### Existence and multiplicity of positive solutions for a class of

### nonlinear boundary value problems

Tsung-fang Wu1

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

Abstract

In this paper, we study a class of nonlinear boundary value problems in RN+. By

means of minimax method and the Lusternik-Schnirelman category, the criteria of the existence, multiplicity and nonexistence of positive solutions are established. Keywords: Ljusternik-Schnirelmann category; Positive solutions; Nonlinear boundary value problems.

1. Introduction

Let Ω be a domain in RN, N ≥ 2, with smooth boundary ∂Ω and consider the following nonlinear boundary value problem:

−∆u = g (x, u) in Ω, (1.1)

∂u

∂n + f (x, u) = 0 in ∂Ω, (1.2)

where _{∂n}∂ _{is the outer unit normal derivative, g : Ω × R → R is a Carath´edory}
function and f : ∂Ω × R → R is a continuous function.

Equations of the type (1.1) arise in many and diverse contexts like differential geometry (e.g., in the scalar curvature problem and the Yamabe problem) [27], nonlinear elasticity [19], non-Newtonian fluid mechanics [20], glaciology [34], math-ematical biology [5], and elsewhere. As a result, questions concerning the solvability of problem (1.1) have received great attention, particularly after the famous work of Brezis and Nirenberg [13]. Among the vast number of results recorded in the litera-ture so far, the case which has been studied extensively concerns the class of positive or non-negative solutions under a variety of the nonlinear term g(x, u). However, an exhaustive review of the existing bibliography is beyond our present scope and the interested reader should consult the survey in [2], as well as the references cited therein.

Email address: tfwu@nuk.edu.tw (Tsung-fang Wu )

In recent years, problem (1.1), (1.2) have become rather an active area of research;
see for example [6, 7, 14, 16, 17, 18, 22, 25, 26, 35, 37, 40] and references therein. The
existence of positive solutions of the problem in bounded domains is strongly
depen-dent on a priori estimates of the solutions [29], so fewer results are known for N ≥ 2.
On the other hand, many papers deal with the existence of positive solutions of the
problems in unbounded domains. For example, in [16, 18], the authors considered
the existence and nonexistence of positive solutions of the problem (1.1) and (1.2) in
upper half-space of RN _{with g (x, u) = −u + |u|}p−2_{u and f (x, u) = − |u|}q−2_{u. They}

gave the exact form of the solution when p = _{N −2}2N , q = 2(N −1)_{N −2} with N ≥ 3 in [18], and
proved the existence of positive solutions for p ≥ _{N −2}2N , q ≥ 2(N −1)_{N −2} and the
nonexis-tence of positive solutions for some cases of p and q in [16]. The nonexisnonexis-tence results
of [16] in some sense can be regarded as an extension of the results in [26] where
Hu considered the problem with g (x, u) = −u + a |u|p−2u and f (x, u) = − |u|q−2u
in the exterior of a ball in the upper half-space of RN_{. In [40], the author }

con-sider the multiplicity of positive solutions of the problem (1.1) and (1.2) in upper half-space of RN with g (x, u) = −u + |u|p−2u and f (x, u) = b (x) |u|q−2u, with 1 < q < 2 < p < 2N

N −2 and b is a sign-changing continuous function. In [17] the

authors proved that the number of sign-changing solutions strongly depends on the spatial dimension. For the existence and multiplicity of positive solutions by varia-tional methods, see [6, 7, 14, 22, 25, 35, 37].

In this paper, we consider the existence and multiplicity of positive solutions for the following nonlinear boundary value problem:

−∆u + u = gλ(x) |u|p−2u in RN+,
∂u
∂n + f (x) |u|
q−2
u = 0 _{in ∂R}N
+,
(Eλ)
where 1 < q < min {2∗, p} ,
2∗ = 2(N −1)_{N −2} if N ≥ 3, 2∗ = ∞ if N = 2
, 2 < p <
2∗ 2∗ = _{N −2}2N if N ≥ 3, 2∗ = ∞ if N = 2 , RN
+ = (x
0_{, x}
N) ∈ RN −1× R | xN > 0

is an upper half space in RN, the parameter λ ∈ R and gλ(x) = 1 + λa (x) . We

assume that the functions f and a satisfy the following conditions: (D1) f ∈ C ∂RN

+ \ {0} and there exists a positive number rf > 1 such that

0 ≤ f (x) ≤_{b}c exp (−rf|x|) for somebc > 0 and for all x ∈ ∂R

N +;

(D2) a ∈ C_{R}N
+

and there exists a positive number ra < {rf, q} such that

a (x) ≥ c0exp (−ra|x|) for some c0 > 0 and for all x ∈ RN+

and

The following theorem is our main result.

Theorem 1.1. Suppose that the functions f and a satisfy the conditions (D1) and (D2) . Then there exists a positive number λ∗ such that equation (Eλ) has at least

three positive solutions for λ ∈ (0, λ∗) , and at least one positive solution for λ ∈

{0} ∪ [λ∗, ∞).

In the following sections, we proceed to prove Theorem 1.1. We use the variational methods to find positive solutions of equation (Eλ) . Associated with the equation

(Eλ) , we consider the energy functional Jλ in H1 RN+

Jλ(u) = 1 2kuk 2 H1 + 1 q Z ∂RN + f |u|qdσ −1 p Z RN+ gλ|u| p dx,

where dσ is the measure on the boundary and kuk_{H}1 =

R

RN+

|∇u|2+ u2_{dx}1/2 _{is}

the standard norm in H1 _{R}N_{+} . It is well known that the solutions of equation (Eλ)

are the critical points of the energy functional Jλ in H1 RN+(see Rabinowitz [36]).

This paper is organized as follows. In section 2, we give some notations and preliminaries. In section 3, we establish the existence of a positive solution for λ = 0. In section 4, we establish the existence of a positive solution for λ > 0. In sections 5, 6, we prove Theorem 1.1.

2. Notations and Preliminaries

First, we define the Palais–Smale (simply (PS)–) sequences, (PS)–values, and (PS)–conditions in H1

RN+ for Jλ as follows.

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

for Jλ if Jλ(un) = β + o(1) and Jλ0(un) = o(1) strongly in H−1 RN+ as n → ∞.

(ii) Jλ satisfies the (PS)β–condition in H1 RN+ if every (PS)β–sequence in H1 RN+

for Jλ contains a convergent subsequence.

As the energy functional Jλ is not bounded below on H1 RN+ , it is useful to

consider the functional on the Nehari manifold Nλ =u ∈ H1 RN+ \ {0} | hJ

0

λ(u) , ui = 0 .

Thus, u ∈ Nλ if and only if

kuk2_{H}1 +
Z
∂RN
+
f |u|qdσ −
Z
RN+
gλ|u|
p
dx = 0.

Lemma 2.2. The energy functional Jλ is coercive and bounded below on Nλ. Proof. If u ∈ Nλ, then Jλ(u) = p − 2 2p kuk 2 H1 + p − q pq Z ∂RN + f |u|qdσ ≥ p − 2 2p kuk 2 H1. (2.1)

Thus, Jλ is coercive and bounded below on Nλ.

Define
ψλ(u) = kuk2_{H}1 +
Z
∂RN
+
f |u|qdσ −
Z
RN+
gλ|u|pdx.
Then for u ∈ Nλ,
hψ0_{λ}(u) , ui = 2 kuk2_{H}1 + q
Z
∂RN
+
f |u|qdσ − p
Z
RN+
gλ|u|
p
dx
= (2 − p) kuk2_{H}1 + (q − p)
Z
∂RN
+
f |u|qdσ < 0.
Furthermore, we have the following result.

Lemma 2.3. Suppose that u0 is a local minimizer for Jλ on Nλ. Then Jλ0 (u0) = 0

in H−1 _{R}N . Furthermore, if u0 is a non-trivial nonnegative function in Ω, then

u0 is a positive solution of equation (Eλ) .

Proof. By a similar argument to that in the proof of Brown and Zhang [15,
Theorem 2.3] (or see Binding, Dr´abek and Huang [8]), we have J_{λ}0 (u0) = 0 in

H−1(Ω) , this implies that u0 is a weak solution of equation (Eλ) . Now, if u0 is a

non-trivial nonnegative function in Ω, then by the maximum principle, u0 is positive

in Ω.

To get a better understanding of the Nehari manifold, we consider the function
mu : R+→ R defined by
mu(t) = t2−qkuk2_{H}1− t
p−q
Z
RN+
gλ|u|pdx for t > 0.

Clearly, tu ∈ Nλ if and only if mu(t) +

R

∂RN + f |u|

q

dσ = 0 and mu btλ(u) = 0, where

b
tλ(u) =
kuk2_{H}1
R
RN+ gλ|u|
p
dx
!_{p−2}1
> 0.
Moreover,
m0_{u}(t) = t1−q
"
(2 − q) kuk2_{H}1 − (p − q)t
p−2
Z
RN+
gλ|u|pdx
#
.

