### Multiple positive solutions for a class of

### concave-convex elliptic problems in R

N### involving sign-changing weight

### Tsung-fang Wu

Department of Applied Mathematics,

National University of Kaohsiung, Kaohsiung 811, Taiwan

Abstract

In this paper, we study the multiplicity of positive solutions for the following concave-convex elliptic equation:

−∆u + u = fλ(x) uq−1+ gµ(x) up−1 in RN,
u ≥ 0 in RN,
u ∈ H1
RN ,
where 1 < q < 2 < p < 2∗
2∗= _{N −2}2N if N ≥ 3, 2∗ = ∞ if N = 1, 2
and the
parameters λ, µ ≥ 0. We assume that fλ(x) = λf+(x)+f−(x) is sign-changing and

gµ(x) = a (x) + µb (x) , where the functions f±, a and b satisfy suitable conditions.

Key words: Semilinear elliptic equations, Sign-changing weight, Multiple positive solutions

1 Introduction

In this paper, we consider the multiplicity results for positive solutions of the following concave-convex elliptic equation:

−∆u + u = fλ(x) uq−1+ gµ(x) up−1 in RN,
u ≥ 0 _{in R}N,
u ∈ H1
RN
,
(Efλ,gµ)

where 1 < q < 2 < p < 2∗ 2∗ = 2N

N −2 if N ≥ 3, 2

∗ _{= ∞ if N = 1, 2}

and the parameters λ, µ ≥ 0. We assume that fλ(x) = λf+(x) + f−(x) and gµ(x) = a (x) + µb (x) where the functions f±, a and b satisfy the following conditions: (D1) f ∈ Lq∗

RN

(q∗ = _{p−q}p ) with f±(x) = ± max {±f (x) , 0} 6≡ 0 and there
exists a positive number rf− such that

f−(x) ≥ −c expb

−rf−|x|

for somec > 0 and for all x ∈ Rb

N_{;}

(D2) a, b ∈ C_{R}N_{and there are positive numbers r}

a, rbwith rb < min

n

rf−, ra, q

o

such that

1 ≥ a (x) ≥ 1 − c0exp (−ra|x|) for some c0 < 1 and for all x ∈ RN and

b (x) ≥ d0exp (−rb|x|) for some d0 > 0 and for all x ∈ RN; (D3) b (x) → 0 and a (x) → 1 as |x| → ∞.

Elliptic problems in bounded domains involving concave and convex terms have been studied extensively since Ambrosetti-Brezis-Cerami [3] considered the following equation:

−∆u = λuq−1_{+ u}p−1 _{in Ω,}
u > 0 in Ω,
u ∈ H1
0(Ω) ,
(Eλ)

where 1 < q < 2 < p ≤ 2∗_{, λ > 0 and Ω is a bounded domain in R}N_{. They}
found that there exists λ0 > 0 such that the equation (Eλ) admits at least
two positive solutions for λ ∈ (0, λ0) , a positive solution for λ = λ0 and no
positive solution exists for λ > λ0 (see also Ambrosetti-Azorezo-Peral [2] for
more references therein). Actually, Adimurthi-Pacella-Yadava [5],
Damascelli-Grossi-Pacella [13], Ouyang-Shi [22] and Tang [25] proved that there exists
λ0 > 0 such that there have exactly two positive solutions of equation (Eλ) in
the unit ball BN_{(0; 1) for λ ∈ (0, λ}

0), exactly one positive solution for λ = λ0
and no positive solution exists for λ > λ0. Generalizations of the result of
equation (Eλ) (involving sign-changing weight) were done by Brown-Wu [9,10],
de Figueiredo-Gossez-Ubilla [16] and Wu [29,30]. However, little has been done
for this type of problem in RN. We are only aware of the works [12,17,21,28]
which studied existence of solutions for some related concave-convex elliptic
problem in RN _{(not involving sign-changing weight). Furthermore, we do not}
know of any results for concave-convex elliptic problems in RN involving
sign-changing weight functions. In this paper, we will study this topic. The following
theorems are our main results.

Theorem 1.1 Suppose that the functions f±, a and b satisfy the conditions
(D1) − (D3) . Let Λ0 = (2 − q)2−q
p−2
kf+k_{Lq}∗
p−2
_{S}
p
p−q
p−q
, where Sp is a best
Sobolev constant for the imbedding of H1

RN into Lp RN . Then

(i) for each λ > 0 and µ > 0 with λp−2(1 + µ kbk_{∞})2−q <q_{2}p−2Λ0, equation

Efλ,gµ

has at least two positive solutions;

(ii) there exist positive numbers λ0, µ0 with λp−20 (1 + µ0kbk∞) 2−q

<q_{2}p−2Λ0
such that for λ ∈ (0, λ0) and µ ∈ (0, µ0) , equation

Efλ,gµ

has at least three positive solutions.

Note that the positive numbers λ0, µ0 are independent of f−. Therefore, if
kf−k_{L}q∗ is sufficiently small, we have the following result.

Theorem 1.2 If in addition to the condition (D1) − (D3) , we still have (D4) a (x) ≤ 1 on RN with a strict inequality on a set of positive measure; (D5) ra> 2,

then there exist positive numbers λe_{0} ≤ λ_{0},
e

µ0 ≤ µ0 and ν0 such that for λ ∈

0,λe_{0}
, µ ∈ (0,µe0) and kf−k_{L}q∗ < ν0, equation
Efλ,gµ

has at least four positive solutions.

Among other interesting similar problems, Adachi-Tanaka [4] has been in-vestigated the following non-homogenous elliptic equation:

−∆u + u = a (x) up−1
+ h (x) in RN,
u > 0 _{in R}N_{,}
u ∈ H1_{R}N,
(Ea,h)

where h (x) ∈ H−1_{R}N_{\ {0} is nonnegative and a (x) ∈ C}
RN
which
sat-isfy
a (x) 1 = lim
|x|→∞a (x)
and

a (x) ≥ 1 − c0exp (− (2 + δ) |x|) for some c0 < 1, δ > 0 and for all x ∈ RN. Using the equationEa,0

does not admit any ground state solution and Bahri-Li’s minimax argument [6], they proved that the equationEa,h

has at least
four positive solutions under the assumption khk_{H}−1 is sufficiently small.

In the following sections, we proceed to prove Theorems 1.1, 1.2. We use the variational methods to find positive solutions of equationEfλ,gµ

. Asso-ciated with the equation Efλ,gµ

H1
RN
Jfλ,gµ(u) =
1
2kuk
2
H1 −
1
q
Z
RN
fλ|u|qdx −
1
p
Z
RN
gµ|u|pdx,
where kuk_{H}1 =
R
RN|∇u|
2

+ u2_{dx}1/2 _{is the standard norm in H}1
RN

. It is well known that the solutions of equation Efλ,gµ

are the critical points of the energy functional Jfλ,gµ in H

1 RN

(see Rabinowitz [23]).

This paper is organized as follows. In section 2, we give some notations and preliminaries. In section 3, we establish the existence of a local minimum for Jfλ,gµ. In section 4, we give an estimate of energy. In section 5, we discussion

some concentration behavior in the Nehari manifold. In sections 6, 7, we prove Theorems 1.1, 1.2.

2 Notations and Preliminaries

Throughout this paper, we denote by Sp the best Sobolev constant for the
embedding of H1
RN
into Lp
RN
is given by
Sp = inf
u∈H1_{(R}N_{)\{0}}
kuk2_{H}1
(R
RN|u|
p
dx)2/p > 0.
In particular,
Z
RN
|u|pdx
1_{p}
≤ S
−1
2

p kuk_{H}1 for all u ∈ H

1

RN

\ {0} .

First, we define the Palais–Smale (simply by (PS)) sequences, (PS)–values,
and (PS)–conditions in H1_{R}Nfor Jfλ,gµ as follows.

Definition 2.1 (i) For β ∈ R, a sequence {un} is a (PS)β–sequence in H1

RN

for Jfλ,gµ if Jfλ,gµ(un) = β + o(1) and J

0 fλ,gµ(un) = o(1) strongly in H −1 RN as n → ∞. (ii) β ∈ R is a (PS)–value in H1RN

for Jfλ,gµ if there exists a (PS)β–

sequence in H1 RN

for Jfλ,gµ.

(iii) Jfλ,gµ satisfies the (PS)β–condition in H

1 RN if every (PS)β–sequence in H1 RN

As the energy functional Jfλ,gµ is not bounded below on H

1 RN

, it is useful to consider the functional on the Nehari manifold

Nfλ,gµ =
n
u ∈ H1_{R}N\ {0} | DJ_{f}0
λ,gµ(u) , u
E
= 0o.
Thus, u ∈ Nfλ,gµ if and only if

kuk2_{H}1 −
Z
RN
fλ|u|qdx −
Z
RN
gµ|u|pdx = 0.

Note that Nfλ,gµ contains every non-zero solution of equation

Efλ,gµ

. Fur-thermore, we have the following results.

Lemma 2.2 The energy functional Jfλ,gµ is coercive and bounded below on

Nfλ,gµ.

