 2–nodal solutions of equation (Ep,q). Now, we state our main result in this paper. Take r > 0 and m ≥
1, we assume that the domains Θ0, Θ1(r) , Θ2(r) , · · · , Θm(r) are satisfying the
(D1) there exists a positive number R0 such that BN(z; R0) \Θ0 6= ∅ for all
z ∈ RN, where BN(z; r) =©x ∈ RN | |x − z| < rª;
(Dr) there exist points z1, z2, . . . , zm in RN such that
BN(zi; r) ⊆ Θi(r) ⊆ BN(zi; r + 1) for all i ∈ {1, 2, . . . , m} , |zi− zj| > 3 (r + 1) and Θ0∪ hm ∪ i=1Θi(r) i is a smooth domain in RN. Let Θ (r) = Θ0∪ hm ∪ i=1Θi(r) i
. Then we have the following result.
Theorem 1.1 For each positive number ε ≤ min n p p−2α+ ¡ RN¢, q q−2α− ¡ RN¢o
and unbounded domain Θ0 which satisfies the condition (D1) , there exists r0 > 0
such that if r ≥ r0 and the domains Θ0, Θ1(r) , Θ2(r) , · · · , Θm(r) are
satisfy-ing the condition (Dr), then equation (Ep,q) in Θ (r) has m2 2–nodal solutions
©
ui,j0 ªi,j∈{1,2,...,m} and m positive solutions u1
0, u20, . . . , um0 with Z Θc i(r) ¯ ¯ ¯¡ui,j0 ¢+ ¯ ¯ ¯p < ε, Z Θc j(r) ¯ ¯ ¯¡ui,j0 ¢− ¯ ¯ ¯q < ε for all i, j ∈ {1, 2 . . . , m} , and Z Θc i(r) ¯ ¯ui 0 ¯ ¯p < ε 2 for all i ∈ {1, 2 . . . , m} ,
where ¡ui,j0 ¢+= max©ui,j0 , 0ª and ¡ui,j0 ¢−= ui,j0 −¡ui,j0 ¢+.
By the condition (Dr) we have for i, j ∈ {1, 2 . . . , m} with i 6= j
dist (Θi(r) , Θj(r)) → ∞ as r → ∞.
Thus, if Θ0 is a bounded domain and m ≥ 2, then the condition (Dr) cannot be
use. Next, we modify conditions (D1) and (Dr) such that the result of Theorem 1.1 also holds even if Θ0 is a bounded domain.
¡
D1¢ there exists a positive number b such that Θ0 ⊂ S (b) = © (x0, xN) ∈ RN −1× R | − b < zN < b ª ; ¡
Dr¢ there exist points z1, z2, . . . , zm in S (b) such that
BN(zi; r) ⊆ Θi(r) ⊆ BN(zi; r + 1) for all i ∈ {1, 2, . . . , m} , |zi− zj| > 2 (r + 1) and Θ0∪ hm ∪ i=1Θi(r) i is a smooth domain in RN.
Furthermore, we have the following result which is the same in Theorem 1.1. Theorem 1.2 For each positive number ε ≤ minn p
p−2α+ ¡ RN¢, q q−2α− ¡ RN¢o
and domain Θ0 which satisfies the condition
¡
D1¢, there exists r0 > 0 such that if
r ≥ r0and the domains Θ0, Θ1(r) , Θ2(r) , · · · , Θm(r) are satisfying the condition
¡
Dr¢, then equation (Ep,q) in Θ (r) has m2 2–nodal solutions
©
ui,j0 ªi,j∈{1,2,...,m}and m positive solutions u1 0, u20, . . . , um0 with Z Θc i(r) ¯ ¯ ¯¡ui,j0 ¢+ ¯ ¯ ¯p < ε; Z Θc j(r) ¯ ¯ ¯¡ui,j0 ¢− ¯ ¯ ¯q < ε for all i, j ∈ {1, 2 . . . , m} , and Z Θc i(r) ¯ ¯ui 0 ¯ ¯p < ε 2 for all i ∈ {1, 2 . . . , m} .
Proof. Similar to the argument in Theorem 1.1 and is omitted here. ¤ Among other interesting results, Del Pino-Felmer [15], Del Pino-Felmer-Wei [16], Noussair-Wei [20] and Wei [22] have considered the effect of domain topology on the existence of single–peak positive solutions, multi–peak positive solutions or nodal solutions. Roughly speaking, if Ω has a ”rich” topology, then the singular perturbation problem
½
−ε∆u + u = |u|p−2u in Ω
u = 0 on ∂Ω
has single–peak positive solutions, multi–peak positive solutions or nodal solu-tions provided that ε is sufficiently small.
This paper is organized as follows. In section 2, we describe various prelimi-naries. In section 3, we construct the Palais–Smale (simply by (PS)) sequences. In section 4, we prove Theorem 1.1.
