Ms. Ref. No.: NA-D-09-00582R1
Title: Existence of multiple positive solutions for a semilinear elliptic equation in R^{N} Nonlinear Analysis Series A: Theory, Methods & Applications
Dear Mr. Tsung-fang Wu,
I am pleased to confirm that your paper "Existence of multiple positive solutions for a semilinear elliptic equation in R^{N}" has been accepted for publication in Nonlinear Analysis Series A: Theory, Methods & Applications.
Comments from the Editor and Reviewers can be found below. Thank you for submitting your work to this journal.
With kind regards, V. Lakshmikantham Editor-in-Chief
Nonlinear Analysis Series A: Theory, Methods & Applications Comments from the Editors and Reviewers:
For further assistance, please visit our customer support site at http://epsupport.elsevier.com. Here you can search for solutions on a range of topics, find answers to frequently asked questions and learn more about EES via interactive tutorials. You will also find our 24/7 support contact details should you need any further assistance from one of our customer support representatives.
來源: "Journal of Nonlinear Analysis - A <[email protected]> 收信: [email protected]
日期: 10 Dec 2009 21:25:22 +0000 標題: Your Submission
Existence of multiple positive solutions for a
semilinear elliptic equation in R
NTsung-fang Wu
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
Abstract
In this paper, we study a class of semilinear elliptic equations in RN. By means of the Lusternik-Schnirelman category and Bahri-Li’s minimax argument, multiple positive solutions are obtained.
Key words: Multiple positive solutions; Nehari manifold; semilinear elliptic equations
1 Introduction and main results
In this paper, we consider the multiplicity of positive solutions of the fol-lowing semilinear elliptic equation:
−∆u + u = g (x) |u|p−2u in RN, lim|x|→∞u = 0, (E)
where 2 < p < N −22N (N ≥ 3) and g (x) ∈ CRN, R is positive such that
lim|x|→∞g (x) = g∞> 0.
During the past years there has been a considerable interest in problems like (E) due essentially to two reasons: such problems arise naturally in various branches of Mathematical Physics, indeed the solutions of (E) can be seen as stationary states in nonlinear equation of the Klein-Gordon or Schr¨odinger type (for a discussion see for example Berestycki-Lions [7]), and, on the other
hand, they present specific mathematical difficulties that make them challeng-ing to the researchers.
It is well known that if g (x) ≡ g∞, then equation (E) has a unique posi-tive solution (see Kwong [18]) and infinitely many radially symmetric nodal solutions. When g (x) 6≡ g∞, the existence of positive solutions has been estab-lished by several authors under various conditions. Berestycki-Lions [7] and Lions [19] proved that if g (x) ≥ lim|x|→∞g (x) = g∞, then equation (E) has
a positive ground state solution in RN. Bahri–Li [4], Bahri-Lions [5] and Li
[20] proved that there is at least one positive solution of equation (E) in RN when lim|x|→∞g (x) = g∞ and g (x) ≥ g∞ − C exp (−δ |x|) for some δ > 0.
Cao [10] proved that there is at least one positive solution of equation (E) in RN when lim|x|→∞g (x) = g∞ and g (x) ≥ 2(p−2)/2g∞. Zhu [26] has studied
the multiplicity of solutions of equation (E) in RN as follows. Assume that
N ≥ 5, g (x) ≥ lim|x|→∞g (x) = g∞and there exist positive constants C, γ and
R0 such that g (x) ≥ g∞+ C/ |x| γ
for |x| ≥ R0. Then equation (E) has at least
one positive solution and one nodal solution. Hsu [17] has studied the multi-plicity of solutions of equation (E) in unbounded cylinder domain Ω = ω × Rn
as follows. Assume that ω ⊂ Rm a bounded domain, m + n ≥ 3(m ≥ 2 and n ≥ 1), g (y, z) ≥ lim|z|→∞g (y, z) = g∞and there exist positive constants C, δ
and R0 such that g (y, z) ≥ g∞+ C exp (−δ |z|) for |z| ≥ R0, uniformly for
y ∈ ω. Then equation (Eµ) has at least one positive solution and one nodal
solution.