Thus, if q ≥ 2, then

m0_{u}(t) < 0 for all t > 0,

which implies that mu is strictly decreasing on (0, ∞) with limt→0+m_{u}(t) = ∞
and limt→∞mu(t) = −∞, and if q < 2, then mu has a unique critical point at

t =etλ(u) <btλ(u) , where

etλ(u) =
(2 − q) kuk2_{H}1
(p − q)R
RN+ gλ|u|
p
dx
!_{p−2}1
> 0
such that
mu etλ(u)
= kukq_{H}1
p − 2
p − q
2 − q
p − q
2−q_{p−2} _{kuk}p
H1
R
RN+ gλ|u|
p
dx
!2−q_{p−2}
> 0,

which implies that muis strictly increasing on (0,etλ(u)) and muis strictly decreasing

on (etλ(u) , ∞) with limt→∞mu(t) = −∞. Therefore, we can conclude that for each

u ∈ H1 _{R}N_{+} \ {0} there exist 0 < etλ(u) < btλ(u) such that mu(t) > 0 for all

t ∈ 0,btλ(u) and mu is strictly decreasing on [etλ(u) , ∞) with mu(t) < 0 for all

t ∈ btλ(u) , ∞ . Moreover, we have the following lemma.

Lemma 2.4. Suppose that λ ≥ 0. Then for each u ∈ H1

RN+ \ {0} we have the

following.

(i) There is a unique tλ(u) ≥btλ(u) such that tλ(u) u ∈ Nλ. Furthermore,

Jλ(tλ(u) u) = sup t≥0

Jλ(tu) = sup t≥btλ(u)

Jλ(tu) . (2.2)

(ii) tλ(u) is a continuous function for u ∈ H1 RN+ \ {0} .

(iii) tλ(u) = _{kuk}1
H1
tλ
u
kuk_{H1}
.
(iv) Nλ =
n
u ∈ H1
RN+ \ {0} |
1
kuk_{H1}tλ
u
kuk_{H1}
= 1o.
Proof. Fix u ∈ H1
RN+ \ {0} .
(i) Let
hu(t) = Jλ(tu) =
t2
2 kuk
2
H1+
tq
q
Z
∂RN
+
f |u|qdσ − t
p
p
Z
RN+
gλ|u|
p
dx.
Then
h0_{u}(t) = t kuk2_{H}1 + tq−1
Z
∂RN
+
f |u|qdσ − tp−1
Z
RN+
gλ|u|
p
dx
= tq−1 mu(t) +
Z
∂RN
+
f |u|qdσ
!
.

Thus, by R ∂RN + f |u| q dσ ≥ 0, the equation mu(t) + R ∂RN + f |u| q dσ = 0 has a unique solution tλ(u) ≥btλ(u) , which implies that h0u(tλ(u)) = 0 and tλ(u) u ∈ Nλ.

More-over, hu is strictly increasing on (0, tλ(u)) and strictly decreasing on (tλ(u) , ∞).

Therefore, (2.2) holds.

(ii) By the uniqueness of tλ(u) and the extremal property of tλ(u) , we have tλ(u)

is a continuous function for u ∈ H1 _{R}N_{+} \ {0} .
(iii) Let v = _{kuk}u

H1. Then, by part (i), there is a unique tλ(v) > 0 such that
tλ(v) v ∈ Nλ or tλ
u
kuk_{H1}
u

kuk_{H1} ∈ Nλ. Thus, by the uniqueness of tλ(v) , we can

conclude that tλ(u) = kuk1_{H1}tλ

u
kuk_{H1}

. (iv) For u ∈ Nλ. By parts (i), (iii), tλ

u
kuk_{H1}

u

kuk_{H1} ∈ Nλ. Since u ∈ Nλ, we have

tλ
u
kuk_{H1}
1

kuk_{H1} = 1, which implies that

Nλ ⊂
u ∈ H1 _{R}N_{+} | 1
kuk_{H}1
tλ
u
kuk_{H}1
= 1
.
Conversely, let u ∈ H1
RN+ such that
1
kuk_{H1}tλ
u
kuk_{H1}

= 1. Then, by part (iii) ,

tλ
u
kuk_{H}1
u
kuk_{H}1
∈ Nλ.
Thus,
Nλ =
u ∈ H1 _{R}N_{+} \ {0} | 1
kuk_{H}1
tλ
u
kuk_{H}1
= 1
.
This completes the proof.

Now we consider the following elliptic problems:
−∆u + u = |u|p−2
u in RN+,
∂u
∂n = 0 on ∂R
N
+.
(E∞)
and
−∆u + u = |u|p−2
u in RN_{,}
lim|x|→∞u = 0.
( eE∞)
Associated with the equations (E∞) and Ee∞

, we consider the energy functionals J∞ in H1 RN+ and Je∞ in H1 RN J∞(u) = 1 2 Z RN+ |∇u|2+ u2dx − 1 p Z RN+ |u|pdx and e J∞(u) = 1 2 Z RN |∇u|2+ u2dx −1 p Z RN |u|pdx,

respectively. Consider the minimizing problems: inf

u∈N∞J ∞

(u) = α∞ and inf

u∈ eN∞
e
J∞(u) =α_{e}∞,
where
N∞ =u ∈ H1 _{R}N_{+} \ {0} | (J∞)0(u) , u = 0
and
e
N∞=
u ∈ H1 _{R}N \ {0} |
e
J∞
0
(u) , u
= 0
.
It is known that equations (E∞) and Ee∞

have unique positive radial solutions
w (x) and w (x) , respectively such that J_{e} ∞(w) = α∞ and eJ∞(w) =_{e} α_{e}∞ (see [21,
28, 32, 33]). Without loess of generality, we may assume that

w (0) = max x∈∂RN + w (x) = max x∈RN + w (x) and e w (0) = max x∈∂RN + e w (x) = max x∈RNw (x) .e

Thus, we observe that solution w (x) can construct solutionw (x) of equation_{e}

e E∞

by reflection with respect to ∂RN

+. Then αe

∞ _{= e}_{J}∞_{(}

e

w) = 2J∞(w) = 2α∞ (or see [21, p. 889]). For λ ≥ 0, similarly as in [10, 11, 30, 31], we have the following results. Proposition 2.5. Let {un} be a (PS)β–sequence in H1 RN+

for Jλ. Then there

exist a subsequence {un} , m,m ∈ N ∪ {0} , sequences {xe

i

n} ⊂ ∂RN+ and {xe

j

n} ⊂ RN,

function v0 ∈ H1 RN+ , 0 6= wi ∈ H1 RN+ , for 1 ≤ i ≤ m and 0 6=we

j _{∈ H}1

RN ,
for 1 ≤ j ≤m such that_{e}

(i) |xi_{n}| → ∞ and |xi
n− xkn| → ∞ as n → ∞, for 1 ≤ i 6= k ≤ m;
(ii) |x_{e}j
n| → ∞ and |ex
j
n−xe
k
n| → ∞ as n → ∞, for 1 ≤ j 6= k ≤m;e
(iii) |xi
n−ex
j
n| → ∞ as n → ∞, for 1 ≤ i ≤ m and 1 ≤ j ≤m;e
(iv)
−∆v0+ v0 = gλ(x) |v0|p−2v0 in RN+,
∂v0
∂n + f (x) |v0|
q−2
v0 = 0 on ∂RN+;
(v)
−∆wi_{+ w}i _{= |w}i_{|}p−2_{w}i
in RN+,
∂wi
∂n = 0 on ∂R
N
+;
(vi)
−∆w_{e}j _{+}
e
wj _{= |}
e
wj_{|}p−2
e
wj _{in R}N_{;}

(vii) un = v0+
m
P
i=1
wi(· − xi_{n}) + e
m
P
j=1e
wj(· −x_{e}j_{n}) + o(1) strongly in H1 _{R}N ;
(viii) kunk2_{H}1_{(}
RN+) = kv0k
2
H1_{(}
RN+) +
m
P
i=1
kwi_{k}2
H1_{(}
RN+) +
e
m
P
j=1
kw_{e}j_{k}2
H1_{(R}N_{)}+ o(1);
(viiii) Jλ(un) = Jλ(v0) +
m
P
i=1
J∞(wi) +
e
m
P
j=1
e
J∞(w_{e}j) + o(1);
In addition, if un ≥ 0, then v0 ≥ 0, wi ≥ 0 and we

j _{≥ 0 for each 1 ≤ i ≤ m and}

1 ≤ j ≤m._{e}

Proof. Since {un} is a a (PS)β–sequence in H1 RN+ for Jλ, by Lemma 2.2, there

is a subsequence {un} and v0 in H1 RN+ such that

un* v0 weakly in H1 RN+

and in Lp _{R}N_{+} (2.3)

and v0 is a solution of equation (Eλ) . Let ubn = un− v0. Then, by (2.3) , b

un * 0 weakly in H1 RN+

and in Lp _{R}N_{+} .