2
Preliminaries
In this section, we recall several known results will be used for later sections. First, we define the (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 → ∞.
For any β ∈ R, a (PS)β–sequence in H01(Ω) for J is bounded.
Lemma 2.2 Let β ∈ R and {un} be a (PS)β–sequence in H01(Ω) for J, then
Proof. See Willem [23]. ¤ Now, we consider the minimization problems
α±(Ω) = inf
u∈M±(Ω)J (u) ; θ (Ω) = infu∈N(Ω)J (u) ,
where M±(Ω) =©u ∈ H1 0(Ω) \{0} | hJ0(u) , ui = 0, ±u ≥ 0 ª and N (Ω) =©u ∈ H1 0 (Ω) | u+ ∈ M+(Ω) , u− ∈ M−(Ω) ª .
Clearly, α+(Ω) + α−(Ω) ≤ θ (Ω) . Then we have the following results.
Lemma 2.3 Let β > 0 and {un} ⊂ H01(Ω) \{0} be a sequence for J such that
J(un) = β + o(1) and kunk2 =
R
Ω|u+n|p+
R
Ω|u−n|q+ o (1) . Furthermore,
(i) if un ≥ 0, then there is a sequence {s+n} ⊂ R+ such that s+n = 1 + o(1),
{s+
nun} ⊂ M+(Ω) and J(s+nun) = β + o(1);
(ii) if un ≤ 0, then there is a sequence {s−n} ⊂ R+ such that s−n = 1 + o(1),
{s−
nun} ⊂ M−(Ω) and J(s−nun) = β + o(1).
Proof. See Willem [23, p. 75]. ¤ Lemma 2.4 (i) Let {un} ⊂ H01(Ω) and un ≥ 0. Then {un} is a (PS)α+(Ω)–
sequence in H1
0(Ω) for J if and only if J (un) = α+(Ω) + o (1) and kunk2 =
R
Ω|u+n|
p+ o (1) ;
(ii) Let {un} ⊂ H01(Ω) and un ≤ 0. Then {un} is a (PS)α−(Ω)–sequence in H01(Ω)
for J if and only if J (un) = α−(Ω) + o (1) and kunk2 =
R
Ω|u−n|p + o (1) .
Proof. See Wang-Wu [21]. ¤
Let Ω be any unbounded domain and ξ ∈ C∞([0, ∞)) such that 0 ≤ ξ ≤ 1,
ξ(t) = 0 for t ∈ [0, 1] and ξ(t) = 1 for t ∈ [2, ∞). For n ∈ N, we define ξn(x) = ξ(
2|x|
n ). (1)
Then we have the following useful lemma, whose proof can be found in Wu [24]. Lemma 2.5 Suppose that {un} is a (PS)β–sequence in H01(Ω) for J satisfies
u±
n * 0 weakly in H01(Ω) . Let vn±= ξnu±n. Then there exists a subsequence {un}
such that ku± n − vn±k = o(1) as n → ∞. Furthermore, (i) if u+ n * 0 weakly in H01(Ω) , then kvn+k 2 =R Ω|vn+|p+ o (1) ; (ii) if u− n * 0 weakly in H01(Ω) , then kvn−k2 = R Ω|vn−|q+ o (1) . Lemma 2.6 If u ∈ H1
0(Ω) is a nodal solution of equation (Ep,q) in Ω and J (u) <
θ(Ω) + min {α+(Ω), α−(Ω)} , then u is a 2–nodal solution of equation (E
p,q) in Ω.
3
Palais–Smale Sequences
Throughout this section, we assume that the domains Θ0, Θ1(r) , Θ2(r) , · · · , Θm(r)
are satisfying conditions (D1) and (Dr) . For each i, j ∈ {1, 2, . . . , m} and 0 <
ε ≤ min n p p−2α+ ¡ RN¢, q q−2α− ¡ RN¢o, we denote M+ i (ε, r) = ½ u ∈ M+(Θ (r)) | Z [Θi(r)]c ¯ ¯u+¯¯p < ε ¾ ; ∂M+ i (ε, r) = ½ u ∈ M+(Θ (r)) | Z [Θi(r)]c ¯ ¯u+¯¯p = ε ¾ ; Mi−(ε, r) = ½ u ∈ M−(Θ (r)) | Z [Θi(r)]c ¯ ¯u−¯¯q < ε ¾ ; ∂Mi−(ε, r) = ½ u ∈ M−(Θ (r)) | Z [Θi(r)]c ¯ ¯u−¯¯q = ε ¾ ; Ni,j(ε, r) = © u ∈ H1 0(Θ (r)) | u+∈ Mi+(ε, r) and u− ∈ Mj−(ε, r) ª ; ∂Ni,j(ε, r) = n u ∈ H01(Θ (r)) | u+ ∈ Mi+(ε, r), u− ∈ Mj−(ε, r) and either u+ ∈ ∂Mi+(ε, r) or u− ∈ ∂Mj−(ε, r)ª,
where u+ = max {u, 0}, u− = u − u+ and M±
i (ε, r) is a closure of Mi±(ε, r) .