By the above results, we can see authors proved that equation (E) has at least one positive solution. The main purpose of this paper is using the shape of the graph of g (x) to prove the multiplicity of positive solutions of equation (E) . Here we consider the following semilinear elliptic equation:
−∆u + u = gµ(x) |u|p−2u in RN, lim|x|→∞u = 0, (Eµ)
where 2 < p < N −22N (N ≥ 3) and the parameter µ ≥ 0. We assume that gµ(x) = a (x) + µb (x) , where the functions a and b satisfy the following
conditions:
(D1) a, b ∈ CRN and there are positive numbers q, r with r < min {p, q} such
that
1 ≥ a (x) ≥ 1 − c0exp (−q |x|) for some c0 < 1 and for all x ∈ RN
and
b (x) ≥ d0exp (−r |x|) for some d0 > 0 and for all x ∈ RN;
(D2) a (x) ≤ 1 on RN with a strict inequality on a set of positive measure;
Then we have the following multiplicity results.
Theorem 1.1 Suppose that the functions a and b satisfy (D1) − (D3) . Then there exists a positive number µ0 such that
(i) for every µ ∈ [µ0, ∞), equation (Eµ) has at least one positive solution;
(ii) for every µ ∈ (0, µ0) , equation (Eµ) has at least two positive solutions.
Theorem 1.2 Suppose that the functions a and b satisfy (D1) − (D3) and in addition to q > 2. Then there exists a positive number µ0 such that for every µ ∈ (0, µ0) , equation (Eµ) has at least three positive solutions.
For the other similarly problems, some other papers discuss cases in which the function g (x) possesses nondegenerate critical points and depends on a pa-rameter, i.e. it appear like gε(x) := g (εx) , and contain results of multiplicity
of positive solutions under restriction on the size of ε (see Cao-Noussair-Yan [12] and Noussair-Yan [23], and for g (x) of general form Cao-Noussair [11]).
This paper is organized as follows. In section 2, we give some notations and preliminaries. In section 3, we prove Theorem 1.1. In section 4, we prove Theorem 1.2.
2 Preliminaries
In the following sections, we give the proofs of Theorems 1.1, 1.2. We use the variational methods to find positive solutions of equation (Eµ) . Associated
with the equation (Eµ) , we consider the energy functional Jµ in H1 RN Jµ(u) = 1 2kuk 2 H1 − 1 p Z RN gµ(x) |u| p dx where kukH1 = R RN|∇u| 2
+ u2dx1/2 is the standard norm in H1
RN
. It is well known that Jµ∈ C2
H1RN, R and the solutions of equation (Eµ)
are the critical points of the energy functional Jµ in H1
RN
. The minimax method is a typical approach for solving problem of this kind, that is
e
αµ= inf
γ∈Γt∈[0,1]maxJµ(γ(t)), (2.1)
where
Γ = {γ ∈ C([0, 1], H1RN) | γ(0) = 0, γ(1) = e},
J (e) = 0 and e 6= 0. By the well-known mountain pass lemma due to Ambrosetti– Rabinowitz [2], we called the nonzero critical point u ∈ H1
RN
of Jµ is a
ground state solutions of equation (Eµ) can also be obtained by the Nehari minimization problem αµ= inf u∈Mµ Jµ(u) , where Mµ = n
u ∈ H1RN\{0} | DJµ0 (u) , uE= 0o. Note that αeµ = αµ > 0
(see Willem [25]).
Now we define the Palais–Smale (simply by (PS)) sequences, (PS)–values, and (PS)–conditions in H1RNfor Jµ as follows.
Definition 2.1 (i) For β ∈ R, a sequence {un} is a (PS)β–sequence in H1
RN
for Jµif Jµ(un) = β+o(1) and Jµ0(un) = o(1) strongly in H−1 RN as n → ∞; (ii) β ∈ R is a (PS)–value in H1 RN
for Jµ if there exists a (PS)β–sequence
in H1RN for Jµ;
(iii) Jµ satisfies the (PS)β–condition in H1 RN if every (PS)β–sequence in H1 RN
for Jµ contains a convergent subsequence.