Suppose that _{b}un6→ 0 strongly in H1 RN+ (otherwise, the result is automatic). By

a similar argument to that in the proof of [11, Lemma 3.1], there exist δ > 0 and xn = (yn, zn) ∈ RN −1× R such that |xn| → ∞ and

Z
BN_{(1)+x}
n
|_{b}un|
2
dx > δ,

where BN(1) = x ∈ RN | |x| < 1 . Moreover, we may assume that one of the following two cases occurs:

(a) {zn} is bounded;

(b) {zn} is unbounded.

Case (a) : Without loss of generality, we may assume that zn = 0. Set x1n = xn =

(yn, 0) . Then |x1n| → ∞ andbun(x + x

1

n) ∈ H1 RN+ for all n. Adopting the approach

employed in the proof of [10, Proposition II.1] (or see [22, Proposition 2.1]), there
exists w1 _{∈ H}1
RN+ \ {0} such that
b
un x + x1n * w1 weakly in H1 RN+
and
−∆w1_{+ w}1 _{= |w}1_{|}p−2_{w}1 _{in R}N
+,
∂wi
∂n = 0 on ∂R
N
+.
Case (b) : Set x_{e}1
n = xn = (yn, zn) . Then |xe
1
n| → ∞ and bun(x +ex
1
n) ∈ H1 RN for

all n. Again, using a similar procedure to that in the proof of [10, Proposition II.1]
(or see [11, Lemma 3.1]), there exists w_{e}1 ∈ H1

RN \ {0} such that b

un x + x1n * w

1 _{weakly in H}1

and

−∆w_{e}1+w_{e}1 = |w_{e}1|p−2w_{e}1 _{in R}N.

Following the same lines of the proof in [10, Proposition II.1] (or see [11, Lemma 3.1]), we repeat the argument above, each iteration will likewise give rise to two cases and the procedure will terminate after some finite steps; the procedure will also lead us to conclude that (i) − (viiii) hold.

For λ ≥ 0, we define

αλ = inf u∈Nλ

Jλ(u) .

Then, by Proposition 2.5, we have the following compactness result. Corollary 2.6. Suppose that {un} is a (PS)β–sequence in H1 RN+

for Jλ with

0 < β < α∞+ min {α∞, αλ} and β 6= α∞. Then there exists a subsequence {un} and

a non-zero v0 in H1 RN+ such that un → v0 strongly in H1 RN+ and Jλ(v0) = β.

Furthermore, v0 is a non-zero solution of equation (Eλ) .

Proof. Sinceα_{e}∞= 2α∞ and

β < α∞+ min {α∞, αλ} ≤ 2α∞,

we have m = 0, which implies that_{e}
kunk2_{H}1_{(}
RN+) = kv0k
2
H1_{(}
RN+) +
m
X
i=1
wi
2
H1_{(}
RN+)
+ o (1)
and
β = Jλ(v0) +
m
X
i=1
J∞(wi).

Since 0 < β < α∞ + min {α∞, αλ} and β 6= α∞, by the uniqueness of positive

solutions of equation (E∞) , we conclude that m = 0. Thus, un → v0 strongly

in H1 _{R}N_{+} and Jλ(v0) = β > 0, which implies that v0 is a non-zero solution of

equation (Eλ) .

3. Existence of positive solutions for λ = 0

Let w (x) be a positive radial solution of equation (E∞) such that J∞(w) = α∞. Then, by Gidas, Ni and Nirenberg [24] and Kwong [28], for any ε > 0, there exist positive numbers Aε and B0 such that

Aεexp (− (1 + ε) |x|) ≤ w (x) ≤ B0exp (− |x|) for all x ∈ RN+. (3.1)

Let y ∈ S =x0 ∈ RN −1 _{| |x}0_{| = 1 and let z}

0 = (δ0, 0, . . . , 0) ∈ RN −1, where

0 < δ0 =

minrf, q,p_{2} − 1

2 minrf, q,p_{2} + 1

Clearly,

1 − δ0 ≤ |y − z0| ≤ 1 + δ0 for all y ∈ S. (3.2)

Define

wy,l(x) = w (x − l (y, 0)) for l ≥ 0 and y ∈ S (3.3)

and

wz0,l(x) = w (x − l (z0, 0)) for l ≥ 0.

Clearly, wy,l and wz0,l are also least energy positive solutions of equation (E
∞_{) for}

all l ≥ 0. Moreover, by Lemma 2.4 for each u ∈ H1

RN+ \ {0} there is a unique

t0(u) ≥bt0(u) such that t0(u) u ∈ N0. Then we have the following results.

Lemma 3.1. For each s0 ∈ (0, 1) there exist l (s0) > 0 and σ (s0) > 1 such that for

any l > l (s0) , we have
bt
p−2
0 (swy,l+ (1 − s) wz0,l) >
σ (s0)
sp−2_{+ (1 − s)}p−2

for all y ∈ S and for all s ∈ (0, 1) with min {s, 1 − s} ≥ s0.

Proof. Since
b
tp−2_{0} (swy,l+ (1 − s) wz0,l) =
kswy,l+ (1 − s) wz0,lk
2
H1
R
RN+ |swy,l+ (1 − s) wz0,l|
p
dx
= s
2_{kwk}2
H1 + (1 − s)
2
kwk2_{H}1 + 2s (1 − s) hwy,l, wz0,li
R
RN+ |swy−z0,l+ (1 − s) w|
p
dx (3.4)

for all s ∈ [0.1] and for all y ∈ S. Moreover, by (3.1) , the triangle inequality and 1 − δ0 ≤ |y − z0| ≤ 1 + δ0 for all y ∈ S, (3.5)

we have
hwy,l, wz0,li =
Z
RN+
wp−1wy−z0,ldx
≤ Bp_{0}
Z
|x|<(1+δ0)l
exp (− (|x| + |x − l (z0− y, 0)|)) dx
+B_{0}p
Z
|x|≥(1+δ0)l
exp (− (|x| + |x − l (z0− y, 0)|)) dx
= Bp_{0}lN
Z
|x|<(1+δ0)
exp (−l (|x| + |x − (z0− y, 0)|)) dx
+B_{0}p
Z
|x|≥(1+δ0)l
exp (− |x|) exp (− |x − l (z0− y, 0)|) dx
≤ Bp_{0}lN
Z
|x|<(1+δ0)
exp (−l (|x| + |x − (z0− y, 0)|)) dx
+B_{0}pexp (− (1 + δ0) l)
Z
|x|≥(1+δ0)l
exp (− (|x − l (z0− y, 0)|)) dx
≤ Bp_{0}lN
Z
|x|<(1+δ0)
exp (−l (|y − z0|)) dx
+B_{0}pexp (− (1 + δ0) l)
Z
RN
exp (− |x|) dx
≤ Bp_{0}lNexp (− (1 − δ0) l)
Z
|x|<(1+δ0)
1dx + d0B
p
0exp (− (1 + δ0) l)
≤ C0B0pl
N_{exp (−l (1 − δ}

0)) for all l ≥ 1 and for all y ∈ S,

which implies that

lim

l→∞hwy,l, wz0,li = 0 uniformly in y ∈ S. (3.6)

By (3.1) , (3.5) and Br´ezis-Lieb lemma [12], for any s ∈ [0.1] we have
lim
l→∞
Z
RN+
|swy−z0,l+ (1 − s) w|
p _{− |sw}
y−z0,l|
p
dx
=
Z
RN+
|(1 − s) w|p_{dx uniformly in y ∈ S.} (3.7)
Thus, by (3.4) , (3.6) and (3.7) , for any s ∈ [0.1]

lim
l→∞bt
p−2
0 (swy,l+ (1 − s) wz0,l) =
s2+ (1 − s)2 kwk2_{H}1
(sp_{+ (1 − s)}p_{)}R
RN+
|w|pdx
= s
2_{+ (1 − s)}2
sp_{+ (1 − s)}p uniformly in y ∈ S. (3.8)

Since
s2 _{+ (1 − s)}2
sp−2_{+ (1 − s)}p−2
sp_{+ (1 − s)}p = 1 +
s2_{(1 − s)}p−2
+ (1 − s)2sp−2
sp _{+ (1 − s)}p
≥ 1 + s
2
0(1 − s0)p−2+ (1 − s0)2sp−20
sp_{0}+ (1 − s0)
p (3.9)

for all s ∈ (0, 1) with min {s, 1 − s} > s0, by (3.8) and (3.9) , there exist l (s0) > 0

and σ (s0) > 1 such that for any l > l (s0) , we have

bt

p−2

0 (swy,l+ (1 − s) wz0,l) >

σ (s0)

sp−2_{+ (1 − s)}p−2

for all y ∈ S and for all s ∈ (0, 1) with min {s, 1 − s} ≥ s0. This completes the

proof.