It is easy to verify that Mi±(ε, r) and Ni,j(ε, r) are nonempty sets for all i, j ∈
{1, 2, . . . , m}. Note that, if Mi±(ε, r) and Ni,j(ε, r) is denoted the closure of
M±
i (ε, r) and Ni,j(ε, r) , respectively, then we have Mi±(ε, r) = Mi±(ε, r) ∪
∂Mi±(ε, r) , Ni,j(ε, r) = Ni,j(ε, r) ∪ ∂Ni,j(ε, r) and ∂Mi±(ε, r) , ∂Ni,j(ε, r) is
the boundary of Mi±(ε, r), Ni,j(ε, r), respectively. Furthermore, we have the
fol-lowing results.
Lemma 3.1 For each r ≥ 2, we have (i) Ni,j(ε, r) are disjoint;
(ii) Mi±(ε, r) and Mj±(ε, r) are disjoint for all i 6= j.
Proof. (i) Since the proof of all cases are similar. Thus, we only need to prove the case “1, 1” and “1, 2”. Assume the contrary, there exists v0 ∈ N1,1(ε, r)∩N1,2(ε, r)
such that Z [Θ1(r)]c ¯ ¯v− 0 ¯ ¯q < ε and Z [Θ2(r)]c ¯ ¯v− 0 ¯ ¯q < ε.
Since v− 0 ∈ M−(Θ (r)) , we have 2q q − 2α −(Θ (r)) ≤ Z Θ(r) ¯ ¯v− 0 ¯ ¯q ≤ Z [Θ1(r)]c ¯ ¯v− 0 ¯ ¯q + Z [Θ2(r)]c ¯ ¯v− 0 ¯ ¯q < 2ε ≤ 2q q − 2α −¡RN¢, which is a contradiction.
The proof of (ii) is similar and omitted here. ¤ Define the minimization problems in Mi±¡ε
2, r ¢ , ∂Mi±¡ε 2, r ¢ , Ni ¡ε 2, r ¢ and ∂Ni,j ¡ε 2, r ¢ for J, βi±(r) = inf v∈Mi±(ε2,r) J (v) ; eβi±(r) = inf v∈∂Mi±(ε2,r) J (v) and γi,j(r) = inf v∈Ni,j(ε2,r) J (v) ; eγi,j(r) = inf v∈∂Ni,j(ε2,r) J (v) . Clearly, βi±(r) ≥ α±(Θ (r)) , γ i,j(r) ≥ α+(Θ (r)) + α−(Θ (r)) . Furthermore, we
have the following results.
Lemma 3.2 For each positive number σ < min©α+¡RN¢, α−¡RN¢ªthere exists
r0 > 0 such that (i) γi,j(r) < α+ ¡ RN¢+ α−¡RN¢+ σ; (ii) β± i (r) < α± ¡ RN¢+ σ,
for all i, j ∈ {1, 2, . . . , m} and r ≥ r0.
Proof. (i) Let x = (x0, x
N) ∈ RN −1× R and
B+N(0; r) =©x ∈ BN(0; r) | xN > 0
ª
and B−N(0; r) = ©x ∈ BN(0; r) | xN < 0
ª
be half N–balls in RN. By the Lien-Tzeng-Wang [18, Lemma 2.2],
α±¡BN ± (0; r)
¢
& α±¡RN¢ as r % ∞.
Thus, there exists r1 > 0 such that α±
¡ BN ± (0; r1) ¢ < α±¡RN¢+σ 2. Moreover, by
Ambrosetti-Rabinowitz [1], equation (Ep,q) in BN+ (0; r1) and in B−N(0; r1) has a
positive solution v+ and a negative solution v−, respectively, such that J (v±) =
α±¡BN ± (0; r1)
¢
. By Lien-Tzeng-Wang [18, Theorem 2.10], if Ω is a domain in
RN, then α±(Ω) is invariant by rigid motions. Thus,
J (v±) = α± ¡£ BN ± (0; r1) + x ¤¢ < α±¡RN¢+σ 2 for all x ∈ R N.
Set vi(x) = v+(x − zi) and vj(x) = v−(x − zj) . Clearly, vi ∈ M+(Θ (r)) , vj ∈ M−(Θ (r)) and Z [BN(zi;r1)]c ¯ ¯v+ i ¯ ¯p = Z [BN(zj;r1)]c ¯ ¯v− j ¯ ¯q = 0. Thus, vi ∈ Mi+ ¡ε 2, r ¢ and vj ∈ Mj− ¡ε 2, r ¢
for all r ≥ r1. Set vi,j = vi + vj, we
obtain vi,j ∈ Ni,j
¡ε 2, r ¢ and γi,j(r) ≤ J (vi,j) < α+ ¡ RN¢+ α−¡RN¢+ σ for all i, j ∈ {1, 2, · · · , m} and r ≥ r1.