Proof. If u ∈ Nfλ,gµ, then, by the H¨older and Sobolev inequalities,

Jfλ,gµ(u) =
1
2 −
1
p
!
kuk2_{H}1 −
1
q −
1
p
!
Z
RN
(λf++ f−) |u|qdx
≥ 1
2 −
1
p
!
kuk2_{H}1 −
1
q −
1
p
!
Z
RN
λf+|u|qdx
≥ p − 2
2p
!
kuk2_{H}1 − λ
p − q
pq
!
kf+k_{L}q∗S
−q_{2}
p kukq_{H}1. (2.1)

Thus, Jfλ,gµ is coercive and bounded below on Nfλ,gµ.

The Nehari manifold Nfλ,gµ is closely linked to the behavior of the function

of the form hu : t → Jfλ,gµ(tu) for t > 0. Such maps are known as fibering

maps and were introduced by Dr´abek-Pohozaev in [14] and are also discussed in Brown-Zhang [11] and Brown-Wu [9,10]. If u ∈ H1

RN
, we have
hu(t) =
t2
2 kuk
2
H1 −
tq
q
Z
RN
fλ|u|qdx −
tp
p
Z
RN
gµ|u|pdx;
h0_{u}(t) = t kuk2_{H}1 − t
q−1Z
RN
fλ|u|qdx − tp−1
Z
RN
gµ|u|pdx;
h00_{u}(t) = kuk2_{H}1 − (q − 1) tq−2
Z
RN
fλ|u|qdx − (p − 1) tp−2
Z
RN
gµ|u|pdx.
It is easy to see that

th0_{u}(t) = ktuk2_{H}1−
Z
RN
fλ|tu|
q
dx −
Z
RN
gµ|tu|
p
dx

and so, for u ∈ H1 RN

\ {0} and t > 0, h0

u(t) = 0 if and only if tu ∈ Nfλ,gµ,

In particular, h0_{u}(1) = 0 if and only if u ∈ Nfλ,gµ. Thus, it is natural to

split Nfλ,gµ into three parts corresponding to local minima, local maxima and

points of inflection. Accordingly, we define

N+_{f}
λ,gµ=
n
u ∈ Nfλ,gµ | h
00
u(1) > 0
o
;
N0_{f}
λ,gµ=
n
u ∈ Nfλ,gµ | h
00
u(1) = 0
o
;
N−_{f}
λ,gµ=
n
u ∈ Nfλ,gµ | h
00
u(1) < 0
o
.

We now derive some basic properties of N+_{f}

λ,gµ, N

0

fλ,gµ and N

− fλ,gµ.

Lemma 2.3 Suppose that u0 is a local minimizer for Jfλ,gµ on Nfλ,gµ and that

u0 ∈ N/ 0fλ,gµ. Then J 0 fλ,gµ(u0) = 0 in H −1 RN .

Proof. The proof is essentially the same as that in Brown-Zhang [11, The-orem 2.3] (or see Binding-Dr´abek-Huang [7]).

For each u ∈ Nfλ,gµ, we have

h00_{u}(1) = kuk2_{H}1 − (q − 1)
Z
RN
fλ|u|
q
dx − (p − 1)
Z
RN
gµ|u|
p
dx
= (2 − p) kuk2_{H}1 − (q − p)
Z
RN
fλ|u|
q
dx (2.2)
= (2 − q) kuk2_{H}1 − (p − q)
Z
RN
gµ|u|
p
dx. (2.3)

Then we have the following result.
Lemma 2.4 (i) For any u ∈ N+_{f}

λ,gµ∪ N
0
fλ,gµ, we have
R
RNfλ|u|
q
dx > 0.
(ii) For any u ∈ N−_{f}

λ,gµ, we have

R

RNgµ|u| p

dx > 0.

Proof. The results now follows immediately from (2.2) and (2.3) . Let Λ0 = (2 − q)2−q p − 2 kf+kLq∗ !p−2 Sp p − q !p−q . Then we have the following result.

Lemma 2.5 For each λ > 0 and µ ≥ 0 with λp−2_{(1 + µ kbk}
∞)

2−q

< Λ0, we have N0

fλ,gµ = ∅.

Proof. Suppose the contrary. Then there exist λ > 0 and µ ≥ 0 with
λp−2(1 + µ kbk_{∞})2−q < Λ0

such that N0

fλ,gµ 6= ∅. Then, for u ∈ N

0

fλ,gµ, by (2.2) and the H¨older and

Sobolev inequalities we have
kuk2_{H}1 =
p − q
p − 2
Z
RN
fλ|u|qdx ≤ λS
−q
2
p
p − q
p − 2kf+kLq∗ kuk
q
H1
and so
kuk2_{H}1 ≤ S
q
q−2
p
"
λ kf+k_{L}q∗
p − q
p − 2
#_{2−q}2
.
Similarly, using (2.3) and the Sobolev inequality we have

2 − q
p − qkuk
2
H1 =
Z
RN
(a + µb) |u|pdx ≤ (1 + µ kbk_{∞}) S
−p
2
p kukp_{H}1,
which implies
kuk2_{H}1 ≥ S
p
p−2
p
"
2 − q
(1 + µ kbk_{∞}) (p − q)
# 2
p−2
for all µ ≥ 0.

Hence, we must have

λp−2(1 + µ kbk_{∞})2−q ≥ (2 − q)2−q p − 2
kf+k_{L}q∗
!p−2
Sp
p − q
!p−q
= Λ0

which is a contradiction. This completes the proof.

In order to get a better understanding of the Nehari manifold and fibering maps, we consider the function mu : R+ → R defined by

mu(t) = t2−qkuk2_{H}1 − t

p−qZ RN

gµ|u|pdx for t > 0. (2.4) Clearly tu ∈ Nfλ,gµ if and only if mu(t) =

R
RNfλ|u|
q
dx. Moreover,
m0_{u}(t) = (2 − q)t1−qkuk2_{H}1 − (p − q)tp−q−1
Z
RN
gµ|u|pdx (2.5)
and so it is easy to see that, if tu ∈ Nfλ,gµ, then t

q−1_{m}0
u(t) = h
00
u(t). Hence,
tu ∈ N+_{f}
λ,gµ( or N
−
fλ,gµ) if and only if m
0
u(t) > 0( or < 0).
Suppose u ∈ H1
RN

\ {0} . Then, by (2.5), mu has a unique critical point at t = tmax,µ(u) where

tmax,µ(u) =
(2 − q) kuk2_{H}1
(p − q)R
RNgµ|u|
p
dx
!_{p−2}1
> 0 (2.6)

and clearly mu is strictly increasing on (0, tmax,µ(u)) and strictly decreasing on (tmax,µ(u) , ∞) with limt→∞mu(t) = −∞. Moreover, if λp−2(1 + µ kbk∞)

2−q < Λ0, then

mu(tmax,µ(u)) =
2 − q
p − q
!2−q_{p−2}
− 2 − q
p − q
!p−q_{p−2}
kuk
2(p−q)
p−2
H1
(R
RNgµ|u|
p
dx)2−qp−2
= kukq_{H}1
p − 2
p − q
!
2 − q
p − q
!2−q
p−2 _{kuk}p
H1
R
RN gµ|u|
p
dx
!2−q
p−2
≥ p − 2
λ kf+k_{L}q∗
Sp
p − q
!p−q_{p−2}
2 − q
1 + µ kbk_{∞}
!2−q_{p−2}
Z
RN
fλ|u|qdx
>
Z
RN
fλ|u|qdx.

Thus, we have the following lemma. Lemma 2.6 For each u ∈ H1

RN

\ {0} we have the following. (i) If R

RN fλ|u| q

dx ≤ 0, then there is a unique t− = t−(u) > tmax,µ(u) such
that t−u ∈ N−_{f}
λ,gµ and hu is increasing on (0, t
−_{) and decreasing on (t}−_{, ∞).}
Moreover,
Jfλ,gµ
t−u= sup
t≥0
Jfλ,gµ(tu) . (2.7)
(ii) If R
RNfλ|u|
q

dx > 0, then there are unique

0 < t+ = t+(u) < tmax,µ(u) < t− = t−(u)
such that t+_{u ∈ N}+

fλ,gµ, t

−_{u ∈ N}−

fλ,gµ, hu is decreasing on (0, t

+_{), increasing}
on (t+_{, t}−_{) and decreasing on (t}−_{, ∞). Moreover,}

Jfλ,gµ t+u= inf 0≤t≤tmax,µ(u) Jfλ,gµ(tu) ; Jfλ,gµ t−u= sup t≥t+ Jfλ,gµ(tu) . (2.8)

