1 Introduction and Main Results
In this paper, we are concerned with the existence of ground state solutions for the following generalized quasilinear Schrödinger equation:
(1.1)
-
div
(
g
2
(
u
)
∇
u
)
+
g
(
u
)
g
′
(
u
)
|
∇
u
|
2
+
V
(
x
)
u
=
h
(
u
)
,
x
∈
ℝ
N
,
where N≥3, g:ℝ→ℝ+ is an even differentiable function such that g′(t)≥0 for all t≥0, h∈C1(ℝ,ℝ) is a nonlinear function, and the potential V(x):ℝN→ℝ is positive.
Solutions of (1.1) are related to the solitary wave solutions for quasilinear Schrödinger equations of the form
(1.2)
i
∂
t
z
=
-
Δ
z
+
W
(
x
)
z
-
h
(
z
)
-
Δ
l
(
|
z
|
2
)
l
′
(
|
z
|
2
)
z
,
where z:ℝ×ℝN→ℂ, W:ℝN→ℝ is a given potential, and l,h:ℝ→ℝ are suitable functions. The quasilinear equation (1.2) models several physical phenomena and has been the subject of extensive study in recent years. When l(s)=s, (1.2) models the time evolution of the condensate wave function in super-fluid film [20, 21]. This equation has been called the superfluid film equation in fluid mechanics by Kurihara [20]. In the case l(s)=(1+s)12, problem (1.2) models the self-channeling of a high-power ultrashort laser in matter; the propagation of a high-irradiance laser in a plasma creates an optical index depending nonlinearly on the light intensity and leads to new interesting nonlinear wave equations (see [2, 4, 7, 30]).
Set z(t,x)=exp(-iEt)u(x), where E∈ℝ and u is a real function. Then (1.2) can be reduced to the following equation of elliptic type:
(1.3)
-
Δ
u
+
V
(
x
)
u
-
Δ
l
(
u
2
)
l
′
(
u
2
)
u
=
h
(
u
)
,
x
∈
ℝ
N
.
If we take
g
2
(
u
)
=
1
+
(
l
(
u
2
)
′
)
2
2
,
then (1.3) turns into (1.1).
Problem (1.1) also arises in biological models and propagation of laser beams when g(u) is a positive constant. If we set g2(u)=1+2u2, i.e., l(s)=s, we get the superfluid film equation in plasma physics
(1.4)
-
Δ
u
+
V
(
x
)
u
-
Δ
(
u
2
)
u
=
h
(
u
)
,
x
∈
ℝ
N
.
If we set g2(u)=1+u22(1+u2), i.e., l(s)=(1+s)12, we get the equation
-
Δ
u
+
V
(
x
)
u
-
[
Δ
(
1
+
u
2
)
1
2
]
u
2
(
1
+
u
2
)
1
2
=
h
(
u
)
,
x
∈
ℝ
N
,
which models the self-channeling of a high-power ultrashort laser in matter.
Recent studies have been focused on problem (1.4) with h(u)∼|u|p-2u at infinity for 4≤p<22* and N≥3, where 2*=2N/(N-2) is the Sobolev critical exponent. In the spirit of [25], this case is called subcritical growth. A positive ground state solution of problem (1.4) has been found in [28] and [23]. To obtain it, a constrained minimization argument was used, which gives a solution to the equation with an unknown Lagrange multiplier λ in front of the nonlinear term. After that, Liu, Wang and Wang in [25] dealt with more general quasilinear problems like (1.4); the positive and sign-changing solutions have been obtained by using the Nehari manifold argument. In [24], by a change of variables, the quasilinear problem was transformed to a semilinear one, an Orlicz space framework was used as the working space, and a positive solution of problem (1.4) was obtained by using the mountain pass lemma (see, e.g., [1]). This argument was also used later in [6] by working in the usual Sobolev space H1(ℝN). Along this line, there have been a lot of works about the existence and multiplicity of solutions for (1.4) with subcritical growth; we refer the readers to [12, 33, 35] and the references therein.
However, to our knowledge, in the case of h(u)=|u|p-2u with lower power p∈(2,4), problem (1.4) has been less studied. In [31], Ruiz and Siciliano considered (1.4) under some hypotheses on V(x) and h(u)=|u|p-2u with 2<p<22*; a positive ground state solution was obtained by using a constrained minimization argument. Recently, Liu and Zhao in [5] dealt with (1.4) for some general hypotheses on the nonlinearity h and the potential V. Their results can be regarded as complements of the main result in [31].