The proof of (ii) is similar and omitted here. ¤ Lemma 3.3 There exist positive numbers δ, r2such that for each i, j ∈ {1, 2, . . . , m} ,
(i) eγi,j(r) > α+ ¡ RN¢+ α−¡RN¢+ δ for all r ≥ r 2; (ii) eβi(r) > α± ¡ RN¢+ δ for all r ≥ r 2.
Proof. (i) Fix i, j ∈ {1, 2, . . . , m} . Assume the contrary, there exist {rn} ⊂ R+
with rn → ∞ as n → ∞, {zi,n} , {zj,n} ⊂ RN and {un} ⊂ ∂Ni,j
¡ε 2, rn ¢ such that J (un) → c ≤ 2α+ ¡ RN¢+ α−¡RN¢, (2) Z Θ(rn) ¯ ¯∇u+ n ¯ ¯2 +¡u+ n ¢2 = Z Θ(rn) ¯ ¯u+ n ¯ ¯p , (3) Z Θ(rn) ¯ ¯∇u− n ¯ ¯2 +¡u− n ¢2 = Z Θ(rn) ¯ ¯u− n ¯ ¯q , (4) and either u+ n ∈ ∂Mi+ ¡ε 2, rn ¢ or u− n ∈ ∂Mj− ¡ε 2, rn ¢ . Since J (un) = J (u+n)+J (u−n) and J (u± n) ≥ α± ¡ RN¢. Thus, J (u± n) → α± ¡ RN¢. Moreover, by (3) , (4) and Lemma 2.4, {u± n} is (PS)α±(RN)–sequences in H1 ¡
RN¢ for J. By (3) and the
Sobolev imbedding theorem, there exists d > 0 such thatRΘ(rn)|∇u±
n|2+(u±n)2 > d
for all n. From the concentration compactness principle of Lions [19], there exist positive numbers R, d and {y±
n} ⊂ RN such that Z BN(y+ n;R) ¯ ¯u+ n ¯ ¯q ≥ d and Z BN(y− n;R) ¯ ¯u− n ¯ ¯q ≥ d for all n.
Without loss of generality, we may assume that u+
n ∈ ∂Mi+ ¡ε 2, rn ¢ , that is Z [Θi(rn)]c ¯ ¯u+ n ¯ ¯p = ε 2 ≤ 1 2min ½ p p − 2α +¡RN¢, q q − 2α −¡RN¢ ¾ . (5)
Let ¯un(x) = u+n (x + y+n). Then ¯un ∈ M+ ¡ RN¢ and there is a u 0 ∈ H1 ¡ RN¢ such that ¯ un * u0 weakly in H1 ¡ RN¢ as n → ∞, ¯ un → u0 a.e. in RN as n → ∞ and Z BN(0;R) |¯un|p → Z BN(0;R) |u0|p ≥ d as n → ∞.
Moreover, by Bahri-Lions [2] and the strong maximum principle ¯
un → u0 strongly in H1
¡
RN¢ as n → ∞,
u0 is a positive solution of equation (Ep,q) in RN and J (u0) = α+
¡
RN¢. Now,
we consider the sequence {zi,n− yn+} . By passing to a subsequence if necessary,
we may assume that one of the following cases occurs: case (I) {zi,n− yn+} is bounded;
case (II) {zi,n− y+n} is unbounded and for each R > 0 there exists n (R) ∈ N
such that
BN(0; R) ∩£Θ
i(rn) − yn+
¤
= ∅ for all n ≥ n (R) ; case (III) {zi,n− yn+} is unbounded and there exists R1 > 0 such that
BN(0; R1) ∩
£
Θi(rn) − y+n
¤
6= ∅ for all n.
Since k¯un → u0k → 0 as n → ∞. By the Sobolev imbedding theorem and the
Vitali convergence theorem, there exists R¡ε
2 ¢ > R0 such that Z |z|>R(ε 2) |¯un|p < ε 2 for all n. (6) In case (I) : We may assume zi,n− y+n → z0. By rn→ ∞ as n → ∞, there exists
n0 ∈ N such that BN
¡ 0; R¡ε
2
¢¢
⊂ [Θi(rn) − yn+] for all n ≥ n0. Thus, for each
n ≥ n0 Z [Θi(rn)]c ¯ ¯u+ n ¯ ¯p = Z [Θi(rn)−yn+] c|¯un| p ≤ Z |z|>R(ε 2) |¯un|p < ε 2, this contradicts (5) .