(iii) t−(u) is a continuous function for u ∈ H1
RN
\ {0} .
(iv) N−_{f}
λ,gµ =
n
u ∈ H1_{R}N | 1
kuk_{H1}t
− u
kuk_{H1}
= 1o.
Proof. Fix u ∈ H1
RN
\ {0} .
(i) Suppose R
RNfλ|u|
q
dx ≤ 0. Then mu(t) =
R
RNfλ|u|
q

dx has a unique so-lution t− > tmax,µ(u) such that m0u(t

−_{) < 0 and h}0
u(t
−_{) = 0. Hence, by}
tq−1_{m}0
u(t) = h
00

u(t), hu has a unique critical point at t = t− and h00u(t

−_{) < 0.}
Thus, t−u ∈ N−_{f}
λ,gµ and (2.7) holds.
(ii) SupposeR
RNfλ|u|
q

dx > 0. Since mu(tmax,µ(u)) >R_{R}Nfλ|u|qdx, the
equa-tion mu(t) =

R

RNfλ|u| q

dx has exactly two solutions t+ _{< t}

max,µ(u) < t−
such that m0_{u}(t+) > 0 and m0_{u}(t−) < 0. Hence, there are exactly two
multi-ples of u lying in Nfλ,gµ, that is, t

+_{u ∈ N}+

fλ,gµ and t

−_{u ∈ N}−

fλ,gµ. Thus, by

tq−1_{m}0

u(t) = h00u(t), huhas critical points at t = t+ and t = t−with h00u(t+) > 0
and h00_{u}(t−) < 0. Thus, hu is decreasing on (0, t+) , increasing on (t+, t−) and
decreasing on (t+_{, ∞) . Therefore, (2.8) must hold.}

t−(u) is a continuous function for u ∈ H1 RN

\ {0} .
(iv) For u ∈ N−_{f}_{λ}_{,g}_{µ}. Let v = _{kuk}u

H1. By parts (i), (ii), there is a unique

t−(v) > 0 such that t−(v) v ∈ N−_{f}
λ,gµ or t
−
u
kuk_{H1}
1
kuk_{H1}u ∈ N
−
fλ,gµ. Since
u ∈ N−_{f}
λ,gµ, we have t
− _{u}
kuk_{H1}
_{1}

kuk_{H1} = 1, and this implies

N−_{f}
λ,gµ ⊂
(
u ∈ H1_{R}N | 1
kuk_{H}1
t− u
kuk_{H}1
!
= 1
)
.

Conversely, let u ∈ H1_{R}N such that _{kuk}1

H1t
− u
kuk_{H1}
= 1. Then
t− u
kuk_{H}1
!
u
kuk_{H}1
∈ N−_{f}_{λ}_{,g}_{µ}.
Thus,
N−_{f}
λ,gµ =
(
u ∈ H1_{R}N | 1
kuk_{H}1
t− u
kuk_{H}1
!
= 1
)
.
This completes the proof.

Remark 2.1 (i) If λ = 0, then, by Lemma 2.6 (i) N+_{f}

0,gµ = ∅, and so Nf0,gµ =

N−_{f}

0,gµ for all µ ≥ 0.

(ii) If λp−2(1 + µ kbk_{∞})2−q < Λ0, then, by (2.2) , for each u ∈ N+fλ,gµ we have

kuk2_{H}1 <
p − q
p − 2
Z
RN
fλ|u|
q
dx ≤ Λ1/(p−2)_{0} S
−q
2
p
p − q
p − 2kf+kLq∗ kuk
q
H1,
and so
kuk_{H}1 ≤ Λ
1/(p−2)
0 S
−q
2
p
p − q
p − 2kf+kLq∗
!1/(2−q)
for all u ∈ N+_{f}
λ,gµ. (2.9)

3 Existence of a first solution

First, we remark that it follows from Lemma 2.5 that Nfλ,gµ = N

+

fλ,gµ∪ N

− fλ,gµ

for all λ > 0 and µ ≥ 0 with λp−2_{(1 + µ kbk}
∞)

2−q

< Λ0. Furthermore, by
Lemma 2.6 it follows that N+_{f}

λ,gµ and N

−

fλ,gµ are non-empty and, by Lemma

2.2, we may define
α+_{f}
λ,gµ = inf
u∈N+_{fλ,gµ}
Jfλ,gµ(u) and α
−
fλ,gµ = inf
u∈N−_{fλ,gµ}
Jfλ,gµ(u) .

Theorem 3.1 We have the following:
(i) α+_{f}

λ,gµ < 0 for all λ > 0 and µ ≥ 0 with λ

p−2_{(1 + µ kbk}
∞)
2−q
< Λ0.
(ii) If λp−2_{(1 + µ kbk}
∞)
2−q

<q_{2}p−2Λ0, then α−fλ,gµ > c0 for some c0 > 0.

In particular, for each λ > 0 and µ ≥ 0 with λp−2(1 + µ kbk_{∞})2−q <q_{2}p−2Λ0,
we have α_{f}+

λ,gµ = infu∈Nfλ,gµJfλ,gµ(u) .

Proof. (i) Let u ∈ N+_{f}

λ,gµ. Then, by (2.2) ,
kuk2_{H}1 <
p − q
p − 2
Z
RN
fλ|u|qdx.

Hence, by (2.1) and Lemma 2.4,

Jfλ,gµ(u) =
p − 2
2p kuk
2
H1 −
p − q
pq
Z
RN
fλ|u|qdx
< −(p − q) (2 − q)
2pq
Z
RN
fλ|u|
q
dx < 0
and so α+_{f}
λ,gµ < 0.
(ii) Let u ∈ N−_{f}

λ,gµ. Then, by (2.3) and the Sobolev inequality,

2 − q
p − qkuk
2
H1 <
Z
RN
gµ|u|
p
dx ≤ (1 + µ kbk_{∞}) S
−p
2
p kukp_{H}1,
which implies
kuk_{H}1 > S
p
2(p−2)
p
2 − q
(1 + µ kbk_{∞}) (p − q)
!1/(p−2)
for all u ∈ N−_{f}
λ,gµ. (3.1)
By (2.1) and (3.1) , we have
Jfλ,gµ(u)
≥ kukq_{H}1
p − 2
2p kuk
2−q
H1 − λ
p − q
pq
!
kf+kLq∗ S
−q
2
p
!
> S
pq
2(p−2)
p
2 − q
(1 + µ kbk_{∞}) (p − q)
!_{p−2}q
·
p − 2
2p S
p(2−q)
2(p−2)
p
2 − q
(1 + µ kbk_{∞}) (p − q)
!2−q
p−2
− λ p − q
pq
!
kf+kLq∗ S
−q_{2}
p
.
Thus, if λp−2_{(1 + µ kbk}
∞)
2−q
<q_{2}p−2Λ0, then
α−_{f}
λ,gµ > c0 for some c0 > 0.

This completes the proof.

Now, we consider the following semilinear elliptic problem:

−∆u + u = |u|p−2_{u in R}N_{,}
u ∈ H1_{R}N.
(E∞)

Associated with the equation (E∞) , we consider the energy functional J∞ in H1 RN J∞(u) = 1 2kuk 2 H1 − 1 p Z RN |u|pdx. Consider the minimizing problem:

inf
u∈N∞J
∞
(u) = α∞
where
N∞ =nu ∈ H1_{R}N\ {0} | D(J∞)0(u) , uE= 0o.

It is known that equation (E∞) has a unique positive radially solution w0(x) such that J∞(w0) = α∞ (see [8,19]). Then the following proposition provides a precise description for the (PS)–sequence of Jfλ,gµ.

Proposition 3.2 (i) If {un} is a (PS)β–sequence in H1

RN

for Jfλ,gµ with

β < α+_{f}_{λ}_{,g}_{µ} + α∞, then there exists a subsequence {un} and a non-zero u0
in H1_{R}N such that un → u0 strongly in H1

RN

and Jfλ,gµ(u0) = β.

Moreover, u0 is a solution of equation

Efλ,gµ
.
(ii) If {un} ⊂ N−fλ,gµ is a (PS)β–sequence in H
1
RN
for Jfλ,gµ with
α+_{f}_{λ}_{,g}_{µ}+ α∞< β < α−_{f}_{λ}_{,g}_{µ}+ α∞,

then there exists a subsequence {un} and a non-zero u0 in H1

RN such that un → u0 strongly in H1 RN

and Jfλ,gµ(u0) = β. Moreover, u0 is a solution

of equation Efλ,gµ

.

Proof. Similarly the argument in Wu [28, Proposition 4.6] (or see Adachi-Tanaka [4, Proposition 1.9]).

Theorem 3.3 For each λ > 0 and µ ≥ 0 with λp−2(1 + µ kbk_{∞})2−q < Λ0, the
functional Jfλ,gµ has a minimizer u

+
λ,µ in N
+
fλ,gµ and it satisfies
(i) Jfλ,gµ
u+_{λ,µ}= α+_{f}
λ,gµ,

(ii) u+_{λ,µ} is a positive solution of equation Efλ,gµ

,
(iii)
u
+
λ,µ
_{H}1 → 0 as λ → 0.

Proof. By the Ekeland variational principle [15] (or see Wu [29, Proposition 1]), there exists {un} ⊂ N+fλ,gµ such that it is a (PS)α+

fλ,gµ–sequence for Jfλ,gµ.