Proposition 3.2. There exists l1 > 0 such that for any l ≥ l1

sup

t≥0

J0(t [swy,l+ (1 − s) wz0,l]) < 2α ∞

for all y ∈ S,

where J0 = Jλ for λ = 0. Furthermore, there is a unique t0(swy,l+ (1 − s) wz0,l) > 0 such that

t0(swy,l+ (1 − s) wz0,l) [swy,l+ (1 − s) wz0,l] ∈ N0.

Proof. When s = 0 or 1, by a similar argument to that in the proof of Wu [40, Proposition 2], there exists et1 > 0 such that

max sup t≥0 J0(twy,l), sup t≥0 J0(twz0,l) ≤ α∞+et1C0 q exp (− min {rf, q} l) (3.10) for all y ∈ S, this implies that there exists el1 > 0 such that for any l > el1,

max sup t≥0 J0(twy,l), sup t≥0 J0(twz0,l) ≤ 3 2α ∞ for all y ∈ S. (3.11) Therefore, by J0 ∈ C2 H1 RN+ , R and (3.11) , there exist positive constants s0, el

such that for any l > el, sup

t≥0

J0(t [swy,l+ (1 − s) wz0,l]) < 2α ∞

for all y ∈ S and for all min {s, 1 − s} ≤ s0. In the following we always assume that
min {s, 1 − s} ≥ s0. Since
J0(t [swy,l+ (1 − s) wz0,l])
= t
2
2 s
2_{kwk}2
H1 + (1 − s)
2
kwk2_{H}1 + 2s (1 − s) hwy,l, wz0,li
+t
q
q
Z
∂RN
+
f (swy,l+ (1 − s) wz0,l)
q
dσ
−t
p
p
Z
RN+
(swy,l+ (1 − s) wz0,l)
p
dx (3.13)
≤ t
2
2 s
2
+ 2s (1 − s) + (1 − s)2 kwk2_{H}1
+bct
q_{2}q−1
q (s
q_{+ (1 − s)}q
)
Z
∂RN+
wqdσ − t
p
p max {s
p_{, (1 − s)}p_{}}Z
RN+
wpdx
≤ t2kwk2_{H}1 +
bct
q_{2}q−1
q
Z
∂RN
+
wqdσ − t
p
p2p
Z
RN+
wpdx

for all 0 ≤ s ≤ 1 and y ∈ S. Then there exists t1 > 0 such that for any t ≥ t1,

J0(t [swy,l+ (1 − s) wz0,l]) < 2α ∞

for all 0 ≤ s ≤ 1 and for all y ∈ S. (3.14) Moreover, by Lemma 2.4 (i) and Lemma 3.1,

sup

t≥0

J0(t [swy,l+ (1 − s) wz0,l]) = sup t≥t2

J0(t [swy,l+ (1 − s) wz0,l]) (3.15)

for all y ∈ S, where t2 =

σ(s0)
sp−2_{+(1−s)}p−2

1/(p−2)

and σ (s0) > 1 is as in Lemma 3.1.

Thus, by (3.14) and (3.15) , we only need to show that there exists l1 ≥ el such that

for any l > l1, sup t2≤t≤t1 J0(t [swy,l+ (1 − s) wz0,l]) < 2α ∞ for all y ∈ S. (3.16) By lemma 2.1 in Bahri and Li [9], there exists Cp > 0, such that for any nonnegative

real numbers a, b,

Then, by (3.13) , (3.17) and Lemma 3.1,
J0(t [swy,l+ (1 − s) wz0,l])
≤ 2α∞+ s (1 − s)t2− tp_{s}p−2_{− t}p_{(1 − s)}p−2
Z
RN+
w_{y,l}p−1wz0,ldx
+t
q
1
q
Z
∂RN
+
f (swy,l+ (1 − s) wz0,l)
q
dσ + t
p
1Cp
p
Z
RN+
w_{y,l}p/2w_{z}p/2
0,ldx
≤ 2α∞− C_{0}2[σ (s0) − 1]
Z
RN+
wp−1_{y,l} wz0,ldx
+t
q
1
q
Z
∂RN
+
f (swy,l+ (1 − s) wz0,l)
q
dσ + t
p
1Cp
p
Z
RN+
w_{y,l}p/2w_{z}p/2
0,ldx (3.18)

for all y ∈ S, where we have used the result
Z
RN+
wp−1_{y,l} wz0,ldx = hwy,l, wz0,li =
Z
RN+
wy,lwp−1z0,ldx.
We first estimate R
RN+ w
p−1
y,l wz0,ldx. Setting
C0 = min
x∈BN
+(eN,1_{2})
wp−1(x) > 0,
where BN
+ eN,1_{2} = x ∈ RN+ | |x − eN| ≤ 1_{2} and eN = (0, . . . , 0, 1) ∈ RN. Then,

by (3.1) and (3.2) , for any ε > 0,
Z
RN+
wp−1_{y,l} wz0,ldx =
Z
RN+
wp−1(x) w (x − l (z0− y, 0)) dx
≥ C0Aε
Z
BN
+(eN,1_{2})
exp (− (1 + ε) |x − l (z0− y, 0)|) dx
≥ C0Aε
Z
BN
+(eN,1_{2})
exp (− (1 + ε) |x| − l (1 + ε) |y − z0|) dx
≥ C0Aεexp (−l (1 + ε) |y − z0|)
≥ C0Aεexp (−l (1 + ε) (1 + δ0)) . (3.19)

From (3.2) we have
Z
RN+
w_{y,l}p/2w_{z}p/2
0,ldx
≤ B_{0}p
Z
|x|<(1+δ0)l
exp−p
2(|x| + |x − l (z0− y, 0)|)
dx
+B_{0}p
Z
|x|≥(1+δ0)l
exp−p
2(|x| + |x − l (z0− y, 0)|)
dx
≤ B_{0}plN
Z
|x|<(1+δ0)
exp−p
2l (|x| + |x − (z0− y, 0)|)
dx
+B_{0}pexp
−(1 + δ0) pl
2
Z
|x|≥(1+δ0)l
exp−p
2(|x − l (y − z0, 0)|)
dx
≤ B_{0}plN
Z
|x|<(1+δ0)
exp
−pl
2 |y − z0|
dx +_{b}c0B0pexp
−pl
2 |y − z0|
≤ C0B0pl
N_{exp}
−pl
2 |y − z0|
≤ C0B0plNexp
− minnrf, q,
p
2
o
(1 − δ0) l
for l ≥ 1. (3.20)
By (D2) , we also have
Z
∂RN
+
f (swy,l+ (1 − s) wz0,l)
q
dσ
≤ 2q−1
Z
∂RN+
f w_{y,l}q dσ +
Z
∂RN+
f wq_{z}
0,ldσ
!
≤ Cb_{0}B_{0}plNexp (− min {r_{f}, q} l |y − z_{0}|)
≤ Cb0B0pl
N
exp
− minnrf, q,
p
2
o
(1 − δ0) l
for l ≥ 1. (3.21)
Since
1 + δ0 = 1 +
minrf, q,p_{2} − 1
2 minrf, q,p_{2} + 1
< min
n
rf, q,
p
2
o
1 − minrf, q,
p
2 − 1
2 minrf, q,p_{2} + 1
!
= minnrf, q,
p
2
o
(1 − δ0) ,

we may take 0 < ε << 1 such that

(1 + ε) (1 + δ0) < min n rf, q, p 2 o (1 − δ0) .

Then, by (3.18) − (3.21), there exists l1 ≥ max

n el, 1

o

such that (3.16) holds. There-fore, by (3.12) and (3.14) − (3.16), we can conclude that for any l > l1,

sup

t≥0

J0(t [swy,l+ (1 − s) wz0,l]) < 2α ∞

for all 0 ≤ s ≤ 1 and for all y ∈ S. Moreover, by Lemma 2.4, there is a unique t0(swy,l+ (1 − s) wz0,l) > 0 such that

t0(swy,l+ (1 − s) wz0,l) [swy,l+ (1 − s) wz0,l] ∈ N0. This completes the proof.

Theorem 3.3. We have α0 = inf u∈N0 J0(u) = inf u∈N∞J ∞ (u) = α∞.

Furthermore, equation (E0) does not admit any solution u0 such that J0(u0) = α0.