In case (II) : By the hypothesis, there exists n0 = n
¡ R¡ε 2 ¢¢ ∈ N such that BN ³ 0; R³ ε 2 ´´ ∩£Θi(rn) − y+n ¤ = ∅ for all n ≥ n0. Thus, Z [Θi(rn)−y+n] |¯un|p ≤ Z |z|>R(ε2) |¯un|p < ε 2 for all n ≥ n0. (7)
Since {u+ n} ⊂ M+(Θ (rn)), this means Z Θ(rn) ¯ ¯u+ n ¯ ¯p > 2p p − 2α +¡RN¢ for all n. (8)
From (7) and (8) , we obtain Z [Θi(rn)]c ¯ ¯u+ n ¯ ¯p = Z Θ(rn) ¯ ¯u+ n ¯ ¯p − Z [Θi(rn)−y+n] |¯un|p > 3p 2 (p − 2)α +¡RN¢
for all n ≥ n0, this contradicts (5) .
In case (III) : First, we claim that for each R ≥ max©R1, R
¡ε 2 ¢ª there exists n¡R¢ ∈ N such that BN¡0; R¢∩£Θ j(rn) − yn+ ¤ = ∅ (9)
for all j ∈ {1, 2, . . . , i − 1, i + 1, . . . , m} and n ≥ n¡R¢. By the condition (D1) ,
BN¡0; R¢\£Θ 0− yn+ ¤ 6= ∅ for all n. (10) Since ¯un ≡ 0 in [Θ (rn) − y+n]c, ¯ un → u0 a.e. in RN as n → ∞
and u0 is a positive solution of equation (Ep,q) in RN, we have
lim n→∞ £ Θ (rn) − y+n ¤ = RN. (11)
Since |zi,n− zj,n| > 3 (rn+ 1) and rn → ∞ as n → ∞, there exists n
¡
R¢ ∈ N
such that
dist (Θi(rn) , Θj(rn)) > 2R for all i 6= j and n ≥ n
¡ R¢. Moreover, BN¡0; R¢∩£Θ i(rn) − yn+ ¤ 6= ∅ for all n.
Therefore, (9) holds. By (9) and (11) , for each R ≥ max©R1, R
¡ε 2 ¢ª there exists n ∈ N such that BN(0; R) ⊆£Θi(rn) − y+n ¤ for all n ≥ n. (12) From (6) and (12), we can conclude that for n ≥ n
Z Θi(rn) ¯ ¯u+ n ¯ ¯p = Z [Θi(rn)−yn+] |¯un|p ≥ Z |z|≤R(ε2) |¯un|p > 2p p − 2α +¡RN¢− ε 2
or Z [Θi(rn)]c ¯ ¯u+ n ¯ ¯p < ε 2, this contradicts (5) .
The proof of (ii) is similar and omitted here. ¤ By Lemmas 3.2, 3.3, there exists r0 > 0 such that for r > r0
γi,j(r) < min © α+¡RN¢+ α−¡RN¢+ min©α+¡RN¢, α−¡RN¢ª, eγ i,j(r) ª and βi±(r) < min n α+¡RN¢+ α−¡RN¢, eβi±(r) o
for all i, j ∈ {1, 2, . . . , m} . Furthermore, we will use the idea of Bartsch-Weth [3] and Clapp-Weth [10] to get the following results.
Lemma 3.4 There exists µ0 > 0 such that, for each v ∈ Ni,j
¡ε 2, r ¢ and u ∈ H1 0(Θ (r)) with kv − uk < µ0, we have ¯ ¯ ¯ ¯ Z [Θi(r)]c ¯ ¯u+¯¯p− Z [Θi(r)]c ¯ ¯v+¯¯p ¯ ¯ ¯ ¯ < ε2 and ¯ ¯ ¯ ¯ ¯ Z [Θj(r)]c ¯ ¯u−¯¯q− Z [Θj(r)]c ¯ ¯v−¯¯q ¯ ¯ ¯ ¯ ¯< ε 2. Proof. If not, then there exist {vn} ⊂ Ni,j
¡ε 2, r ¢ and {un} ⊂ H01(Θ (r)) such that kvn− unk → 0, but ¯ ¯ ¯ ¯ Z [Θi(r)]c ¯ ¯u+ n ¯ ¯p − Z [Θi(r)]c ¯ ¯v+ n ¯ ¯p ¯ ¯ ¯ ¯ ≥ 2ε (13) or ¯¯ ¯ ¯ ¯ Z [Θj(r)]c ¯ ¯u− n ¯ ¯q − Z [Θj(r)]c ¯ ¯v− n ¯ ¯q ¯ ¯ ¯ ¯ ¯≥ ε 2. (14) Since Z [Θi(r)]c ¯ ¯u+ n − v+n ¯ ¯p ≤ Z Θ(r) ¯ ¯u+ n − v+n ¯ ¯p ≤ Z Θ(r) |un− vn|p → 0 and Z [Θj(r)]c ¯ ¯u− n − vn− ¯ ¯q ≤ Z Θ(r) ¯ ¯u− n − v−n ¯ ¯q ≤ Z Θ(r) |un− vn|q→ 0.