Then, by Proposition 3.2, there exist a subsequence {un} and u+λ,µ ∈ N + fλ,gµ

a non-zero solution of equation Efλ,gµ

such that un → u+λ,µ strongly in
H1_{(R}N) and Jfλ,gµ
u+_{λ,µ} = α+_{f}_{λ}_{,g}_{µ}. Since Jfλ,gµ
u+_{λ,µ} = Jfλ,gµ
u
+
λ,µ
and
u
+
λ,µ
∈ N
+

fλ,gµ, by Lemma 2.3 we may assume that u

+

λ,µ is a positive solution of equationEfλ,gµ

. Finally, by (2.2) and the H¨older and Sobolev inequalities,

u
+
λ,µ
2−q
H1 < λ
p − q
p − 2kf+kLq∗S
−q
2
p
and so
u
+
λ,µ
_{H}1 → 0 as λ → 0.

4 The estimate of energy

First, we let w0(x) be a unique radially symmetric positive solution of equa-tion (E∞) such that J∞(w0) = α∞. Then, by the result in Gidas-Ni-Nirenberg [18], for any ε > 0, there exist positive numbers Aε, B0 and Cε such that

Aεexp (− (1 + ε) |x|) ≤ w0(x) ≤ B0exp (− |x|) (4.1)

and

|∇w0(x)| ≤ Cεexp (− (1 − ε) |x|) . (4.2) Let

wl(x) = w0(x + le) , for l ∈ R and e ∈ SN −1, (4.3)
where SN −1=n_{x ∈ R}N | |x| = 1o. Then we have the following results.

Proposition 4.1 For each λ > 0 and µ > 0 with λp−2_{(1 + µ kbk}
∞)
2−q
< Λ0,
we have
α−_{f}
λ,gµ < α
+
fλ,gµ+ α
∞
.

Proof. Let u+_{λ,µ} be a positive solution of equation Efλ,gµ

as in Theorem 3.3. Then

Jfλ,gµ(u
+
λ,µ+ twl)
=1
2
u
+
λ,µ+ twl
2
H1 −
1
q
Z
RN
fλ
u
+
λ,µ+ twl
q
dx − 1
p
Z
RN
gµ
u
+
λ,µ+ twl
p
dx
≤ Jfλ,gµ
u+_{λ,µ}+ J∞(twl) +
1
p
Z
RN
tpwp_{l}dx − 1
p
Z
RN
gµtpwpldx
−
Z
RN
(λf++ f−)
Z twl
0
(u+_{λ,µ}+ η)q−1−u+_{λ,µ}q−1
dη
dx
−1
p
Z
RN
(u+_{λ,µ}+ twl)p −
u+_{λ,µ}p− tp_{w}p
l − p
u+_{λ,µ}p−1twl
dx
≤ α+
fλ,gµ+ J
∞
(twl) +
1
p
Z
RN
(1 − g0) tpwpldx
−µ
p
Z
RN
btpwp_{l}dx +
Z
RN
|f−|
Z twl
0
ηq−1dη
dx
−1
p
Z
RN
(u+_{λ,µ}+ twl)p −
u+_{λ,µ}p− tp_{w}p
l − p
u+_{λ,µ}p−1twl
dx
= α+_{f}
λ,gµ+ J
∞
(twl) +
tp
p
Z
RN
(1 − g0) wlpdx −
µtp
p
Z
RN
bw_{l}pdx + t
q
q
Z
RN
|f−| w_{l}qdx
−1
p
Z
RN
(u+_{λ,µ}+ twl)p −
u+_{λ,µ}p− tp_{w}p
l − p
u+_{λ,µ}p−1twl
dx. (4.4)

By Brown-Zhang [11] and Willem [27], we know that

J∞(twl) ≤ α∞ for all l ∈ R. (4.5) Thus, by (4.4) and (4.5), we have

Jfλ,gµ(u
+
λ,µ+ twl)
≤ α+
fλ,gµ+ α
∞_{+}tp
p
Z
RN
(1 − g0) wpldx −
µtp
p
Z
RN
bw_{l}pdx +t
q
q
Z
RN
|f−| wlqdx
−1
p
Z
RN
(u+_{λ,µ}+ twy)p−
u+_{λ,µ}p − tp_{w}p
l − p
u+_{λ,µ}p−1twl
dx. (4.6)
Since
Jfλ,gµ(u
+
λ,µ+ twl) → Jfλ,gµ(u
+
λ,µ) = α
+
fλ,gµ < 0 as t → 0
and
Jfλ,gµ(u
+
λ,µ+ twl) → −∞ as t → ∞,
we can easily find 0 < t1 < t2 such that

Jfλ,gµ(u

+

λ,µ+ twl) < αf+λ,gµ+ α

∞

for all t ∈ [0, t1] ∪ [t2, ∞). (4.7) Thus, we only need to show that there exists l0 > 0 such that for l > l0,

sup t1≤t≤t2 Jfλ,gµ(u + λ,µ+ twl) < α+fλ,gµ+ α ∞ . (4.8)

We also remark that

(u + v)p− up_{− v}p_{− pu}p−1_{v ≥ 0 for all (u, v) ∈ [0, ∞) × [0, ∞).}
Thus,
Z
RN
(u+_{λ,µ}+ twl)p−
u+_{λ,µ}p− tp_{w}p
l − p
u+_{λ,µ}p−1twl
dx ≥ 0. (4.9)

From the condition (D2) and (4.1)

Z RN (1 − g0) tpwpldx ≤ c0B0p Z RN

exp (−ra|x|) exp (−p |x + le|) dx ≤ c0B

p 0

Z

|x|<lexp (− min {ra, p} (|x| + |x + le|)) dx +c0B

p 0

Z

|x|≥lexp (− min {ra, p} (|x| + |x + le|)) dx ≤ c0B

p 0l

NZ

|x|<1exp (− min {ra, p} l (|x| + |x + e|)) dx +c0Bp0exp (− min {ra, p} l)

Z

|x|≥lexp (− min {ra, p} (|x + le|)) dx ≤ c0B0pl

NZ

|x|<1exp (− min {ra, p} l) dx + C0B p

0exp (− min {ra, p} l) ≤ C0B0plNexp (− min {ra, p} l) for l ≥ 1 (4.10)

and
Z
RN
bw_{l}pdx =
Z
RN
b (x − le) w_{0}p(x) dx ≥ min
x∈BN_{(1)}w
p
0(x)
!
Z
BN_{(1)}b (x − le) dx
≥ min
x∈BN_{(1)}w
p
0(x)
!
d0
Z
BN_{(1)}exp (−rb|x| − rbl |e|) dx
= min
x∈BN_{(1)}w
p
0(x)
!
D0exp (−rbl) . (4.11)

From the condition (D1) and the same argument of inequality (4.10) , we also
have
Z
RN
|f−| wq_{l}dx ≤cBb
q
0
Z
RN
exp−rf−|x|
exp (−q |x + le|) dx
≤cBb
q
0lN exp
− minnrf−, q
o
l for l ≥ 1. (4.12)

Since rb < min n rf−, ra, q o ≤ minnrf−, ra, p o and t1 ≤ t ≤ t2, by (4.6)−(4.12) , we can find l1 ≥ max {l0, 1} such that

sup t≥0 Jfλ,gµ(u + λ,µ+ twl) < α+fλ,gµ+ α ∞ for all l > l1.

To complete the proof of Proposition 4.1, it remains to show that there exists a positive number t∗ such that u+λ,µ+ t∗wl ∈ N−fλ,gµ. Let

U1=
(
u ∈ H1_{R}N
1
kuk_{H}1
t− u
kuk_{H}1
!
> 1
)
∪ {0} ;
U2=
(
u ∈ H1_{R}N
1
kuk_{H}1
t− u
kuk_{H}1
!
< 1
)
.

Then N−_{f}_{λ}_{,g}_{µ} separates H1_{R}N into two connected components U1 and U2,
and H1
RN
\N−_{f}
λ,gµ = U1 ∪ U2. For each u ∈ N
+
fλ,gµ, we have

1 < tmax,µ(u) < t−(u) .
Since t−(u) = _{kuk}1

H1t
− u
kuk_{H1}
, then N+_{f}
λ,gµ ⊂ U1. In particular, u
+
λ,µ ∈ U1.
We claim that there exists t0 > 0 such that u+λ,µ + t0wl ∈ U2. First, we
find a constant c > 0 such that 0 < t−

u+_{λ,µ}+twl
ku+_{λ,µ}+twlk
H1
< c for each t ≥ 0.
Suppose the contrary. Then there exists a sequence {tn} such that tn→ ∞ and
t−
u+_{λ,µ}+tnwl
ku+_{λ,µ}+tnwlk
H1
→ ∞ as n → ∞. Let vn =
u+_{λ,µ}+tnwl
ku+_{λ,µ}+tnwlk
H1
. Since t−(vn) vn ∈
N−_{f}

λ,gµ, by the Lebesgue dominated convergence theorem,

Z
RN
gµvnpdx =
1
u
+
λ,µ+ tnwl
p
H1
Z
RN
gµ
u+_{λ,µ}+ tnwl
p
dx
=
1
u+_{λ,µ}
tn + wl
p
H1
Z
RN
gµ
u+_{λ,µ}
tn
+ wl
!p
dx
→
R
RNgµw
p
ldx
kwlkpH1
as n → ∞,
we have
Jfλ,gµ
t−(vn) vn
= 1
2
h
t−(vn)
i2
− [t
−_{(v}
n)]q
q
Z
RN
fλvnqdx −
[t−(vn)]p
p
Z
RN
gµvnpdx
→ −∞ as n → ∞,

this contradicts the fact that Jfλ,gµ is bounded below on Nfλ,gµ. Let
t0 =
p − 2
2pα∞
c
2_{−}
u
+
λ,µ
2
H1
!1_{2}
+ 1.
Then
u
+
λ,µ+ t0wl
2
H1=
u
+
λ,µ
2
H1 + t
2
0kwlk2_{H}1 + o (1)
>
u
+
λ,µ
2
H1 +
c2−
u
+
λ,µ
2
H1
+ o (1)
> c2+ o (1) >
t
−
u+_{λ,µ}+ t0wl
u
+
λ,µ+ t0wl
_{H}1
2
+ o (1) as l → ∞.