Proof. By Lemma 2.4, there is a unique t0(wy,l) > 0 such that t0(wy,l) wy,l ∈ N0

for all y ∈ S, that is

kt0(wy,l) wy,lk2_{H}1 +
Z
∂RN
+
f |t0(wy,l) wy,l|qdσ =
Z
RN+
|t0(wy,l) wy,l|pdx.
Since
Z
∂RN
+
f |wy,l|
q
dσ → 0 as l → ∞,
and
kwy,lk
2
H1 =
Z
RN+
|wy,l|
p
dx = 2p
p − 2α
∞

for all l ≥ 0 and for all y ∈ S, we have t0(wy,l) → 1 as l → ∞. Thus,

lim

l→∞J0(t0(wy,l) wy,l) = liml→∞J ∞

(t0(wy,l) wy,l) = α∞ for all y ∈ S,

which implies that

α0 ≤ inf u∈N∞J

∞

(u) = α∞.

Let u ∈ N0. Then, by Lemma 2.4, J0(u) = supt≥0J0(tu) . Moreover, there is a

unique t∞> 0 such that t∞u ∈ N∞. Thus,

J0(u) ≥ J0(t∞u) ≥ J∞(t∞u) ≥ α∞ and so α0 ≥ α∞. Therefore, α0 = inf u∈N0 J0(u) = inf u∈N∞J ∞ (u) = α∞.

Next, we will show that equation (E0) does not admit any solution u0 such that

J0(u0) = α0. Suppose the contrary. Then we can assume that there exists u0 ∈ N0

such that J0(u0) = α0. Then, by Lemma 2.4 (i) , J0(u0) = supt≥0J0(tu0) . Moreover,

there is a unique t∞(u0) > 0 such that t∞(u0) u0 ∈ N∞. Thus,

α∞ = inf
u∈N0
J0(u) = J0(u0) ≥ J0(t∞(u0) u0)
≥ α∞_{+} [t
∞_{(u}
0)]q
q
Z
∂RN
+
f |u0|qdσ,

which implies that R

∂RN +

f |u0|qdσ = 0 and so u0 ≡ 0 in x ∈ ∂RN+ | f (x) 6= 0 ,

form the condition (D1) . Therefore, α∞ = inf

u∈N∞J ∞

(u) = J∞(tu0u0) .

Since |t∞(u0) u0| ∈ N∞ and J∞(|t∞(u0) u0|) = J∞(t∞(u0) u0) = α∞, by Willem

[39, Theorem 4.3] and the maximum principle, we can assume that tu0u0is a positive solution of (E∞) , this contradicts

u0 ≡ 0 in x ∈ ∂RN+ | f (x) 6= 0 .

This completes the proof.

By Theorem 3.3, equation (E0) does not admit any solution u0 such that J0(u0) =

α0 and
α0 = inf
u∈N0
J0(u) = inf
u∈N∞J
∞_{(u) = α}∞_{.}

Furthermore, we have the following result.

Lemma 3.4. Suppose that {un} is a minimizing sequence for J0 in N0. Then

Z

∂RN +

f |un|qdσ = o (1) .

Furthermore, {un} is a (PS)α∞–sequence for J∞ in H1 _{R}N_{+} .

Proof. For each n, there is a unique tn > 0 such that tnun ∈ N∞, that is

t2_{n}kunk2_{H}1 = t
p
n
Z
RN+
|un|pdx.

Then, by Lemma 2.4 (i) ,

J0(un) ≥ J0(tnun) = J∞(tnun) + tq n q Z ∂RN + f |un|qdσ ≥ α∞+t q n q Z ∂RN + f |un| q dσ.

Since J0(un) = α∞+ o (1) from Theorem 3.3, we have tq n q Z ∂RN + f |un|qdσ = 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 J0(un) = α∞+ o (1) , by

Lemma 2.2, we have 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 dσ = o (1) ,

which implies that

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

Moreover, by Wang and Wu [38, Lemma 7], we have {un} is a (PS)α∞–sequence for J∞ in H1

RN+ .

For u ∈ H1 _{R}N_{+}, we define the center mass function from N0 to the unit ball of

RN −1
m(u) = 1
kukp_{L}p_{(R}N
+)
Z
RN+
x0
|x0_{|}|u(x
0
, xN)|
p
dx0dxN.

Clearly, m is continuos from N0 to BN −1(0, 1) and |m (u)| < 1. Let

θ0 = inf {J0(u) | u ∈ N0, u ≥ 0, m(u) = 0} .

Then we have the following result.

Lemma 3.5. There exists ξ0 > 0 such that α∞< ξ0 ≤ θ0.

Proof. Suppose the contrary. Then there exists a sequence {un} ⊂ N0 and

m(un) = 0 for each n, such that J0(u) = α∞ + o (1) . By Lemma 3.4, we have

{un} is a (PS)α∞–sequence in H1 _{R}N_{+} for J∞. By the concentration–compactness
principle (see Lions [30, 31] or del Pino and Flores [22, proof of proposition 2.1])
and the fact that α∞ = α_{e}∞/2 > 0, there exist a subsequence {un} , a sequence

{(x0

n, 0)} ⊂ ∂RN+, and a positive solution w ∈ H1 RN+ of equation (E∞) such that

Now we will show that |(x0_{n}, 0)| → ∞ as n → ∞. Suppose the contrary. Then we
may assume that {(x0_{n}, 0)} is bounded and (x0_{n}, 0) → (x0_{0}, 0) for some (x0_{0}_{, 0) ∈ ∂R}N_{+}.
Thus, by (3.22) ,
Z
∂RN
+
f |un|
q
dσ =
Z
∂RN
+
f (x) |w (x − (x0_{n}, 0))|qdσ + o (1)
=
Z
∂RN
+
f (x + (x0_{0}, 0)) |w (x)|qdσ + o (1) ,
this contradicts the result of Lemma 3.4: R_{∂R}N

+ f |un| q

dσ = o (1) . Hence we may assume that x0n

|x0

n| → e as n → ∞, where e ∈ S. Then, by (3.22) and the Lebesgue dominated convergence theorem, we have

0 = m(un)
= kunk
−p
Lp_{(R}N
+)
Z
RN+
x0
|x0_{|}|un(x
0
, xN)|
p
dx0dxN
= kwk−p_{L}p_{(R}N
+)
Z
RN+
x0 + x0_{n}
|x0 _{+ x}0
n|
|w(x0, xN)|
p
dx0dxN + o(1)
= e + o(1) as n → ∞,

which is a contradiction. Therefore, there exists ξ0 > 0 such that α∞ < ξ0 ≤ θ0.

By Lemma 2.4 and Proposition 3.2, for each y ∈ S and l > l1there exists t0(wy,l) >

0 such that t0(wy,l) wy,l ∈ N0. Moreover, we have the following result.

Lemma 3.6. There exists l0 ≥ l1 such that for any l ≥ l0,

(i) α∞ < J0(t0(wy,l) wy,l) < ξ0 for all y ∈ S;

(ii) hm(t0(wy,l) wy,l), yi > 0, for all y ∈ S.

Proof. (i) Follow from (4.4) − (4.6) and Theorem 3.3.
(ii) For x0 _{∈ R}N −1 _{with x}0 _{+ ly 6= 0, we have}

( x
0_{+ ly}
|x0_{+ ly|}, ly) = |x
0_{+ ly| −} 1
|x0_{+ ly|}(x
0_{+ ly, x}0_{)}
≥ |x0+ ly| − |x0| ≥ l |y| − 2 |x0| = l − 2 |x0| .
Then
hm(t0(wy,l) wy,l), yi =
1
l kwy,lk
p
Lp_{(R}N
+)
Z
RN+
x0
|x0_{|}, ly
|wy,l|pdx0dxN
= 1
l kwkp_{L}p_{(R}N
+)
Z
RN+
x0_{+ ly}
|x0_{+ ly|}, ly
|w|pdx0dxN
≥ 1
l kwkp_{L}p_{(R}N
+)
(l
Z
RN+
|w|pdx0dxN − 2
Z
RN+
|x0| |w|pdx0dxN)
= 1 − 2c0
l ,

where c0 = kwk
−p
Lp_{(R}N
+)
R
RN+ |x
0_{| |w|}p

dx0dxN. Thus, there exists l0 ≥ l1 such that

hm(t0(wy,l) wy,l), yi ≥ 1 −

2c0

l > 0 for all l ≥ l0. This completes the proof.

In the following, we will use Bahri-Li’s minimax argument [9]. Let B =u ∈ H1 RN+ \ {0} | u ≥ 0 and kukH1 = 1 . We define

I0(u) = sup t≥0

J0(tu) : B → R.

Then, by Lemma 2.4 (iii), for each u ∈ H1 _{R}N_{+} \ {0} there exists
t0(u) =
1
kuk_{H}1
t0
u
kuk_{H}1
> 0
such that t0(u) u ∈ N0 and

I0(u) = J0(t0(u) u) = J0
t0
u
kuk_{H}1
u
kuk_{H}1
(3.23)
Next, we define a map h0 from S to B by

h0(y) =
w (x − l (y, 0))
kw (x − l (y, 0))k_{H}1
= wy,l
kt0(wy,l) wy,lk_{H}1
,

where y ∈ S. Then, by (3.10) and (3.23) , for l > l0 sufficiently large, we have

I0(h0(y)) = J0(t0(wy,l) wy,l) < θ0 for all y ∈ S.