Thus, by the Minkowski inequality ¯ ¯ ¯ ¯ Z [Θi(r)]c ¯ ¯u+ n ¯ ¯p − Z [Θi(r)]c ¯ ¯v+ n ¯ ¯p ¯ ¯ ¯ ¯ → 0 and ¯ ¯ ¯ ¯ ¯ Z [Θj(r)]c ¯ ¯u− n ¯ ¯q − Z [Θj(r)]c ¯ ¯v− n ¯ ¯q ¯ ¯ ¯ ¯ ¯→ 0 this contradicts (13) and (14) . ¤
Lemma 3.5 For each v0 ∈ Ni,j
¡ε
2, r
¢
there exists a map h : H1
0 (Θ (r)) → R2
such that
(i) h¡s1v0++ s2v0−
¢
= (s1, s2) for s1, s2 ≥ 0;
(ii) h (u) = (1, 1) if and only if u ∈ Ni,j
¡ε
2, r
¢
.
Proof. Similar to the method used in Clapp-Weth [10, Lemma 13]. ¤ Let b = α+¡RN¢+ α−¡RN¢+ min©α+¡RN¢, α−¡RN¢ª. Then we have the
following results.
Proposition 3.6 Let λ0 = b − γi,j(r) . Then for each λ ∈ (0, λ0) and µ ∈ (0, µ0)
there exists u0 ∈ H01(Θ (r)) such that
(i) dist¡u0, Ni,j
¡ε
2, r
¢¢
≤ µ;
(ii) J (u0) ∈ [γi,j(r) , γi,j(r) + λ);
(iii) k∇J (u0)k ≤ max n√ λ,λ µ o ; (iv) R[Θ i(r)]c ¯ ¯u+ 0 ¯ ¯p < ε and R[Θ j(r)]c ¯ ¯u− 0 ¯ ¯q < ε.
Proof. Fix v0 ∈ Ni,j
¡ε
2, r
¢
such that J (v0) < γi,j(r) + λ, and fix l0 > 1 such that
J¡l0v±0 ¢ ≤ 0. Let h : H1 0 (Θ (r)) → R2as in Lemma 3.5. We put K = [0, l0]×[0, l0] and define η : K → H1 0 (Θ (r)) , η (s1, s2) = s1v+0 + s2v−0. Then h ◦ η = id : K → K, in particular deg (h ◦ η, K, (1, 1)) = 1. (15) Notice also that
J (η (s1, s2)) ≤ J (v0) < γi,j(r) + λ for all (s1, s2) ∈ K. (16)
Now we choose a Lipschitz continuous function χ : R → R such that 0 ≤ χ ≤ 1, χ (s) = 1 for s ≥ 0 and χ (s) = 0 for s ≤ −1. Since J ∈ C2(H1
0 (Θ (r)) , R) , there is a semiflow ϕ : [0, ∞) × H1 0 (Θ (r)) → H01(Θ (r)) satisfying ½ ∂ ∂tϕ (t, u) = −χ (J (ϕ (t, u))) ∇J (ϕ (t, u)) , ϕ (0, u) = u.
We will frequently write ϕt in place of ϕ (t, ·) . Since
J¡v± 0 ¢ < γi,j(r) + λ − α± ¡ RN¢< α+¡RN¢+ α−¡RN¢ and J¡l 0v±0 ¢ ≤ 0, it follows that sup J (η (∂K)) < α+¡RN¢+ α−¡RN¢. Hence ¡ ϕt◦ η¢(∂K) ∩ N (Θ (r)) = ∅ for all t ≥ 0.
By Lemma 3.5, this implies ¡
h ◦ ϕt◦ η¢(y) 6= (1, 1) for all y ∈ ∂K, t ≥ 0.
Equality (15) and the global continuation principle of Leray-Schauder (see e.g. Zeider [27, p.629]) imply that there exists a connected subset Z ⊂ K × [0, 1] such that (1, 1, 0) ∈ Z; ϕt(η (s 1, s2)) ∈ N (Θ (r)) for all (s1, s2, t) ∈ Z; Z ∩ (K × {1}) 6= ∅. We put e Z =©ϕt(η (s1, s2)) ∈ N (Θ (r)) | (s1, s2, t) ∈ Z ª . By inequality (16) , sup u∈ eZ J (u) < γi,j(r) + λ < b.
Therefore, since Z is connected, we obtain that eZ ⊂ Ni,j
¡ε 2, r ¢ . Now we pick (¯s1, ¯s2, 1) ∈ Z ∩ (K × {1}) and write v1 := η (¯s1, ¯s2) , v2 := ϕ1(v1) . Then v2 ∈ eZ ⊂ Ni,j ¡ε 2, r ¢
. We distinguish two cases.