Thus, there exists l2 ≥ l1 such that for l > l2,
1
u
+
λ,µ+ t0wl
_{H}1
t−
u+_{λ,µ}+ t0wl
u
+
λ,µ+ t0wl
_{H}1
< 1

or u+_{λ,µ}+ t0wl ∈ U2. Define a path γl(s) = u+λ,µ+ st0wl for s ∈ [0, 1] . Then
γl(0) = u+λ,µ∈ U1, γl(1) = u+λ,µ+ t0wl ∈ U2.
Since _{kuk}1
H1
t−_{kuk}u
H1

is a continuous function for non-zero u and γl([0, 1]) is connected, there exists sl ∈ (0, 1) such that u+λ,µ + slt0wl ∈ N−fλ,gµ. This

completes the proof.

Then we have the following result.

Theorem 4.2 For each λ > 0 and µ > 0 with λp−2(1 + µ kbk_{∞})2−q <q_{2}p−2Λ0,
equation Efλ,gµ

has positive solution u−_{λ,µ} ∈ N−_{f}

λ,gµ such that Jfλ,gµ

u−_{λ,µ}=
α−_{f}

λ,gµ.

Proof. Analogous to the proof of Wu [30, Proposition 9], one can show that by the Ekeland variational principle (see [15]), there exist minimizing sequences {un} ⊂ N−fλ,gµ such that

Jfλ,gµ(un) = α − fλ,gµ+ o (1) and J 0 fλ,gµ(un) = o (1) in H −1 RN .

Since α−_{f}_{λ}_{,g}_{µ} < α+_{f}_{λ}_{,g}_{µ}+ α∞, by Theorem 3.1 (ii) and Proposition 3.2 there exist
a subsequence {un} and u−λ,µ∈ N

−

fλ,gµ a non-zero solution of equation

Efλ,gµ such that un→ u−λ,µ strongly in H 1 (RN).

Since Jfλ,gµ
u−_{λ,µ}= Jfλ,gµ
u
−
λ,µ
and
u
−
λ,µ
∈ N
−
fλ,gµ, by Lemma 2.3, we may

assume that u−_{λ,µ} is a positive solutions of equation Efλ,gµ

.

5 Concentration Behavior

We need the following lemmas. Lemma 5.1 We have

inf

u∈N_{f0,g0}Jf0,g0(u) = infu∈N∞J

∞

(u) = α∞.

Furthermore, equation (Ef0,g0) does not admit any solution u0 such that Jf0,g0(u0) =

infu∈N_{f0,g0} Jf0,g0(u) .

Proof. Let wlbe as in (4.3) . Then, by Lemma 2.6, there is a unique t−(wl) >

_{2−q}

p−q

1/(p−2)

such that t−(wl) wl∈ Nf0,g0 for all l > 0, that is

t − (wl) wl 2 H1 = Z RN f− t − (wl) wl q dx + Z RN g0 t − (wl) wl p dx. Since kwlk 2 H1 = Z RN |wl| p dx = 2p p − 2α ∞ for all l ≥ 0, Z RN f−|wl| q dx → 0 and Z RN (1 − g0) |wl| p dx → 0 as l → ∞, we have t−(wl) → 1 as l → ∞. Thus, lim l→∞Jf0,g0 t−(wl) wl = lim l→∞J ∞ t−(wl) wl = α∞. Then inf

u∈N_{f0,g0}Jf0,g0(u) ≤ infu∈N∞J

∞

(u) = α∞.

Let u ∈ Nf0,g0. Then, by Lemma 2.6 (i) , Jf0,g0(u) = supt≥0Jf0,g0(tu) .

More-over, there is a unique t∞> 0 such that t∞u ∈ N∞. Thus, Jf0,g0(u) ≥ Jf0,g0(t

∞

u) ≥ J∞(t∞u) ≥ α∞
and so infu∈N_{f0,g0}Jf0,g0(u) ≥ α

∞_{. Therefore,}
inf

u∈N_{f0,g0}Jf0,g0(u) = infu∈N∞J

∞

(u) = α∞.

Next, we will show that equation (Ef0,g0) does not admit any solution u0 such

that Jf0,g0(u0) = infu∈N_{f0,g0}Jf0,g0(u) . Suppose the contrary. Then we can

Then, by Lemma 2.6 (i) , Jf0,g0(u0) = supt≥0Jf0,g0(tu0) . Moreover, there is a

unique tu0 > 0 such that tu0u0 ∈ N

∞_{. Thus,}
α∞= inf
u∈N_{f0,g0}Jf0,g0(u) = Jf0,g0(u0) ≥ Jf0,g0(tu0u0)
≥ J∞(tu0u0) −
tq_{u}_{0}
q
Z
RN
f−|u0|qdx ≥ α∞−
tq_{u}_{0}
q
Z
RN
f−|u0|qdx.
This implies R
RNf−|u0|
q
dx = 0 and so u0 ≡ 0 in
n
x ∈ RN _{| f}
−(x) 6= 0
o
form
the condition (D1) . Therefore,

α∞= inf u∈N∞J

∞

(u) = J∞(tu0u0) .

By the Lagrange multiplier and the maximum principle, we can assume that tu0u0 is a positive solution of (E ∞ ) . This contradicts u0 ≡ 0 in n x ∈ RN | f−(x) 6= 0 o

and completes the proof.

Lemma 5.2 Suppose that {un} is a minimizing sequence in Nf0,g0 for Jf0,g0.

Then (i) R RN f−|un| q dx = o (1) ; (ii) R RN (1 − g0) |un| p dx = o (1) . Furthermore, {un} is a (PS)α∞–sequence in H1 RN for J∞.

Proof. For each n, there is a unique tn > 0 such that tnun∈ N∞, that is
t2_{n}kunk2_{H}1 = tp_{n}

Z

RN

|un|pdx.

Then, by Lemma 2.6 (i) ,

Jf0,g0(un) ≥ Jf0,g0(tnun) = J∞(tnun) + tp n p Z RN (1 − g0) |un|pdx − tq n q Z RN f−|un|qdx ≥ α∞+ t p n p Z RN (1 − g0) |un|pdx − tq n q Z RN f−|un|qdx. Since Jf0,g0(un) = α

∞_{+ o (1) from Lemma 5.1, we have}
tq
n
q
Z
RN
f−|un|qdx = o (1)
and
tp
n
p
Z
RN
(1 − g0) |un|pdx = o (1) .

We will show that there exists c0 > 0 such that tn > c0 for all n. Suppose the contrary. Then we may assume tn → 0 as n → ∞. Since Jf0,g0(un) =

α∞+ o (1) , by Lemma 2.2, kunk is uniformly bounded and so ktnunkH1 → 0

or J∞(tnun) → 0 and this contradicts J∞(tnun) ≥ α∞ > 0. Thus,

Z RN f−|un| q dx = o (1) and Z RN (1 − g0) |un|pdx = o (1) , which this implies

kunk2_{H}1 =
Z
RN
|un|pdx + o (1)
and
J∞(un) = α∞+ o (1) .

Moreover, by Wang–Wu [26, Lemma 7], we have {un} is a (PS)α∞–sequence

in H1 RN

for J∞.

The following lemma is a key lemma in proving our main result.

Lemma 5.3 There exists d0 > 0 such that if u ∈ Nf0,g0 and Jf0,g0(u) ≤

α∞+ d0, then Z RN x |x| |∇u|2 + u2dx 6= 0.

Proof. Suppose the contrary. Then there exists sequence {un} ⊂ Nf0,g0 such

that Jf0,g0(u) = α
∞_{+ o (1) and}
Z
RN
x
|x|
|∇un|2+ u2n
dx = 0.