Consider the equation
−∆u + u = |u|p−2u in RN, u ∈ H1RN, (E∞)
and its associated energy functional J∞ defined by
J∞(u) = 1 2kuk 2 H1 − 1 p Z RN |u|pdx, u ∈ H1RN.
By [18,19], equation (E∞) has a unique positive solution w0 such that
J∞(w0) = α∞= inf n
J∞(u) | u ∈ H1RN\ {0} ,D(J∞)0(u) , uE= 0o.
Note that α0 = α∞. From [19,21], for each µ ≥ 0 we have the following
decomposition lemma.
Lemma 2.2 Let {un} is a (PS)β–sequence in H1
RN
for Jµ. Then there
exist a subsequence {un} , a positive integer m, a sequence {yni} ∞
functions u0 ∈ H1 RN and wi ∈ H1 RN
\ {0} for 1 ≤ i ≤ m such that |yi n− yjn| → ∞ for 1 ≤ i, j ≤ m and i 6= j; −∆u0+ u0 = gµ(x) |u0|p−2u0 in RN; −∆wi+ wi = |wi|p−2wi in RN; un = u0+Pmi=1wi(x − yni) + o (1) strongly in H1 RN ; Jµ(un) = Jµ(u0) +Pmi=1J ∞(wi) + o (1) .
Furthermore, we have the following result.
Proposition 2.3 For each minimizing sequence {un} in M0 for J0, there
exist a subsequences {un} such that it is a (PS)α∞–sequence in H1
RN
for J∞.
Proof. Let {un} be a minimizing sequence {un} in M0 for J0. By Wang-Wu
[24], {un} is (PS)α0–sequences in H
1
(RN) for J0. Since α0 = α∞ and equation
(E0) does not admits any solution u0 such that J0(u0) = α0. Thus, by Lemma
2.2 there exist a subsequences {un} such that it is a (PS)α∞–sequence in
H1
RN
for J∞.
3 Proof of Theorem 1.1
First, let w0(x) be a unique radially symmetric positive solution of equation
(E∞) such that J∞(w0) = α∞. Then by Gidas-Ni-Nirenberg [16], for any
ε > 0, there exist positive numbers Aε, B0 and Cε such that
Aεexp (− (1 + ε) |x|) ≤ w0(x) ≤ B0exp (− |x|) (3.1)
and
|∇w0(x)| ≤ Cεexp (− (1 − ε) |x|) (3.2)
for all x ∈ RN. Let e ∈ SN −1 = nx ∈ RN | |x| = 1o and w
l(x) = w0(x − le)
for l > 0. Then we have the following result.
Proposition 3.1 For each µ > 0 there exists l0 > 0 such that for l ≥ l0 we
have
sup
t≥0
Jµ(twl) < α∞ for all e ∈ SN −1.
Proof. Since Jµ(twl) = 1 2ktwlk 2 H1− 1 p Z RN gµ|twl| p dx < t 2 2 kw0k 2 H1− tp p Z BN(0,1)amin|w0| p dx where BN(0, 1) =nx ∈ RN | |x| < 1o and amin = inf n a (x) | x ∈ RNo> 0.