Case 1. kϕt(v
1) − v2k ≤ µ for all t ∈ [0, 1] . Then by Lemma 3.4, we have
Z [Θi(r)]c ¯ ¯ ¯¡ϕt(v1) ¢+¯¯ ¯p < ε and Z [Θj(r)]c ¯ ¯ ¯¡ϕt(v1) ¢−¯¯ ¯q < ε
for all t ∈ [0, 1]. We choose t0 ∈ [0, 1] with
° °∇J¡ϕt0(v 1) ¢° ° = min 0≤t≤1 ° °∇J¡ϕt(v 1) ¢°°
and put u0 = ϕt0(v1) . Thus,
λ ≥ J (v1) − J (v2) = − Z 1 0 ∂ ∂tJ ¡ ϕt(v 1) ¢ dt = Z 1 0 ° °∇J¡ϕt(v 1) ¢°°2 dt ≥ k∇J (u0)k2.
We obtain u0 has the desired properties.
Case 2. There exists ¯t ∈ [0, 1] such that °°ϕt¯(v
1) − v2 ° ° > µ. Then let t1 = sup © t ≥ ¯t | °°ϕt(v 1) − v2 ° ° > µª.
By Lemma 3.4, Z [Θi(r)]c ¯ ¯ ¯¡ϕt(v1) ¢+¯¯ ¯p < ε and Z [Θj(r)]c ¯ ¯ ¯¡ϕt(v1) ¢−¯¯ ¯q < ε
for all t ∈ [t1, 1] . We choose t0 ∈ [t1, 1] with
° °∇J¡ϕt0(v 1) ¢° ° = min t1≤t≤1 ° °∇J¡ϕt(v 1) ¢°°
and put u0 = ϕt0(v1) . Then
µ ≤ Z 1 t1 ° ° ° °∂t∂ ϕt(v1) ° ° ° ° dt ≤ Z 1 t1 ° °∇J¡ϕt(v 1) ¢° ° dt and λ ≥ J¡ϕt1(v 1) ¢ − J (v2) = Z 1 t1 ° °∇J¡ϕt(v 1) ¢°°2 dt ≥ k∇J (u0)k Z 1 t1 ° °∇J¡ϕt(v 1) ¢° ° dt.
We conclude that k∇J (u0)k ≤ µλ. Thus, u0 has the desired properties. ¤
Corollary 3.7 For each r > r0 there exists a sequence {ui,jn } ⊂ H01(Θ (r)) such
that
(i) dist¡ui,j n , Ni,j ¡ε 2, r ¢¢ → 0; (ii) J (ui,j n ) → γi,j(r) < α+ ¡ RN¢+ α−¡RN¢+ min©α+¡RN¢, α−¡RN¢ª; (iii) J0(ui,j n ) = o(1) strongly in H−1(Θ (r)) ; (iv) R[Θ i(r)]c ¯ ¯ ¯(ui,j n ) +¯¯ ¯p < ε and R[Θ j(r)]c ¯ ¯ ¯(ui,j n ) −¯¯ ¯q < ε.
For the set Mi
¡ε
2, r
¢
, by a similar argument in Wu [24] (or see Cao-Noussair
[8]), we have
Proposition 3.8 For each r > r0 there exists a sequence {uin} ⊂ Mi
¡ε 2, r ¢ such that J (ui n) → βi±(r) < min n 2α±¡RN¢, eβ i(r) o and J0(ui n) = o(1) strongly in H−1(Θ (r)) .
4
Proof of Theorem 1.1
In this section we establish the existence of m positive solutions and m2 2–nodal
solutions of equation (Ep,q) in Θ (r) provided that r is sufficiently large. For each
Proposition 4.1 For each sequence {ui,j
n } ⊂ H01(Θ (r)) which satisfies
(i) dist¡ui,j n , Ni,j ¡ε 2, r ¢¢ → 0; (ii) J (ui,j n ) → γi,j(r) ; (iii) J0(ui,j n ) = o(1) strongly in H−1(Θ (r)) ; (iv) R[Θi(r)]c ¯ ¯ ¯(ui,j n ) +¯¯ ¯p < ε and R[Θj(r)]c ¯ ¯ ¯(ui,j n ) −¯¯ ¯q < ε, there exist a subsequence {ui,j
n } and ui,j0 ∈ Ni,j(ε, r) such that ui,jn → ui,j0 strongly
in H1
0(Θ (r)). Furthermore, ui,j0 is a 2-nodal solution of equation (Ep,q) in Θ (r) .
Proof. Since {ui,j
n } is bounded in H01(Θ (r)), we have n (ui,j n ) +o and n(ui,j n ) −o
are also bounded in H1
0(Θ (r)) and ° ° °¡ui,j n ¢+°° °2 = Z Θ(r) ¯ ¯ ¯¡ui,j n ¢+¯¯ ¯p+ o (1) and ° ° °¡ui,j n ¢−°° °2 = Z Θ(r) ¯ ¯ ¯¡ui,j n ¢−¯¯ ¯q+ o (1) .