Moreover, by Lemma 5.2, we have {un} is a (PS)α∞–sequence in H1

RN

for J∞. It follows from Lemma 2.2 that there exist a subsequence {un} and u0 ∈ H1

RN

such that un * u0 weakly in H1

RN

. By the concentration– compactness principle (see Lions [20] or Struwe [24, Theorem 3.1]), there exist a sequence {xn} ⊂ RN, and a positive solution w0 ∈ H1

RN
of equation
(E∞) such that
kun(x) − w0(x − xn)k_{H}1 → 0 as n → ∞. (5.1)

Now we will show that |xn| → ∞ as n → ∞. Suppose the contrary. Then we may assume that {xn} is bounded and xn → x0 for some x0 ∈ RN. Thus, by (5.1)

Z RN f−|un|qdx = Z RN f−(x) |w0(x − xn)|qdx + o (1) = Z RN f−(x + xn) |w0(x)|qdx + o (1) = Z RN f−(x + x0) |w0(x)|qdx + o (1) ,

this contradicts the result of Lemma 5.2: R

RNf−|un| q

dx = o (1) . Hence we may assume xn

|xn| → e as n → ∞, where e ∈ S

N −1_{. Then, by the Lebesgue}
dominated convergence theorem, we have

0 =
Z
RN
x
|x|
|∇un|
2
+ u2_{n}dx =
Z
RN
x + xn
|x + xn|
|∇w0|
2
+ w_{0}2dx + o (1)
= 2p
p − 2α
∞
e + o (1) ,

which is a contradiction. This completes the proof.
By (2.3) , (2.6) and Lemma 2.6, for each u ∈ N−_{f}

λ,gµthere is a unique t

−
0 (u) >
0 such that t−_{0} (u) u ∈ Nf0,g0 and

t−_{0} (u) > tmax,0(u) =

(2 − q) kuk2_{H}1
(p − q)R
RNg0|u|
p
dx
!p−21
> 0.
Let
θµ=
(p − q) (1 + µ kb/ak_{∞})
2 − q
1 + kf−k_{L}q∗
(p − q) (1 + µ kb/ak_{∞})
(2 − q) S
p−q
2−q
p
2−q
p−2
p
p−2
.

Then we have the following results.

Lemma 5.4 For each λ > 0 and µ > 0 with λp−2(1 + µ kbk_{∞})2−q <q_{2}p−2Λ0
we have the following.

(i) ht−_{0} (u)ip < θµ for all u ∈ N−fλ,gµ with Jfλ,gµ(u) < α

+
fλ,gµ+ α
∞_{.}
(ii) R
RNg0|u|
p
dx ≥ _{θ} pq
µ(p−q)α
∞ _{for all u ∈ N}−
fλ,gµ with Jfλ,gµ(u) < α
+
fλ,gµ+ α
∞_{.}

Proof. (i) For u ∈ N−_{f}

λ,gµ with Jfλ,gµ(u) < α
+
fλ,gµ+ α
∞_{, we have}
kuk2_{H}1 −
Z
RN
fλ|u|qdx −
Z
RN
gµ|u|pdx = 0
and
(2 − q) kuk2_{H}1 < (p − q)
Z
RN
gµ|u|
p
dx.

We distinguish two cases.

Case (A): t−_{0} (u) < 1. Since θµ > 1 for all µ > 0, we have

h

t−_{0} (u)ip < 1 < θµ.

Case (B): t−_{0} (u) ≥ 1. Since

h
t−_{0} (u)ip
Z
RN
g0|u|pdx =
h
t−_{0} (u)i2kuk2_{H}1 −
h
t−_{0} (u)iq
Z
RN
f−|u|qdx
≤ht−_{0} (u)i2
kuk2_{H}1 +
Z
RN
|f−| |u|qdx
,
we have
h
t−_{0} (u)ip−2 ≤ kuk
2
H1 +
R
RN|f−| |u|
q
dx
R
RNg0|u|
p
dx . (5.2)

Moreover, by (2.3) and the Sobolev inequality,

kuk2_{H}1<
p − q
2 − q
Z
RN
gµ|u|pdx ≤
p − q
2 − q (1 + µ kb/ak∞)
Z
RN
g0|u|pdx (5.3)
≤(p − q) (1 + µ kb/ak∞)
(2 − q) S
p
2
p
kukp_{H}1
and so
kuk_{H}1 ≥
(2 − q) S
p
2
p
(p − q) (1 + µ kb/ak_{∞})
1
p−2
. (5.4)

Thus, by (5.2) − (5.4) and the Sobolev inequality,

h
t−_{0} (u)ip−2
≤ (1 + µ kb/ak_{∞}) p − q
2 − q
!
1 +
R
RN f−|u|
q
dx
kuk2_{H}1
!
≤ (1 + µ kb/ak_{∞}) p − q
2 − q
!
1 +
kf−k_{L}q∗
S
q
2
p kuk2−q_{H}1
≤(p − q) (1 + µ kb/ak∞)
2 − q
1 + kf−k_{L}q∗
(p − q) (1 + µ kb/ak_{∞})
(2 − q) S
p−q
2−q
p
2−q
p−2
or [t−(u)]p ≤ θµ.

α∞≤ Jf0,g0
t−_{0} (u) u
= 1
2 −
1
q
!
h
t−_{0} (u)i2kuk2_{H}1 +
1
q −
1
p
!
h
t−_{0} (u)ip
Z
RN
g0|u|pdx
< 1
q −
1
p
!
h
t−_{0} (u)ip
Z
RN
g0|u|pdx,

and this implies

Z
RN
g0|u|pdx ≥
1
h
t−_{0} (u)ip
pq
p − q
!
α∞.

By part (i) , we can conclude that

Z
RN
g0|u|pdx ≥
pq
θµ(p − q)
α∞.
for all u ∈ N−_{f}
λ,gµ with Jfλ,gµ(u) < α
+
fλ,gµ+ α

∞_{. This completes the proof.}
By the proof of Proposition 4.1, there exist positive numbers t∗ and l2 such
that u+_{λ,µ}+ t∗wl ∈ N−fλ,gµ and
Jfλ,gµ(u
+
λ,µ+ t∗wl) < α+fλ,gµ+ α
∞ _{for all l > l}
2.
Furthermore, we have the following result.

Lemma 5.5 There exist positive numbers λ0 and µ0 with

λp−2_{0} (1 + µ0kbk∞)
2−q
<
_{q}
2
p−2
Λ0

such that for every λ ∈ (0, λ0) and µ ∈ (0, µ0) , we have

Z
RN
x
|x|
|∇u|2+ u2dx 6= 0
for all u ∈ N−_{f}
λ,gµ with Jfλ,gµ(u) < α
+
fλ,gµ+ α
∞_{.}
Proof. For u ∈ N−_{f}
λ,gµ with Jfλ,gµ(u) < α
+
fλ,gµ+ α

∞_{, by Lemma 2.6 (i) there}
exists t−_{0} (u) > 0 such that t−_{0} (u) u ∈ Nf0,g0. Moreover,

Jfλ,gµ(u) = sup
t≥0
Jfλ,gµ(tu) ≥ Jfλ,gµ
t−_{0} (u) u
= Jf0,g0
t−_{0} (u) u− λ
h
t−_{0} (u)iq
q
Z
RN
f+|u|qdx
−µ
h
t−_{0} (u)ip
p
Z
RN
b |u|pdx.

Thus, by Lemma 5.4 and the H¨older and Sobolev inequalities,
Jf0,g0
t−_{0} (u) u
≤ Jfλ,gµ(u) +
λht−_{0} (u)iq
q
Z
RN
f+|u|
q
dx +µ
h
t−_{0} (u)ip
p
Z
RN
b |u|pdx
< α+_{f}_{λ}_{,g}_{µ}+ α∞+λθ
q/p
µ
q kf+kLq∗ S
−q
2
p kukq_{H}1 +
µθµkbk∞
p S
−p
2
p kukp_{H}1.
Since Jfλ,gµ(u) < α
+
fλ,gµ + α

∞ _{< α}∞_{, by (2.1) in Lemma 2.2, for each λ > 0}
and µ > 0 with λp−2_{(1 + µ kbk}

∞) 2−q

< q_{2}p−2Λ0, there exists a positive
number c independent of λ, µ such that kuke H1 ≤ c for all u ∈ Ne

−
fλ,gµ with
Jfλ,gµ(u) < α
+
fλ,gµ+ α
∞_{. Therefore,}
Jf0,g0
t−_{0} (u) u< α+_{f}
λ,gµ+ α
∞
+λθ
q/p
µ
q kf+kLq∗ S
−q_{2}
p ce
q_{+}µθµkbk∞
p S
−p_{2}
p ec
p_{.}

Let d0 > 0 be as in Lemma 5.3. Then there exist positive numbers λ0 and µ0 with λp−20 (1 + µ0kbk∞)

2−q

< q_{2}p−2Λ0 such that for λ ∈ (0, λ0) and µ ∈
(0, µ0) ,

Jf0,g0

t−(u) u< α∞+ d0. (5.5)
Since t−_{0} (u) u ∈ Nf0,g0 and t

−

0 (u) > 0, by Lemma 5.3 and (5.5)

Z
RN
x
|x|
_{}
∇
t−_{0} (u) u
2
+t−_{0} (u) u2
dx 6= 0,

and this implies

Z
RN
x
|x|
|∇u|2+ u2dx 6= 0
for all u ∈ N−_{f}
λ,gµ with Jfλ,gµ(u) < α
+
fλ,gµ+ α
∞_{.}
6 Proof of Theorem 1.1

In the following, we use an idea of Adachi-Tanaka [4]. For c ∈ R+, we denote

h
Jfλ,gµ ≤ c
i
=nu ∈ N−_{f}
λ,gµ | u ≥ 0, Jfλ,gµ(u) ≤ c
o
.
We then try to show for a sufficiently small σ > 0

cathJfλ,gµ ≤ α

+

fλ,gµ+ α

∞_{− σ}i

≥ 2. (6.1)

To prove (6.1) , we need some preliminaries. Recall the definition of Lusternik-Schnirelman category.