Then we have Jµ(twl) → −∞ as t → ∞. Thus, there exists t1 > 0 such that
Jµ(twl) < α∞ for all t ≥ t1. (3.3) Moreover, Jµ(0) = 0 < α∞, Jµ ∈ C2 H1 RN , R and kwlk2H1 = p−22p α∞ for
all l ∈ R, this implies that there exists t2 > 0 such that
Jµ(twl) < α∞ for all 0 ≤ t ≤ t2. (3.4)
Thus, we only need to show that there exists l0 > 0 such that for l > l0,
sup t2≤t≤t1 Jµ(twl) < α∞. Since Jµ(twl) = J∞(tw0) + tp p Z RN (1 − a) wlpdx − µt p p Z RN bwpldx. (3.5) By Brown-Zhang [9] and Willem [25], we know that
J∞(tw0) ≤ α∞ for all t > 0. Thus, by (3.5) , Jµ(twl) ≤ α∞+ tp p Z RN (1 − a) wpldx − µt p p Z RN bwlpdx. (3.6) Setting C0 = C min x∈BN(0,1)w p 0(x) ! > 0. By the condition (D1) µ Z RN bwpldx = µ Z RN b (x + le) wp0(x) dx ≥ µC0 Z BN(0,1)b (x + le) dx (3.7) ≥ µC0d0exp (−r |l|) and
Z
RN
(1 − a) wlpdx ≤
Z
RN
c0exp (−q |x|) B0pexp (−p |x − le|) dx (3.8)
≤ C exp (− min {p, q} |l|) .
Since r < min {p, q} and t2 ≤ t ≤ t1, we can find l0 > 0 such that
tp p Z RN (1 − a) wlpdx < µ p Z RN
bwpldx for all l ≥ l0 and t ∈ [t2, t1] . (3.9)
Thus, by (3.3) , (3.4) , (3.6) and (3.9), we obtain sup
t≥0
Jµ(twl) < α∞ for all l ≥ l0.
This completes the proof.
Remark 3.1 By (3.7) , (3.8) and r < min {p, q} , l0 → ∞ as µ → 0.
Furthermore, we have the following result.
Theorem 3.2 For each µ > 0, equation (Eµ) has a positive ground state
solution.
Proof. Analogously to the proof of Ni-Takagi [22], one can show that the Ekeland variational principle (see [15]), there exists a minimizing sequence {un} ⊂ Mµ such that Jµ(un) = αµ+ o (1) and Jµ0 (un) = o (1) in H−1 RN .
Since αµ < α∞, by Lemma 2.2 there exist a subsequence {un} and u0 ∈ Mµ
is a nonzero solution of equation (Eµ) such that
un → u0 strongly in H01(R
N) and J
µ(u0) = αµ.
Since Jµ(u0) = Jµ(|u0|) and |u0| ∈ Mµ, we may assume that u0 ≥ 0.
More-over, by the maximum principle, we obtain u0 > 0 in RN. This completes the
proof
Note that for each for each µ > 0 and u ∈ Mµ there is a unique
t0(u) = kuk2H1 R RN g0|u| p dx !1/(p−2) > 0 (3.10)
such that t0(u) u ∈ M0. Then we have the following result.
Lemma 3.3 1 < [t0(u)] p
Proof. Clearly, t0(u) = kuk2H1 R RNg0|u| p dx !1/(p−2) > kuk 2 H1 R RN gµ|u| p dx !1/(p−2) = 1. Since kuk2H = Z RN gµ|u|pdx ≤ (1 + µ kb/ak∞) Z RN g0|u|pdx, (3.11)
by (3.10) , (3.11) we have [t0(u)]p ≤ (1 + µ kb/ak∞) p/(p−2)
. This completes the
proof.
In the following, we use an idea of Adachi-Tanaka [3]. For c ∈ R, we denote [Jµ ≤ c] = {u ∈ Mµ| Jµ(u) ≤ c and u ≥ 0} .
We then try to show for a sufficiently small σ > 0
cat ([Jµ ≤ α∞− σ]) ≥ 2. (3.12)
To prove (3.12) , we need some preliminaries. Recall the definition of Lusternik-Schnirelman category.
Definition 3.4 (i) For a topological space X, we say a non-empty, closed subset Y ⊂ X is contractible to a point in X if and only if there exists a continuous mapping
η : [0, 1] × Y → X such that for some x0 ∈ X
η (0, x) = x for all x ∈ Y, and
η (1, x) = x0 for all x ∈ Y.
(ii) We define
cat (X) = min {k ∈ N | there exist closed subsets Y1, ..., Yk ⊂ X such that
Yj is contractible to a point in X for all j and k
∪
j=1Yj = X
.
When there do not exist finitely many closed subsets Y1, ..., Yk ⊂ X such
that Yj is contractible to a point in X for all j and k
∪
j=1Yj = X, we say cat (X) =
∞.