As observed in [25], the number 22* behaves like a critical exponent for (1.4). In fact, it was shown in [25], by using a variational identity given by Pucci and Serrin [29], that (1.4) has no positive solutions in H1(ℝN) with u2|∇u|2∈L1(ℝN) if h(u)=|u|p-2u, p≥22* and V satisfies ∇V(x)⋅x≥0 in ℝN. As pointed by Liu, Wang and Wang in [24], the critical case for (1.4) is very interesting. Concerning this case, Moameni in [27] considered the related singularly perturbed problem and obtained a positive radial solution in the radially symmetric case. Later on, positive solutions were established in [11] via the mountain pass lemma. Recently, Liu, Liu and Wang obtained positive solutions for general quasilinear elliptic equations like (1.4) in [26] by a perturbation method.
We point out that all the results mentioned above mainly focus on the special case l(s)=s. An interesting question naturally arises: Is there a common method to study (1.3) with general functions l(s)?
By introducing a change of variables as follows:
(1.5)
v
=
G
(
u
)
=
∫
0
u
g
(
t
)
𝑑
t
,
Shen and Wang in [32] studied the existence of positive solitary wave solutions for (1.3) with a general function l(s). The existence of positive solutions for problem (1.1) was obtained under some assumptions on g, V and h when h is subcritical. By using the same change of variables and variational argument, the first author, Peng and Yan studied the generalized problem (1.1) with critical growth, and obtained the existence of positive solutions in [9].
In particular, the first author, Peng and Yan in [10] found that the critical exponents for problem (1.1) with general g(u) are α2* if limt→+∞g(t)tα-1=β>0 for some α≥1. Using this fact, they proposed the critical problem for given g(s) (or l(s)) as follows:
(1.6)
-
div
(
g
2
(
u
)
∇
u
)
+
g
(
u
)
g
′
(
u
)
|
∇
u
|
2
+
V
(
x
)
u
=
h
*
(
x
,
u
)
+
|
u
|
α
2
*
-
2
u
,
x
∈
ℝ
N
,
where h*:ℝN×ℝ→ℝ is a continuous function with subcritical growth. They established the existence of positive solutions for problem (1.6) if g satisfies the following assumption:
g
∈
C
1
(
ℝ
)
is an even positive function, with g′(t)≥0 for all t≥0 and g(0)=1. Moreover, there exist some constants α≥1, β>0 and γ∈(-∞,α) such that
(1.7)
g
(
t
)
=
β
t
α
-
1
+
O
(
t
γ
-
1
)
as
t
→
+
∞
and
(
α
-
1
)
g
(
t
)
≥
g
′
(
t
)
t
for all
t
≥
0
.
We point out that, in [10], the authors need the subcritical perburtation h*(x,u) satisfying the variant (AR) condition, i.e., there exists μ∈(2,2*) such that for any u>0 and x∈ℝN, we have
(1.8)
α
μ
∫
0
u
h
*
(
x
,
t
)
𝑑
t
≤
h
*
(
x
,
u
)
u
,
which usually plays a very important role in verifying that the functional has a mountain pass geometry and showing a related (PS) sequence is bounded. To guarantee that the variant (AR) condition holds, they must assume that the power p>2α if the subcritical perturbation h*(x,u)=|u|p-2u.
The main purpose of the present paper is to establish the existence of ground state solutions for problem (1.1) under the assumption (A1), and without assuming the variant (AR) condition (1.8).
We say a nontrivial weak solution u of (1.1) is a ground state solution if I(u)≤I(w) for any nontrivial solution w of (1.1), where I is defined in (1.9).
To establish our main result, we need the nonlinearity h satisfying the following assumptions:
h
∈
C
1
(
ℝ
,
ℝ
)
and h(t)≡0 for all t∈(-∞,0),
lim
t
→
0
+
h
(
t
)
t
=
0
,
lim
t
→
+
∞
h
(
t
)
t
α
2
*
-
1
=
K
>
0
,
there exist D>0 and 2<p<α2* such that h(t)≥Ktα2*-1+Dtp-1 for t≥0.
Moreover, we assume that the potential V∈C1(ℝN,ℝ) satisfies the following conditions:
0
<
V
0
≤
V
(
x
)
≤
V
∞
:=
lim
|
x
|
→
∞
V
(
x
)
for all x∈ℝN,
there exists a positive constant C0<(N-2)22 such that
|
(
∇
V
(
x
)
⋅
x
)
|
≤
C
0
|
x
|
2
for all
x
∈
ℝ
N
∖
{
0
}
.