We define another map h∗ from BN −1_{(0, 1) to N}
0 by

h∗(sy + (1 − s) z0) =

swy,l+ (1 − s) wz0,l kswy,l+ (1 − s) wz0,lkH1

where 0 ≤ s ≤ 1 and y ∈ S. It is clear that h∗|_{S} = h0. It follows from Proposition

3.2 and (3.23) that

I0(h0(y)) = J0(t0(swy,l+ (1 − s) wz0,l) [swy,l+ (1 − s) wz0,l]) < 2α ∞

(3.24) for all y ∈ S. We next define a min-max value. Let

β0 = inf
γ∈Γz∈BmaxN −1_{(0,1)}I0(γ(z)), (3.25)
where
Γ =nγ ∈ CBN −1_{(0, 1), B} _{| γ|}
S = h0
o
. (3.26)

Lemma 3.7. We have

α∞< ξ0 ≤ θ0 ≤ β0 < 2α∞.

Proof. By Lemmas 3.5, 3.6, (3.24) and (3.23), we only need to show θ0 ≤ β0.

For any γ ∈ Γ, there exists t0(γ(z)) > 0 such that t0(γ(z)) γ(z) ∈ N0 and

t0(γ(z)) γ(z) = t0(wz,l) wz,l for all z ∈ S.

Consider the homotopy H(s, z) : [0, 1] × BN −1_{(0, 1) → R defined by}

H(s, z) = (1 − s)m(t0(γ(z)) γ(z)) + sI(z),

where I denotes the identity map. Note that m(t0(γ(z)) γ(z)) = m(t0(wz,l) wz,l) for

all z ∈ S. By Lemma 3.6 (ii), H(s, z) 6= 0 for z ∈ S and s ∈ [0, 1]. Therefore, deg(m (t0(γ) γ) , BN −1(0, 1) , 0) = deg(I, BN −1(0, 1) , 0) = 1.

There exists z0 ∈ BN −1(0, 1) such that

m(t0(γ(z0)) γ(z0)) = 0.

Hence, for each γ ∈ Γ0, we have

θ0 = inf {J0(u) | u ∈ N0, u ≥ 0, m(u) = 0}

≤ I0(γ(z0))

≤ max

z∈BN −1_{(0,1)}I0(γ(z)).
This shows that θ0 ≤ β0.

Now, we are going to assert that the equation (E0) has a positive solution.

Theorem 3.8. Equation (E0) has a positive solution eu0 such that J0(eu0) = β0. Proof. By Lemma 3.7 and the minimax principle (see Ambrosetti and Rabinowitz [3]), there exists a sequence {un} ⊂ B such that

I0(un) = β0+ o (1) ,
kI0
0(un)k_{T}∗
unB ≡ sup {I
0
0(un)φ | φ ∈ TunB, kφkH1 = 1} = o (1) as n → ∞,
where α∞< β0 < 2α∞ and TunB =φ ∈ H
1
RN+
| hφ, uni = 0 . By an argument

similar to the proof of proposition 1.7 in Adachi and Tanaka [4], there exists t0(un) >

0 such that t0(un) un ∈ N0 and

J0(t0(un) un) = β0+ o (1) ,

J_{0}0(t0(un) un) = o (1) in H−1 RN+

as n → ∞.

Thus, by Corollary 2.6, Theorem 3.3 and the maximum principle, we can conclude that the equation (E0) has a positive solution ue0 such that J0(ue0) = β0.

4. Existence of positive solutions for λ > 0

By Lemma 2.4 for each u ∈ H1 _{R}N_{+} \ {0} and λ > 0 there is a unique tλ(u) ≥

b

tλ(u) such that tλ(u) u ∈ Nλ. Let wy,l be as in (3.3) . Then we have the following

results.

Proposition 4.1. For each λ > 0, there exists bl1 = bl1(λ) > 0 such that for any

l ≥ bl1,

sup

t≥0

Jλ(twy,l) < α∞ for all y ∈ S.

Furthermore, there is a unique tλ(wy,l) > 0 such that tλ(wy,l) wy,l ∈ Nλ.

Proof. We have
Jλ(twy,l)
= t
2
2 kwy,lk
2
H1 +
tq
q
Z
∂RN
+
f |wy,l|
q
dσ −t
p
p
Z
RN+
gλ|wy,l|
p
dx
= t
2
2 kwk
2
H1 −
tp
p
Z
RN+
wpdx +t
q
q
Z
∂RN
+
f w_{y,l}q dσ − λt
p
p
Z
RN+
awp_{y,l}dx (4.1)
≤ t
2
2 kwk
2
H1 +b
ctq
q
Z
∂RN
+
wqdσ −t
p
p
Z
RN+
wpdx.

for all λ > 0. This implies that Jλ(twy,l) → −∞ as t → ∞ for all y ∈ S. Thus, there

exists t1 > 0 such that for any l ≥ 0,

Jλ(twy,l) < α∞ for all t ≥ t1 and for all y ∈ S. (4.2)

Moreover, Jλ(0) = 0 < α∞, Jλ ∈ C2 H1 RN , R and kwy,lk 2 H1 =

2p p−2α

∞ _{for all}

l ≥ 0, this implies that there exists t2 > 0 such that for any l ≥ 0,

Jλ(twy,l) < α∞ for all 0 ≤ t ≤ t2 and for all y ∈ S. (4.3)

Moreover, by Brown and Zhang [15] and Willem [39], we know that
J∞(tw) = t
2
2 kwk
2
H1 −
tp
p
Z
RN+
wpdx ≤ α∞ for all t > 0. (4.4)
Thus, by (4.1) ,
Jλ(twy,l) ≤ α∞+
tq
q
Z
∂RN
+
f w_{y,l}q dσ − λt
p
p
Z
RN+

awp_{y,l}dx for all t > 0. (4.5)

By (4.2) and (4.3) , we only need to show that there exists bl1 > 0 such that for any

l > bl1,

sup

t2≤t≤t1

We set
C0 = min
x∈BN
+(eN,12)
wp(x) > 0,
where BN
+ eN,1_{2} = x ∈ RN+ | |x − eN| < 1_{2} and eN = (0, . . . , 0, 1) ∈ RN. Then,
by the conditions (D1) ,
Z
RN
awp_{y,l}dx =
Z
RN
a (x + l (y, 0)) wp(x) dx ≥ C0
Z
BN
+(eN,1_{2})
a (x + l (y, 0)) dx
≥ C0exp (−ral) .

Moreover, by (3.1) and the condition (D1) ,
Z
∂RN
+
f w_{y,l}q dσ ≤ _{b}cB_{0}p
Z
∂RN
+

exp (−rf|x|) exp (−q |x − l (y, 0)|) dσ

≤ C0exp (− min {rf, q} l) (4.6)

Since ra < min {rf, q} and t2 ≤ t ≤ t1, we can find bl1 > 0 such that for any l > bl1,

tq
q
Z
∂RN
+
f w_{y,l}q dσ < λt
p
p
Z
RN+

aw_{y,l}p _{dx for all y ∈ S and for all t ∈ [t}2, t1] . (4.7)

Thus, by (4.2) − (4.5) and (4.7), we obtain that for any l > bl1

sup

t≥0

Jλ(twy,l) < α∞ for all y ∈ S.

Moreover, by Lemma 2.4, there is a unique tλ(wy,l) > 0 such that tλ(wy,l) wy,l ∈ Nλ.

This completes the proof.

Theorem 4.2. For each λ > 0, equation (Eλ) has a positive solution u0 such that

Jλ(u0) = αλ = inf u∈Nλ

Jλ(u) < α∞.

Proof. By analogy with the proof of Ni and Takagi [32], one can show that the Ekeland variational principle (see [23]), there exists a minimizing sequence {un} ⊂

Nλ such that

Jλ(un) = αλ+ o (1) and Jλ0 (un) = o (1) in H−1 RN+ .

Since infu∈NλJλ(u) < α

∞ _{from Proposition 4.1 (ii) , by Lemma 2.2 and Corollary}

2.6 there exist a subsequence {un} and u0 ∈ Nλ is a nonzero solution of equation

(Eλ) such that

un→ u0 strongly in H1(RN+) and Jλ(u0) = αλ.

Since Jλ(u0) = Jλ(|u0|) and |u0| ∈ Nλ, by Lemma 2.3, we may assume that u0 is a

5. Existence of two positive solutions We need the following result.

Lemma 5.1. There exists d0 > 0 such that if u ∈ N0 and J0(u) ≤ α∞+ d0, then

Z
RN+
x0
|x0_{|} |∇u|
2
+ u2 dx0dxN 6= 0.