Definition 6.1 (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

ξ : [0, 1] × Y → X such that for some x0 ∈ X

ξ (0, x) = x for all x ∈ Y, and

ξ (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) = ∞.

We need the following two lemmas.

Lemma 6.2 Suppose that X is a Hilbert manifold and F ∈ C1_{(X, R) . }
As-sume that there are c0 ∈ R and k ∈ N,

(i) F (x) satisfies the Palais–Smale condition for energy level c ≤ c0; (ii) cat ({x ∈ X | F (x) ≤ c0}) ≥ k.

Then F (x) has at least k critical points in {x ∈ X; F (x) ≤ c0} . Proof. See Ambrosetti [1, Theorem 2.3].

Lemma 6.3 Let X be a topological space. Suppose that there are two contin-uous maps

Φ : SN −1 → X, Ψ : X → SN −1

such that Ψ ◦ Φ 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) = (Ψ ◦ Φ) (x) for each x ∈ SN −1, ζ (1, x) = x for each x ∈ SN −1.

Then

cat (X) ≥ 2.

For l > l2, we define a map Φfλ,gµ : S
N −1_{→ H}1
RN
by
Φfλ,gµ(e) (x) = u
+
λ,µ+ slt0wl for e ∈ SN −1,

where u+_{λ,µ}+ slt0wl is as in the proof of Proposition 4.1. Then we have the
following result.

Lemma 6.4 There exists a sequence {σl} ⊂ R+ with σl → 0 as l → ∞ such
that
Φfλ,gµ
SN −1
⊂hJfλ,gµ ≤ α
+
fλ,gµ+ α
∞_{− σ}
l
i
.

Proof. By Proposition 4.1, for each l > l2 we have u+λ,µ+ slt0wl ∈ N−fλ,gµ

and
sup
t≥0
Jfλ,gµ
u+_{λ,µ}+ twl
< α+_{f}
λ,gµ+ α
∞
uniformly in e ∈ SN −1.
Since Φfλ,gµ
SN −1
is compact, Jfλ,gµ
u+_{λ,µ}+ slt0wl
≤ α+
fλ,gµ + α
∞_{− σ}
l, so
that the conclusion holds.

From Lemma 5.5, we define
Ψfλ,gµ :
h
Jfλ,gµ < α
+
fλ,gµ+ α
∞i
→ SN −1
by
Ψfλ,gµ(u) =
R
RN
x
|x|
|∇u|2+ u2_{dx}
R
RN
x
|x|
|∇u|2+ u2_{dx}
.

Then we have the following results.

Lemma 6.5 Let λ0, µ0 be as in Lemma 5.5. Then for each λ ∈ (0, λ0) and µ ∈ (0, µ0) , there exists l∗ ≥ l2 such that for l > l∗, the map

Ψfλ,gµ◦ Φfλ,gµ : S

N −1

→ SN −1 is homotopic to the identity.

Proof. Let Σ = nu ∈ H1
RN
\ {0} | R
RN
x
|x|
|∇u|2+ u2_{dx 6= 0}o_{. We}
define
Ψfλ,gµ : Σ → S
N −1
by
Ψfλ,gµ(u) =
R
RN
x
|x|
|∇u|2+ u2_{dx}
R
RN
x
|x|
|∇u|2+ u2_{dx}

as an extension of Ψfλ,gµ. Since wl ∈ Σ for all e ∈ S

N −1 _{and for l sufficiently}
large, we let γ : [s1, s2] → SN −1 be a regular geodesic between Ψfλ,gµ(wl) and

Ψfλ,gµ Φfλ,gµ(e) such that γ (s1) = Ψfλ,gµ(wl) , γ (s2) = Ψfλ,gµ Φfλ,gµ(e) .

By an argument similar to that in Lemma 5.3, there exists a positive number l∗ ≥ l2 such that for l > l∗,

w0 x + l 2 (1 − θ)e

!

∈ Σ 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) =
γ (2θ (s1 − s2) + s2) for θ ∈ [0, 1/2) ;
Ψfλ,gµ
w0
x + _{2(1−θ)}l e for θ ∈ [1/2, 1) ;
e for θ = 1.
Then ζl(0, e) = Ψfλ,gµ
Φfλ,gµ(e)

= Ψfλ,gµ(Φfλ(e)) and ζl(1, e) = e. By the

standard regularity, we have u+_{λ,µ}∈ C_{R}N_{. First, we claim that lim}

θ→1−ζl(θ, e) =
e and lim
θ→1
2
−ζl(θ, e) = Ψfλ,gµ(w0(x + le)) .
(a) lim
θ→1−ζl(θ, e) = e : since
Z
RN
x
|x|
∇
"
w0 x +
l
2 (1 − θ)e
!#
2
+
"
w0 x +
l
2 (1 − θ)e
!#2
dx
=
Z
RN
x − _{2(1−θ)}l e
x −
l
2(1−θ)e
|∇ [w0(x)]|2 + [w0(x)]2
dx
= 2p
p − 2
!
α∞e + o(1) as θ → 1−,
then lim
θ→1−ζl(θ, e) = e.
(b) lim
θ→1_{2}−
ζl(θ, e) = Ψfλ,gµ(w0(x + le)) : since Ψfλ,gµ ∈ C
Σ, SN −1, we obtain
lim
θ→1_{2}−
ζl(θ, l) = Ψfλ,gµ(w0(x + le)) .
Thus, ζl(θ, e) ∈ C
[0, 1] × SN −1, SN −1 and
ζl(0, e) = Ψfλ,gµ
Φfλ,gµ(e)
for all e ∈ SN −1,
ζl(1, e) = e for all e ∈ SN −1,

provided l > l∗. This completes the proof.

Lemma 6.6 For each λ ∈ (0, λ0) , µ ∈ (0, µ0) and l > l∗, functional Jfλ,gµ has

at least two critical points in hJfλ,gµ < α

+

fλ,gµ + α

Proof. Applying Lemmas 6.3 and 6.5, we have for λ ∈ (0, λ0) , µ ∈ (0, µ0)
and l > l∗,
cathJfλ,gµ ≤ α
+
fλ,gµ+ α
∞_{− σ}
l
i
≥ 2.

By Proposition 3.2, Lemma 6.2, Jfλ(u) has at least two critical points in

h

Jfλ,gµ < α

+

fλ,gµ+ α

∞i_{.}

We can now complete the proof of Theorem 1.1: (i) by Theorems 3.3, 4.2. (ii) for λ ∈ (0, λ0) and µ ∈ (0, µ0) , from Theorem 3.3 and Lemma 6.6, equation (Efλ,gµ) has three positive solutions u

+
λ,µ, u
−
1, u
−
2 such that u+λ,µ ∈ N
+
fλ,gµand
u−_{i} ∈ N−_{f}

λ,gµ for i = 1, 2. This completes the proof of Theorem 1.1.

7 Proof of Theorem 1.2
For c > 0, we define
J0,cg0(u) =
1
2
Z
RN
|∇u|2+ u2dx −1
p
Z
RN
cg0|u|pdx,
N0,cg0=
n
u ∈ H1_{R}N\ {0} |DJ_{0,cg}0 _{0}(u) , uE= 0o.

Recall that for each u ∈ H1 RN

\ {0} there exist a unique t−_{(u) > 0 and}
t0(u) > 0 such that t−(u) u ∈ N−fλ,gµ and t0(u) u ∈ N0,g0. Let

B =

n

u ∈ H1_{R}N\ {0} | u ≥ 0 and kuk_{H}1 = 1

o

. Then we have the following result.

Lemma 7.1 For each u ∈ B we have the following. (i) There is a unique tc

0 = tc0(u) > 0 such that tc0u ∈ N0,cg0 and

sup
t≥0
J0,cg0(tu) = J0,cg0(t
c
0u) =
p − 2
2p
Z
RN
cg0|u|
p
dx
_{p−2}−2
.
(ii) For ρ ∈ (0, 1) ,
Jfλ,gµ
t−(u) u≥ (1 − ρ)
p
p−2
(1 + µ kb/ak_{∞})p−22
J0,g0(t0(u) u)−
2 − q
2q (ρSp)
q
q−2_{(λ kf}
+kLq∗)
2
2−q
and
Jfλ,gµ
t−(u) u≤ (1 + ρ)p−2p _{J}
0,g0(t0(u) u)+
2 − q
2q (ρSp)
q
q−2_{(λ kf}
+k_{L}q∗ + kf−k_{L}q∗)
2
2−q _{.}

Proof. _{(i) For each u ∈ B, let}
K(t) = J0,cg0(tu) =
1
2t
2_{−} 1
pt
pZ
RN
cg0|u|pdx,
then K (t) → −∞ as t → ∞, K0(t) = t − tp−1R
RNcg0|u|
p
dx and K00(t) =
1 − (p − 1) tp−2R
RNcg0|u|
p
dx. Let
tc_{0} = tc_{0}(u) =
Z
RN
cg0|u|pdx
_{2−p}1
> 0.
Then K0(tc_{0}) = 0, tc_{0}u ∈ N0,cg0 and K
00_{(t}c

0) = 2 − p < 0. Thus, there is a unique tc

0 = tc0(u) > 0 such that tc0u ∈ N0,cg0 and

sup
t≥0
J0,cg0(tu) = J0,cg0(t
c
0u) =
p − 2
2p
Z
RN
cg0|u|pdx
_{p−2}−2
.