Lemma 3.5 Suppose that X is a Hilbert manifold and Ψ ∈ C1(X, R) .
As-sume that there are c0 ∈ R and k ∈ N,
(i) Ψ (x) satisfies the Palais–Smale condition for energy level c ≤ c0;
(ii) cat ({x ∈ X | Ψ (x) ≤ c0}) ≥ k.
Then Ψ (x) has at least k critical points in {x ∈ X; Ψ (x) ≤ c0} .
Proof. See Ambrosetti [1, Theorem 2.3].
Lemma 3.6 Let N ≥ 1, SN −1=nx ∈ RN; |x| = 1o, and let X be a
topologi-cal space. Suppose that there are two continuous maps F : SN −1→ X, G : X → SN −1
such that G ◦ F is homotopic to the identity map of SN −1, that is, there exists a continuous map ζ : [0, 1] × SN −1→ SN −1 such that
ζ (0, x) = (G ◦ F ) (x) for each x ∈ SN −1, ζ (1, x) = x for each x ∈ SN −1.
Then
cat (X) ≥ 2.
Proof. See Adachi-Tanaka [3, Lemma 2.5].
By Proposition 3.1, for each l ≥ l0 there exists t∗(l) > 0 such that t∗(l) wl ∈
Mµ and
sup
t≥0
Jµ(twl) = Jµ(t∗(l) wl) < α∞.
Now, we define a map Fl : SN −1→ H1
RN
by
Fl(e) (x) = t∗(l) wl(x) for e ∈ SN −1.
Then we have the following result.
Lemma 3.7 There exists a sequence {σl} ⊂ R+ such that
Fl SN −1⊂ [Jµ ≤ α∞− σl] . Proof. By Proposition 3.1, sup t≥0 Jµ(twl) < α∞ uniformly in e ∈ SN −1. Since Fl SN −1 is compact. Thus, J
µ(t∗(l) wl) ≤ α∞− σl, so that the
Next, we need a generalized barycenter map. By this we mean a continuous map Φ : Lp
RN
\ {0} → RN which is equivalent with respect to the action
of the group of Euclidian motions in RN, that is, for every ξ ∈ RN, orthogonal
N × N matrix A and u ∈ Lp
RN
\ {0} , one has
Φ (u (x − ξ)) = ξ + Φ (u (x)) and Φ (u (Ax)) = Φ (u (x)) . (3.13) Such a map has been constructed in Bartsch-Weth [8, Theorem 2.1] and Cerami-Passaseo [14]. For matters of convenience we also assume equivalence with respect to scaling that is
Φ (u (εx)) = ε−1Φ (u (x)) for all u ∈ LpRN\ {0} and ε > 0. (3.14) This property is easily built into the construction (or see Bartsch-Clapp-Weth [6]). Indeed, if Φ0 satisfies (3.13) , then Φ defined by
Φ (u (x)) = kukp/NLp Φ0
ukukp/NLp x
satisfies (3.13) and (3.14) .
The following lemma is a key lemma to prove our main result.
Lemma 3.8 There exists a positive number δ such that for every u ∈ M0
with J0(u) ≤ α∞+ δ we have |Φ(u)| > 0.
Proof. On the contrary, there exists a sequence {un} ⊂ M0 such that
J0(un) = α∞+ o (1) and |Φ(un)| = 0. By Proposition 2.3 we may assume
that {un} is a (PS)α∞–sequences in H1
RN
for J∞. Then by Lemmas 2.2 there exist a sequence {yn} ⊂ RN with |yn| → ∞ and w0 is a solution of
equation (E∞) such that
kun− w0(x − yn)kH1 → 0 as n → ∞. (3.15)
Then by (3.13) and (3.15)
Φ(un) = Φ(w0(x − yn)) + o (1) = Φ(w0) + yn+ o (1)
and so |Φ(un)| → ∞ as n → ∞, this contradicts |Φ(un)| = 0.