Proof. Suppose the contrary. Then there exists sequence {un} ⊂ N0 such that

J0(u) = α∞+ o (1) and
Z
RN+
x0
|x0_{|} |∇un|
2
+ u2_{n} dx0
dxN = 0.

Moreover, by Lemma 3.4, we have {un} is a (PS)α∞–sequence in H1 _{R}N_{+} for J∞. By
the concentration–compactness principle (see Lions [30, 31] or del Pino and Flores
[22, proof of proposition 2.1]) and α∞ = α_{e}∞/2, there exist a subsequence {un} , a

sequence {(x0_{n}_{, 0)} ⊂ ∂R}N_{+}, and a positive solution w ∈ H1 _{R}N_{+} of equation (E∞)
such that

kun(x) − w (x − (x0n, 0))kH1 → 0 as n → ∞. (5.1)
Now we will show that |(x0_{n}, 0)| → ∞ as n → ∞. Suppose the contrary. Then we
may assume that {(x0_{n}, 0)} is bounded and (x0_{n}, 0) → (x0_{0}, 0) for some (x0_{0}_{, 0) ∈ ∂R}N

+.
Thus, by (5.1) ,
Z
∂RN
+
f |un|qdσ =
Z
∂RN
+
f (x) |w (x − (x0_{n}, 0))|qdσ + o (1)
=
Z
∂RN
+
f (x + (x0_{0}, 0)) |w (x)|qdσ + o (1) ,

which contradicts the result of Lemma 3.4: R_{∂R}N
+ f |un|
q
dσ = o (1) . Hence we may
assume x0n
|x0
n| → e as n → ∞, where e ∈ S = x
0
∈ RN −1 _{| |x}0_{| = 1 . Then, by the}

Lebesgue dominated convergence theorem, we have
0 =
Z
RN+
x0
|x0_{|} |∇un|
2
+ u2_{n} dx0dxN =
Z
RN+
x0+ x0_{n}
|x0_{+ x}0
n|
|∇w|2+ w2 dx + o (1)
= 2p
p − 2α
∞_{e + o (1) ,}

which is a contradiction. This completes the proof.

For u ∈ Nλ, by Lemma 2.4, there is a unique t0(u) > 0 such that t0(u) u ∈ N0.

Let η0 = (1 + λ kak∞) 1

p−q_{. Then we have the following result.}

Proof. Let u ∈ Nλ. Then we have
kuk2_{H}1 +
Z
∂RN
+
f |u|qdσ =
Z
RN+
gλ|u|
p
dx.
We distinguish two cases.

Case (A) : t0(u) < 1. Since θ0 > 1, we have

t0(u) < 1 < η0.

Case (B) : t0(u) ≥ 1. Since

[t0(u)]
pZ
RN+
|u|pdx = [t0(u)]
2
kuk2_{H}1 + [t (u)]
qZ
∂RN
+
f |u|qdσ
≤ [t0(u)]q kuk2_{H}1 +
Z
∂RN
+
f |u|qdσ
!
= [t0(u)]q
Z
RN+
gλ|u|pdx
≤ [t0(u)]
q
(1 + λ kak_{∞})
Z
RN+
|u|pdx,
we have
t0(u) ≤ (1 + λ kak∞)
1
p−q = η
0.

This completes the proof.

By the proof of Proposition 4.1, there exist positive numbers tλ(wy,l) and bl1 such

that tλ(wy,l) wy,l ∈ Nλ and

Jλ(tλ(wy,l) wy,l) < α∞ for all l > bl1.

Then we have the following result.

Lemma 5.3. There exists a positive number λ0 such that for every λ ∈ (0, λ0) , we

have
Z
RN+
x0
|x0_{|} |∇u|
2
+ u2 dx0dxN 6= 0

for all u ∈ Nλ with Jλ(u) < α∞.

Proof. (i) Let u ∈ Nλ with Jλ(u) < α∞. Then, by Lemma 2.4 (i) , there exists

t0(u) > 0 such that t0(u) u ∈ N0. Moreover,

Jλ(u) = sup t≥0 Jλ(tu) ≥ Jλ(t0(u) u) = J0(t0(u) u) − λ [t0(u)] pZ RN+ a |u|pdx.

Thus, by Lemma 5.2 and the Sobolev inequality, J0(t0(u) u) ≤ Jλ(u) + λ [t0(u)]p

Z RN+ a |u|pdx < α∞+ λc0η0pkak∞kuk p H1 for some c0 > 0. (5.2) Moreover, by (2.1) , α∞> Jλ(u) ≥ p − 2 2p kuk 2 H1, which implies that

kuk_{H}1 <

2pα∞

p − 2
1_{2}

(5.3) for all u ∈ Nλ with Jλ(u) < α∞. Therefore, by (5.2) and (5.3) ,

J0(t0(u) u) < α∞+ λc0η0pkak∞

2pα∞

p − 2
p_{2}

.

Let d0 > 0 be as in Lemma 5.1. Then there exists a positive number λ0 such that

for λ ∈ (0, λ0) ,

J0(t (u) u) < α∞+ d0. (5.4)

Since t0(u) u ∈ N0 and t0(u) > 0, by Lemma 5.1 and (5.4)

Z
RN+
x0
|x0_{|} |∇ (t0(u) u)|
2
+ (t0(u) u)
2_{ dx}0
dxN 6= 0,

which implies that

Z
RN+
x0
|x0_{|} |∇u|
2
+ u2 dx0dxN 6= 0

for all u ∈ Nλ with Jλ(u) < α∞.

In the following, we use an idea of Adachi and Tanaka [4]. For c ∈ R+, we denote [Jλ ≤ c] = {u ∈ Nλ | u ≥ 0, Jλ(u) ≤ c} .

We then try to show for a sufficiently small σ > 0

cat ([Jλ ≤ α∞− σ]) ≥ 2. (5.5)

To prove (5.5) , we need some preliminaries. Recall the definition of Lusternik-Schnirelman category.

Definition 5.4. (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

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 5.5. Suppose that X is a Hilbert manifold and F ∈ C1_{(X, R) . Assume}
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].

Let Sm−1 _{= {x ∈ R}m _{| |x| = 1} be a unit sphere in R}m _{for m ∈ N. Then we have}

the following results.

Lemma 5.6. Let X be a topological space. Suppose that there are two continuous maps

Φ : Sm−1 → X, Ψ : X → Sm−1

such that Ψ ◦ Φ is homotopic to the identity map of Sm−1_{, that is, there exists a}

continuous map ζ : [0, 1] × Sm−1 → Sm−1 _{such that}

ζ (0, x) = (Ψ ◦ Φ) (x) for each x ∈ Sm−1, ζ (1, x) = x for each x ∈ Sm−1.

Then

cat (X) ≥ 2. Proof. See Adachi and Tanaka [4, Lemma 2.5].

For l > bl1, we may define a map Φλ,l : S(N −1)−1 → H1 RN+ by

Φλ,l(y) (x) = tλ(w (x − l (y, 0))) w (x − l (y, 0)) for y ∈ S(N −1)−1,

where tλ(w (x − l (y, 0))) w (x − l (y, 0)) is as in the proof of Proposition 4.1. Note

Lemma 5.7. There exists a sequence {σl} ⊂ R+ with σl → 0 as l → ∞ such that

Φλ,l S(N −1)−1 ⊂ [Jλ ≤ α∞− σl] .

Proof. By Proposition 4.1, for each l > bl1 we have

tλ(w (x − l (y, 0))) w (x − l (y, 0)) ∈ Nλ

and

sup

l>bl1

Jλ(tλ(w (x − l (y, 0))) w (x − l (y, 0))) < α∞ for all y ∈ S(N −1)−1.

Since S = S(N −1)−1 _{and Φ}

λ,l S(N −1)−1 is compact,

Jλ(tλ(w (x − l (y, 0))) w (x − l (y, 0))) ≤ α∞− σl,

so that the conclusion holds. From Lemma 5.3, we define

Ψλ : [Jλ < α∞] → S(N −1)−1
by
Ψλ(u) =
R
RN+
x0
|x0_{|} |∇u|
2
+ u2_{ dx}0_{dx}
N
R
RN+
x0
|x0_{|} |∇u|
2
+ u2_{ dx}0_{dx}
N
.

Then we have the following results.

Lemma 5.8. Let λ0 > 0 be as in Lemma 5.3. Then for each λ ∈ (0, λ0) and there

exists bl0 ≥ bl1 such that for l > bl0, the map

Ψλ◦ Φλ,l : S(N −1)−1→ S(N −1)−1

is homotopic to the identity.