(ii) Let c = (1 + µ kb/ak_{∞}_{) / (1 − ρ) . Then for each u ∈ B and ρ ∈ (0, 1) , we}
get
Z
RN
fλ|tc0u|
q
dx ≤ λS
−q
2
p kf+kLq∗ ktc_{0}uk
q
H1
≤2 − q
2
(ρSp)
−q
2 _{λ kf}
+k_{L}q∗
_{2−q}2
+ q
2
ρq2 ktc
0uk
q
H1
2_{q}
=2 − q
2 (ρSp)
q
q−2(λ kf
+k_{L}q∗)
2
2−q + qρ
2 kt
c
0uk
2
H1. (7.1)

Then, by part (i) and (7.1) ,

sup
t≥0
Jfλ,gµ(tu)
≥ Jfλ,gµ(t
c
0u)
≥1 − ρ
2 kt
c
0uk
2
H1 −
2 − q
2q (ρSp)
q
q−2_{(λ kf}
+kLq∗)
2
2−q
−(1 + µ kb/ak∞)
p
Z
RN
g0|tc0u|
p
dx
= (1 − ρ) J0,cg0(t
c
0u) −
2 − q
2q (ρSp)
q
q−2_{(λ kf}
+k_{L}q∗)
2
2−q
= (p − 2) (1 − ρ)
p
p−2
2p ((1 + µ kb/ak_{∞})R
RNg0|u|
p
dx)p−22
− 2 − q
2q (ρSp)
q
q−2_{(λ kf}
+kLq∗)
2
2−q
= (1 − ρ)
p
p−2
(1 + µ kb/ak_{∞})p−22
J0,g0(t0(u) u) −
2 − q
2q (ρSp)
q
q−2_{(λ kf}
+kLq∗)
2
2−q _{.}

By Lemma 2.6 and Theorem 3.1,
sup
t≥0
Jfλ,gµ(tu) = Jfλ,gµ
t−(u) u.
Thus,
Jfλ,gµ
t−(u) u≥ (1 − ρ)
p
p−2
(1 + µ kb/ak_{∞})p−22
J0,g0(t0(u) u)−
2 − q
2q (ρSp)
q
q−2(λ kf
+k_{L}q∗)
2
2−q .

Moreover, by the H¨older, Sobolev and Young inequalities,

Z
RN
fλ|tu|
q
dx ≤ (λ kf+kLq∗ + kf−k_{L}q∗) S
−q_{2}
p ktukq_{H}1
≤2 − q
2 (ρSp)
q
q−2(λ kf
+k_{L}q∗ + kf−k_{L}q∗)
2
2−q + qρ
2 ktuk
2
H1.
Therefore,
Jfλ,gµ(tu) ≤
(1 + ρ)
2 t
2
+2 − q
2q (ρSp)
q
q−2_{(λ kf}
+k_{L}q∗ + kf−k_{L}q∗)
2
2−q
−1
p
Z
RN
g0|tu|
p
dx
≤ (1 + ρ)p−2p _{J}
0,g0(t0(u) u)
+2 − q
2q (ρSp)
q
q−2_{(λ kf}
+k_{L}q∗ + kf−k_{L}q∗)
2
2−q
and so
Jfλ,gµ
t−(u) u≤ (1 + ρ)p−2p _{J}
0,g0(t0(u) u)
+2 − q
2q (ρSp)
q
q−2_{(λ kf}
+kLq∗ + kf−k_{L}q∗)
2
2−q _{.}

This completes the proof.

Since α−_{f}_{λ}_{,g}_{µ} > 0 for all λ ∈ (0, λ0) and µ ∈ (0, µ0) , we define

Ifλ,gµ(u) = sup
t≥0
Jfλ,gµ(tu) = Jfλ,gµ
t−(u) u> 0,
where t−(u) u ∈ N−_{f}

λ,gµ. We observe that if λ, µ and kf−kLq∗ are sufficiently

small, Bahri-Li’s minimax argument [6] also works for Jfλ,gµ. Let

Γfλ,gµ =
n
γ ∈ CBN_{(0, l), B} _{| γ|}
∂BN_{(0,l)} = w_{l}/ kw_{l}k_{H}1
o
for large l.

Then we define
βfλ,gµ = inf
γ∈Γ_{fλ,gµ}_{x∈R}supN
Ifλ,gµ(γ (x)) and β0,g0 = inf
γ∈Γ_{0,g0}_{x∈R}supN
I0,g0(γ (x)) .

By Lemma 7.1 (ii) , for 0 < ρ < 1, we have

βfλ,gµ ≥
(1 − ρ)p−2p
(1 + µ kb/ak_{∞})p−22
β0,g0 −
2 − q
2q (ρSp)
q
q−2_{(λ kf}
+kLq∗)
2
2−q _{(7.2)}
and
βfλ,gµ ≤ (1 + ρ)
p
p−2β
0,g0 +
2 − q
2q (ρSp)
q
q−2(λ kf
+k_{L}q∗ + kf−k_{L}q∗)
2
2−q . (7.3)

We need the following results. Lemma 7.2 α∞ < β0,g0 < 2α

∞_{.}

Proof. Bahri-Li [6] prove that equation (E0,g0) admits at least one positive

solution u0 and J0,g0(u0) = β0,g0 < 2α

∞_{. Moreover, by the condition (D4) ,}
equation (E0,g0) does not have a positive ground state solution. Hence, α

∞_{<}
β0,g0 < 2α

∞_{.}

Theorem 7.3 Let λ0, µ0 be as in Lemma 5.5. Then there exist positive
num-bers λe_{0} ≤ λ_{0},µ_{e}_{0} ≤ µ_{0} and ν_{0} such that for λ ∈

0,λe_{0}
, µ ∈ (0,µe0) and
kf−k_{L}q∗ < ν0, we have
α+_{f}
λ,gµ+ α
∞
< βfλ,gµ < α
−
fλ,gµ+ α
∞
.
Furthermore, equation Efλ,gµ

has a positive solution vfλ,gµ such that

Jfλ,gµ

vfλ,gµ

= βfλ,gµ.

Proof. By Lemma 7.1 (ii), we also have that for 0 < ρ < 1

α−_{f}
λ,gµ ≥
(1 − ρ)p−2p
(1 + µ kb/ak_{∞})p−22
α∞− 2 − q
2q (ρSp)
q
q−2_{(λ kf}
+k_{L}q∗)
2
2−q
and
α_{f}−
λ,gµ ≤ (1 + ρ)
p
p−2_{α}∞_{+}2 − q
2q (ρSp)
q
q−2_{(λ kf}
+kLq∗ + kf−k_{L}q∗)
2
2−q _{.}

For any ε > 0 there exist positive numbers λe_{1} ≤ λ_{0},
e

µ1 ≤ µ0 and ν1 such that
for λ ∈0,λe_{1}
, µ ∈ (0,µe1) and kf−k_{L}q∗ < ν1, we have
α∞− ε < α_{f}−
λ,gµ < α
∞
+ ε.

Thus,

2α∞− ε < α−_{f}

λ,gµ+ α

∞

< 2α∞+ ε.

Applying (7.2) and (7.3) for any δ > 0 there exist positive numbers λe_{2} ≤

λ0,µe2 ≤ µ0 and ν2 such that for λ ∈
0,λe_{2}
, µ ∈ (0,µe2) and kf−k_{L}q∗ < ν2, we
have
β0,g0 − δ < βfλ,gµ < β0,g0 + δ.

Fix a small 0 < ε < (2α∞− β0,g0) /2, since α

∞ _{< β}

0,g0 < 2α

∞_{, choosing}
a δ > 0 such that for λ < λe_{0} = min

n
e
λ1,λe_{2}
o
, µ < µe0 = min {µe1,µe2} and
kf−k_{L}q∗ < ν0 = min {ν1, ν2} , we get
α+_{f}
λ,gµ+ α
∞
< α∞ < βfλ,gµ < 2α
∞_{− ε < α}−
fλ,gµ+ α
∞
.
Therefore, by Proposition 3.2, we obtain thatEfλ,gµ

has a positive solution vfλ,gµ such that Jfλ,gµ

vfλ,gµ

= βfλ,gµ.

We can now complete the proof of Theorem 1.2: for λ ∈0,λe_{0}

, µ ∈ (0,µe0)

and kf−k_{L}q∗ < ν0, from Theorems 1.1, 7.3, equation (Efλ,gµ) has at least four

positive solutions.

Acknowledgements

The author is grateful for the referee’s valuable suggestions and helps.

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