Lemma 3.9 There exists µ1 > 0 such that for µ ∈ (0, µ1) , we have
|Φ(u)| > 0 for all u ∈ [Jµ < α∞] .
Proof. For u ∈ [Jµ < α∞] , by Lemma 3.3 there exists t0(u) > 0 such that
Jµ(u) = sup t≥0 Jµ(tu) ≥ Jµ(t0(u) u) = J0(t0(u) u) − µ [t0(u)] p p Z RN b (x) |u|pdx. Thus, by Lemma 3.3 and the Sobolev inequality
J0(t0(u) u) ≤ Jµ(u) + µ [t0(u)] p p Z RN b (x) |u|pdx < α∞+ µ (1 + µ kb/ak∞) p/(p−2)kbk ∞ p S −p2 p kukpH1,
where Spis a best Sobolev constant for the imbedding of H1 RN in Lp RN . Moreover, we can find a positive numberc independent of µ such that kuke H1 ≤ e
c for all µ ≥ 0 and for all u ∈ [Jµ < α∞] . Therefore,
J0(t0(u) u) < α∞+ µ (1 + µ kb/ak∞)p/(p−2)kbk∞ p S −p2 p ec p.
Then there exists µ > 0 such that for µ ∈ (0, µ) , J0(t0(u) u) < α∞+ δ.
By Lemma 3.8, |Φ(t0(u) u)| > 0 for all u ∈ [Jµ < α∞] . Moreover, by Lemma
3.3 and the continuity of Φ there exists a positive number µ1 ≤ µ such that
for every µ ∈ (0, µ1) and u ∈ [Jµ < α∞] we have |Φ(u)| > 0. This completes
the proof.
From Lemma 3.9, we define
G : [Jµ< α∞] → SN −1
by
G (u) = Φ(u) |Φ(u)| . Then we have the following results.
Lemma 3.10 There exist µ0 ∈ (0, µ1] and l∗ ∈ [l0, ∞) such that for µ ∈
(0, µ0) and l ∈ (l∗, ∞) , the map
G ◦ Fl : SN −1 → SN −1
is homotopic to the identity. Proof. Let Θ =nu ∈ H1
RN
\ {0} | |Φ(u)| > 0o. We define G : Θ → SN −1
by
G (u) = Φ(u) |Φ(u)| . as an extension of G. By Remark 3.1 for θ ∈ [0, 1/2)
(1 − 2θ) Fl(e) + 2θw0(x − le) = w0(x − le) + o (1) in H1
RN
as µ → 0. By an argument similar to that in Lemma 3.8, there exist µ0 ∈ (0, µ1] and
l∗ ∈ [l0, ∞) such that for µ ∈ (0, µ0) and l ∈ (l∗, ∞) ,
(1 − 2θ) Fl(e) + 2θw0(x − le) ∈ Θ for all e ∈ SN −1 and θ ∈ [0, 1/2)
and
w0 x −
l 2 (1 − θ)e
!
∈ Θ for all e ∈ SN −1 and θ ∈ [1/2, 1) .
We define ζl(θ, e) : [0, 1] × SN −1 → SN −1 by ζl(θ, e) =
G ((1 − 2θ) Fl(e) + 2θw0(x − le)) for θ ∈ [0, 1/2) ;
Gw0
x − 2(1−θ)l e for θ ∈ [1/2, 1) ;
e for θ = 1.
Then ζl(0, e) = G (Fl(e)) = G (Fl(e)) and ζl(1, e) = e. Since vλ ∈ C (Ω) .
First, we claim that lim
θ→1−ζl(θ, e) = e and lim θ→1 2 −ζl(θ, e) = G (w0(x − le)) . (a) lim θ→1−ζl(θ, e) = e : since Φ(w0 x + l 2 (1 − θ)e ! ) = Φ(w0(x)) + l 2 (1 − θ)e = l 2 (1 − θ)e, we have lim θ→1−ζl(θ, e) = e. (b) lim θ→12− ζl(θ, e) = G (w0(x − le)) : since G ∈ C Θ, SN −1, we obtain lim θ→12− ζl(θ, l) = G (w0(x − le)) . Thus, ζl(θ, e) ∈ C [0, 1] × SN −1, SN −1 and
ζl(0, e) = G (Fl(e)) for all e ∈ SN −1,
ζl(1, e) = e for all e ∈ SN −1,
provided µ ∈ (0, µ0) and l ∈ (l∗, ∞). This completes the proof.