Proof. Let Σ = nu ∈ H1 _{R}N_{+} \ {0} | R
RN+
x0
|x0_{|} |∇u|
2
+ u2 dx0dxN 6= 0
o
. We
define
Ψλ : Σ → S(N −1)−1
by
Ψλ(u) =
R
RN+
x0
|x0_{|} |∇u|
2
+ u2_{ dx}0_{dx}
N
R
RN+
x0
|x0_{|} |∇u|
2
+ u2_{ dx}0_{dx}
N
,

an extension of Ψλ. Since w (x − l (y, 0)) ∈ Σ for all e ∈ S(N −1)−1and for l sufficiently

Ψλ(Φλ,l(y)) such that γ (s1) = Ψλ(wy,l) , γ (s2) = Ψλ(Φλ,l(y)) . By an argument

similar to that in Lemma 5.1, there exists a positive number bl0 ≥ bl1 such that for

l > bl0, w x − l (y, 0) 2 (1 − θ)

∈ Σ for all y ∈ S(N −1)−1 _{and θ ∈ [1/2, 1) .}

We define
ζl(θ, y) : [0, 1] × S(N −1)−1 → S(N −1)−1
by
ζl(θ, y) =
γ (2θ (s1 − s2) + s2) for θ ∈ [0, 1/2) ;
Ψλ
wx − _{2(1−θ)}l(y,0) for θ ∈ [1/2, 1) ;
y for θ = 1.

Then ζl(0, y) = Ψλ(Φλ,l(y)) = Ψλ(Φλ,l(y)) and ζl(1, y) = y. First, we claim that

lim
θ→1−ζl(θ, y) = y and lim
θ→1_{2}−
ζl(θ, y) = Ψλ(wy,l) .
(a) lim
θ→1−ζl(θ, y) = y : since
Z
RN+
x0
|x0_{|}
∇
w
x − l (y, 0)
2 (1 − θ)
2
+
w
x − l (y, 0)
2 (1 − θ)
2!
dx0dxN
=
Z
RN+
x0+ _{2(1−θ)}ly
x
0_{+} ly
2(1−θ)
|∇ [w (x)]|2+ [w (x)]2 dx0dxN
=
2p
p − 2
α∞y + o(1) as θ → 1−,
then lim
θ→1−ζl(θ, y) = y.
(b) lim
θ→1_{2}−

ζl(θ, y) = Ψλ(wy,l) : since Ψλ ∈ C Σ, S(N −1)−1 , we obtain lim
θ→1_{2}−

ζl(θ, y) =

Ψλ(wy,l) .

Thus, ζl(θ, y) ∈ C [0, 1] × S(N −1)−1, S(N −1)−1 and

ζl(0, y) = Ψλ(Φλ,l(y)) for all y ∈ S(N −1)−1,

ζl(1, y) = y for all y ∈ S(N −1)−1,

provided l > bl0. This completes the proof.

Theorem 5.9. For each λ ∈ (0, λ0) , functional Jλ has at least two critical points

in [Jλ < α∞] . In particular, equation (Eλ) has two positive solutions u (1)

0 and u (2) 0

Proof. Applying Lemmas 5.6 and 5.8, we have for λ ∈ (0, λ0) ,

cat ([Jλ ≤ α∞− σl]) ≥ 2.

By Corollary 2.6 and Lemmas 5.5 and 5.7, Jλ(u) has at least two critical points in

[Jλ < α∞] . This implies, equation (Eλ) has two nontrivial nonnegative solutions u1

and u2 such that ui ∈ Nλ for i = 1, 2. Moreover, by the maximum principle, we

have ui > 0 in RN+.

6. Proof of Theorem 1.1

Given a positive real number r0 > max

n
2
p−2,
q
p−q
o
. Let
Λ0 = min
r0(p − 2) − 2
2 kak_{∞}(r0+ 1)
, r0(p − q) − q
q kak_{∞}(r0+ 1)
, λ0
,
where λ0 > 0 as in Lemma 5.3. Then we have the following results.

Lemma 6.1. We have
1
2(1 + λ kak∞)
r0 _{−} 1
p(1 + λ kak∞)
r0+1_{−}p − 2
2p > 0
and
1
q(1 + λ kak∞)
r0 _{−}1
p(1 + λ kak∞)
r0+1_{−}p − q
pq > 0
for all λ ∈ (0, Λ0) .
Proof. Let
k (λ) = 1
2(1 + λ kak∞)
r0 _{−} 1
p(1 + λ kak∞)
r0+1_{−} p − 2
2p .
Then k (0) = 0 and
k0(λ) = r0
2 (1 + λ kak∞)
r0−1_{kak}
∞−
r0 + 1
p (1 + λ kak∞)
r0_{kak}
∞
= kak_{∞}(1 + λ kak_{∞})r0−1 r0
2 −
r0+ 1
p (1 + λ kak∞)
> 0
for all λ ∈ (0, Λ0) . This implies that k (λ) > 0 or

1
2(1 + λ kak∞)
l0 _{−} 1
p(1 + λ kak∞)
r0+1_{−} p − 2
2p > 0 for all λ ∈ (0, Λ0) .
Similar to the argument we also have

1
q (1 + λ kak∞)
r0 _{−} 1
p(1 + λ kak∞)
r0+1_{−} p − q
pq > 0 for all λ ∈ (0, Λ0) .

This completes the proof. We define

Iλ(u) = sup t≥0

Jλ(tu) : B → R.

Then we have the following result.

Lemma 6.2. For each λ ∈ (0, Λ0) and u ∈ B we have

(1 + λ kak_{∞})−r0

I0(u) ≤ Iλ(u) ≤ I0(u) .

Proof. _{Let u ∈ B. Then, by Lemmas 2.4 and 6.1, and (3.23)}
Iλ(u) = sup
t≥0
Jλ(tu) ≥ Jλ(t0(u) u)
≥ 1
2
Z
RN+
|∇t0(u) u|2 + (t0(u) u)2dx +
1
q
Z
∂RN
+
f |t0(u) u|qdσ
−(1 + λ kak∞)
p
Z
RN+
|t0(u) u|pdx
= 1
2 −
1 + λ kak_{∞}
p
Z
RN+
|∇t0(u) u|2+ (t0(u) u)2dx
+ 1
q −
1 + λ kak_{∞}
p
Z
∂RN
+
f |t0(u) u|
q
dσ
≥ (1 + λ kak_{∞})−r0
J0(t0(u) u) = (1 + λ kak∞)
−r0
I0(u) .
Moreover,

Jλ(tu) ≤ J0(tu) ≤ I0(u) for all t > 0.

Then Iλ(u) ≤ I0(u) . This completes the proof.

We observe that if λ is sufficiently small, the minimax argument in Section 4 also works for Jλ. Let l > max

n l0, bl0

o

be very large and let βλ = inf

γ∈Γy∈BmaxN −1_{(0,1)}Iλ(γ(y)).

where Γ is as in (3.26) . Then, by (3.25) and Lemma 6.2, for λ ∈ (0, Λ0), we have

(1 + λ kak_{∞})−r0

β0 ≤ βλ ≤ β0. (6.1)

Then we have the following result.

Theorem 6.3. There exists a positive number λ∗ ≤ Λ0 such that for λ ∈ (0, λ∗) ,

α∞< βλ < α∞+ αλ < 2α∞.

Furthermore, equation (Eλ) has a positive solution u (3)

0 such that Jλ

Proof. By Theorems 3.3 and 4.2, and Lemma 6.2, we also have that
(1 + λ kak_{∞})−r0

α∞≤ αλ < α∞.

For any ε > 0 there exists a positive number λ1 ≤ Λ0 such that for λ ∈ 0, λ1 ,

α∞− ε < αλ < α∞.

Thus,

2α∞− ε < α∞+ αλ < 2α∞.

Applying (6.1) for any δ > 0 there exists a positive number λ2 ≤ Λ0 such that for

λ ∈ 0, λ2 ,

β0− δ < βλ ≤ β0.

Moreover, by Lemma 3.7,

α∞< β0 < 2α∞.

Fix a small 0 < ε < 2α∞− β0, choosing a δ > 0 such that for λ ∈ (0, λ∗) we get,

α∞< βλ < 2α∞− ε < α∞+ αλ < 2α∞,

where λ∗ = minλ1, λ2 . Similar to the argument in the proof of Theorem 3.8, we

can conclude that the equation (Eλ) has a positive solution u (3)

0 such that Jλ

u(3)_{0} =
βλ. This completes the proof.

We can now complete the proof of Theorem 1.1: By Theorems 4.2 and 3.8, equa-tion (Eλ) has at least one positive solution for all λ ∈ [0, ∞). Moreover, by Theorems

5.9 and 6.3, there exists a positive number λ∗ such that for λ ∈ (0, λ∗) , equation

(Eλ) has three positive solutions u
(1)
0 , u
(2)
0 and u
(3)
0 with
0 < Jλ
u(i)_{0} < α∞ < Jλ
u(3)_{0} for i = 1, 2.
This completes the proof of Theorem 1.1.

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