Thus, there exist a subsequence {ui,j
n } and ui,j0 in H01(Θ (r)) such that
ui,j n * ui,j0 ; ¡ ui,j n ¢± *¡ui,j0 ¢± weakly in H1 0(Θ (r)) and ui,j n → ui,j0 ; ¡ ui,j n ¢± →¡ui,j0 ¢± a.e. in Θ (r) . (17) Moreover, ui,j0 is a solution of equation (Ep,q) in Θ (r) . We will show that
¡
ui,j0 ¢± 6≡
0. If not, we may assume that ¡ui,j0 ¢+ ≡ 0 and J
³ (ui,j
n )
+´ = c + o (1) for some
c > 0. By Lemma 2.5, there exists a subsequence
n (ui,j n ) +osuch that J (ξ nui,jn ) = c + o (1) and ° ° °ξn ¡ ui,j n ¢+°° °2 = Z Θ(r) ¯ ¯ ¯ξn ¡ ui,j n ¢+¯¯ ¯p+ o (1) ,
where ξn is as in (1). Moreover, by Lemma 2.3, there exist sequences {si,jn } ⊂
R+\ {0} with si,j n = 1 + o (1) such that J ³ si,j n ξn ¡ ui,j n ¢+´ = c + o (1) and ° ° °si,jn ξn ¡ ui,jn ¢+ ° ° °2 = Z Θ(r) ¯ ¯ ¯si,jn ξn ¡ ui,jn ¢+ ¯ ¯ ¯p.
Then there exists n0 ∈ N such that for n > 2n0,
2p p − 2α +(Θ (r)) ≤ Z Θ(r) |si,j n ξn ¡ ui,j n ¢+ |p = Z [Θi(r)]c ¯ ¯ ¯¡ui,jn ¢+ ¯ ¯ ¯p+ o (1) < p p − 2α +¡RN¢+ o (1)
which is a contradiction. Therefore, ¡ui,j0 ¢± 6≡ 0 and ui,j0 is a nodal solution of equation (Ep,q) in Θ (r) . By the Fatou lemma, we have
Z [Θi(r)]c ¯ ¯ ¯¡ui,j0 ¢+ ¯ ¯ ¯p ≤ lim inf Z [Θi(r)]c ¯ ¯ ¯¡ui,j n ¢+¯¯ ¯p < ε, Z [Θj(r)]c ¯ ¯ ¯¡ui,j0 ¢− ¯ ¯ ¯q≤ lim inf Z [Θj(r)]c ¯ ¯ ¯¡ui,j n ¢−¯¯ ¯q< ε, and J¡ui,j0 ¢= µ 1 2− 1 p ¶ Z Θ(r) ¯ ¯ ¯¡ui,j0 ¢+ ¯ ¯ ¯p+ µ 1 2 − 1 q ¶ Z Θ(r) ¯ ¯ ¯¡ui,j0 ¢− ¯ ¯ ¯q ≤ lim inf ·µ 1 2− 1 p ¶ Z Θ(r) ¯ ¯ ¯¡ui,j n ¢+¯¯ ¯p+ µ 1 2− 1 q ¶ Z Θ(r) ¯ ¯ ¯¡ui,j n ¢−¯¯ ¯q ¸ = γi,j(r) .
Thus, ui,j0 ∈ Ni,j(ε, r) and J
¡
ui,j0 ¢ = γi,j(r) . Moreover, by the concentration
compactness principle of Lions [19] and
γi,j(r) < α+
¡
RN¢+ α−¡RN¢+ min©α+¡RN¢, α−¡RN¢ª,
we have ui,j
n → ui,j0 strongly in H01(Θ (r)) and ui,j0 is a 2-nodal solution of equation
(Ep,q) in Θ (r) . ¤
For the set Mi
¡ε
2, r
¢
, by a similar argument, we have
Proposition 4.2 For each sequence {ui
n} ⊂ Mi+ ¡ε 2, r ¢ which satisfies J (ui n) → βi+(r) and J0(ui
n) = o(1) strongly in H−1(Θ (r)) there exist a subsequence {uin}
and ui 0 ∈ Mi+ ¡ε 2, r ¢ such that ui n→ ui0 strongly in H01(Θ (r)). Furthermore, ui0 is
a positive solution of equation (Ep,q) in Θ (r) and J (ui0) = βi+(r) .
Now, we begin to show the proof of Theorem 1.1: By Corollary 3.7 and Propositions 3.8, 4.1, 4.2, equation (Ep,q) in Θ (r) , there exist 2–nodal solutions
ui,j0 ∈ Ni,j(ε, r) and positive solutions ui0 ∈ Mi+
¡ε
2, r
¢
. By Lemma 3.1, ui,j0 6= ui,k0
for all i, j, k ∈ {1, 2, . . . , m} with j 6= k and ui
0 6= uj0 for all i, j ∈ {1, 2, . . . , m}
with i 6= j. Thus, we have proved Theorem 1.1.
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