Theorem 3.11 For µ ∈ (0, µ0) and l ∈ (l∗, ∞) , Jµ(u) has at least two critical
points in
[Jµ < α∞] .
Proof. Applying Lemmas 3.6 and 3.10, we have for µ ∈ (0, µ0) and l ∈
(l∗, ∞) ,
cat ([Jµ ≤ α∞− σl]) ≥ 2.
By Lemmas 2.2, 3.5, Jµ(u) has at least two critical points in [Jµ < α∞] .
We now begin to show the proof of Theorem 1.1: By Theorems 3.2, 3.11.
4 Proof of Theorem 1.2 For c > 0, we define J0c(u) =1 2 Z RN |∇u|2+ u2dx − 1 p Z RN cg0|u| p dx, Mc0=nu ∈ H1RN\ {0} |D(J0c)0(u) , uE= 0o.
Recall that for each u ∈ H1
RN
\ {0} there is a unique t0(u) > 0 such that
t0(u) u ∈ M0. Let S = n u ∈ H1RN\ {0} | u ≥ 0 and kukH1 = 1 o . Then we have the following results.
Lemma 4.1 For each u ∈ S, we have
(i) there is a unique tc = tc(u) > 0 such that tcu ∈ Mc 0 and max t≥0 J c 0(tu) = J c 0(t cu) = p − 2 2p Z RN cg0|u|pdx p−2−2 ; (ii) (1 − µ kb/ak∞)p−2−2 J
0(t0(u) u) ≥ Jµ(u) ≥ (1 + µ kb/ak∞)
−2 p−2J
0(t0(u) u) .
Proof. (i) Similar to the proof of some results in Brown-Zhang [9].
(ii) For each u ∈ Mµ, let, tc(u) > 0 and t0(u) > 0 such that tc(u) u ∈ Mc0
Jµ(u) = sup t≥0 Jµ(tu) ≥ Jµ(tc(u) u) ≥ J1+µkb/ak∞ 0 (tc(u) u) = (1 + µ kb/ak∞)p−2−2 p − 2 2p ! kukpH1 R RN g0|u| p dx !p−22 = (1 + µ kb/ak∞)p−2−2 J 0(t0(u) u) . Moreover, Jµ(tu) ≤ J 1−µkb/ak∞ 0 (tu) ≤ (1 − µ kb/ak∞) −2 p−2J 0(t0(u) u) .
This completes the proof.
For u ∈ H1RN\ {0} , we define Iµ(u) = sup
t≥0
Jµ(tu) = Jµ(t∗(u) u) > 0,
where t∗(u) u ∈ Mµ. We observe that if µ is sufficiently small, Bahri-Li’s
minimax argument also works for Jµ. Let
Γ = ( γ ∈ CBN(0, r) , Σ | γ|∂BN(0,r) = wl kwlkH1 ) for large r = l,
where Σ =nu ∈ H1RN | u ≥ 0 and kuk = 1o. Then we define βµ= inf
γ∈Γy∈RsupN
Iµ(γ (y)) for µ ≥ 0.
By Lemma 4.1 (ii) , for µ ∈ (0, kb/ak−1∞), we have (1 + µ kb/ak∞)p−2−2 β
0 ≤ βµ≤ (1 − µ kb/ak∞)
−2 p−2β
0. (4.1)
Furthermore, we have the following result.
Theorem 4.2 There exists a positive number µ0 such that for µ ∈ (0, µ0) , α∞< βµ< 2α∞.
Furthermore, Jµ has a critical point u (3)
0 such that
Jµ
u(3)0 = βµ.
Proof. Bahri-Li [4] proved that equation (E0) admits at least one positive
solution u0and J0(u0) = β0 < 2α∞. Lien-Tzeng-Wang [21] (or see Chabrowski
solution. Hence α∞ < β0 < 2α∞. Moreover, by (4.1) There exists a positive
number µ0 ≤ minnµ0, kb/ak −1 ∞
o
such that for µ ∈ (0, µ0) , α∞< βµ< 2α∞.
Moreover, by Lemma 2.2, Jµ satisfies (PS)βµ–condition and there exists a
critical point u(3)0 such that Jµ
u(3)0 = βµ.
This completes the proof.
We now begin to show the proof of Theorem 1.2: By Theorems 3.11, 4.2, Jµ(u) has three critical points u
(1) 0 , u (2) 0 and u (3) 0 such that Jµ u(i)0 < α∞ < Jµ u(3)0 = βµ< 2α∞ for all i = 1, 2
and so u(1)0 , u(2)0 and u(3)0 are different nontrivial non-negative solutions of equa-tion (Eµ) . By the maximum principle, u
(i)
0 > 0 in RN for all i = 1, 2, 3.
References
[1] A. Ambrosetti, Critical points and nonlinear variational problems, Bulletin Soc. Math. France, M´emoire, N.49, 1992.
[2] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
[3] S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: −4u + u = a(x)up + f (x) in RN, Calc. Var. Partial Differential Equations 11 (2000), 63–95.
[4] A. Bahri and Y. Y. Li, On the min-max procedure for the existence of a positive solution for certain scalar field equations in RN, Rev. Mat. Iberoamericana 6 (1990), 1–15.
[5] A. Bahri and P. L. Lions, On the existence of positive solutions of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 14 no. 3 (1997), 365–413.
[6] T. Bartsch, M. Clapp and T. Weth, Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schr¨odinger equation, Mathematische Annalen 388 (2007), 147–185.
[7] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–345.
[8] T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 22 (2005), 259–281.
[9] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Equns 193 (2003), 481– 499.
[10] D. M. Cao, Positive solutions and bifurcation from essential spectrum of semilinear elliptic equation on RN, Nonlinear Analysis:T. M. A. 15 (1990), 1045–1052.
[11] D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN, Ann. Inst. H. Poincar´e Anal. Non Lineair´e 13 (1996), 567–588.
[12] D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 235–264. [13] J. Chabrowski, Weak Convergence Methods for Semilinear Elliptic Equations,
World Scientific, 1999.
[14] G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations 17 (2003), 257–281.
[15] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 17 (1974), 324– 353.
[16] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1978), 209–243.
[17] T. S. Hsu, Multiple solutions for semilinear elliptic equations in unbounded cylinder domains, Proc. R. Soc. Edinb. A 134 (2004), 719–731.
[18] M. K. Kwong, Uniqueness of positive solution of ∆u − u + up = 0 in RN, Arch.
Rat. Math. Anal. 105 (1989), 243–266.
[19] P. L. Lions, The concentration-compactness principle in the calculus of variations. The local compact case, Ann. Inst. H. Poincar´e Anal. Non Lineair´e 1 (1984), 102–145 and 223–283.
[20] Y. Li, Remarks on a semilinear elliptic equation on RN, J. Diff. Equns 74 (1988), 34–49.
[21] W. C. Lien, S. Y. Tzeng, and H. C. Wang, Existence of solutions of semilinear elliptic problems in unbounded domains, Differential Integral Equations 6 (1993), 1281–1298.
[22] W. M. Ni and I. Takagi, On the shape of least energy solution to a Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819–851.
[23] E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, Proc. London Math. Soc. 62 (2000), 213–227.
[24] H. C. Wang and T. F. Wu, Symmetry breaking in a bounded symmetry domain, NoDEA-Nonlinear Differential Equations Appl. 11 (2004), 361–377.
[25] M. Willem, Minimax Theorems, Birkh¨auser, Boston 1996.
[26] X. P. Zhu, Multiple entire solutions of a semilinear elliptic equation, Nonlinear Analysis: T. M. A. 12 (1988), 1297–1316.