We observe that the variational functional associated with (1.1)
I
(
u
)
=
1
2
∫
ℝ
N
g
2
(
u
)
|
∇
u
|
2
𝑑
x
+
1
2
∫
ℝ
N
V
(
x
)
u
2
𝑑
x
-
∫
ℝ
N
H
(
u
)
𝑑
x
may be not well defined in H1(ℝN). Moreover, the set {u∈H1(ℝN):∫ℝNg2(u)|∇u|2𝑑x<+∞} is not a linear space. To overcome this difficulty, we make use of the change of variables (1.5) to get
(1.9)
I
(
u
)
=
J
(
v
)
=
1
2
∫
ℝ
N
|
∇
v
|
2
d
x
+
1
2
∫
ℝ
N
V
(
x
)
|
G
-
1
(
v
)
|
2
d
x
-
∫
ℝ
N
H
(
G
-
1
(
v
)
)
d
x
,
where G-1(v) is the inverse function of G(u). Since g is a nondecreasing positive function, we get that |G-1(v)|≤1g(0)|v|. From this, it is clear that J is well defined in H1(ℝN) and J∈C1 if h and V satisfy the assumptions above.
If u is a nontrivial solution of (1.1), then it should satisfy
(1.10)
∫
ℝ
N
[
g
2
(
u
)
∇
u
∇
φ
+
g
(
u
)
g
′
(
u
)
|
∇
u
|
2
φ
+
V
(
x
)
u
φ
-
h
(
u
)
φ
]
𝑑
x
=
0
for all φ∈C0∞(ℝN). Let φ=1g(u)ψ. It can be checked that (1.10) is equivalent to
〈
J
′
(
v
)
,
ψ
〉
=
∫
ℝ
N
[
∇
v
∇
ψ
+
V
(
x
)
G
-
1
(
v
)
g
(
G
-
1
(
v
)
)
ψ
-
h
(
G
-
1
(
v
)
)
g
(
G
-
1
(
v
)
)
ψ
]
𝑑
x
=
0
for all
ψ
∈
C
0
∞
(
ℝ
N
)
.
Therefore, in order to find nontrivial solutions of (1.1), it suffices to study the following semilinear equation:
(1.11)
-
Δ
v
+
V
(
x
)
G
-
1
(
v
)
g
(
G
-
1
(
v
)
)
-
h
(
G
-
1
(
v
)
)
g
(
G
-
1
(
v
)
)
=
0
,
x
∈
ℝ
N
.
Now, our main result can be stated as follows.
Theorem 1.1.
Assume (A1), (A1)–(A2) and (h1)–(h4) hold. Then, for all D>0, there exists a positive ground state solution for problem (1.1), which is equivalent to (1.11), if either p>α(N+2)N-2+γ+ for 3≤N<6 or p>max{α(N+2)N-2+γ+,2α} for N≥6, where γ+=max{γ,0} and γ∈(-∞,α) is given by (1.7). Moreover, for problem (1.1) there still exists a positive ground state solution provide that 2<p<α2* for N≥3 and D>0 sufficiently large.
As a special case of Theorem 1.1, for h(t)=K|t|α2*-2t+D|t|p-2t, we can verify the following corollary.
Corollary 1.2.
Assume (A1), (A1)–(A2) and let h(t)=K|t|α2*-2t+D|t|p-2t. Then there exists a positive ground state solution for problem (1.1), which equivalent to (1.11), for all p∈(2,α2*) if D>0 is sufficiently large.
To prove our main result, we make use of the monotone method due to Jeanjean [17], where the author studied the problem of the form
-
Δ
u
+
K
¯
u
=
f
(
x
,
u
)
,
x
∈
ℝ
N
,
where K¯>0 is a constant and f(x,t) is asymptotically linear in t at infinity and periodic in xi, 1≤i≤N. To this end, we introduce a family of C1-functionals defined as
J
λ
(
v
)
=
1
2
∫
ℝ
N
(
|
∇
v
|
2
+
V
(
x
)
|
G
-
1
(
v
)
|
2
)
𝑑
x
-
λ
∫
ℝ
N
H
(
G
-
1
(
v
)
)
𝑑
x
for all
λ
∈
[
1
2
,
1
]
.
By (A1) and Proposition 2.6 below, for a.e. λ∈[12,1], there exists a bounded (PS)cλ sequence {vn}⊂H1(ℝN) for Jλ, where cλ is given in Lemma 4.1 below. To prove the convergence of the bounded (PS)cλ sequence for Jλ and obtain a nontrivial critical point vλ of Jλ with Jλ(vλ)=cλ for a.e. λ∈[12,1], we need to establish a version of the global compactness lemma related to the functional Jλ and its limiting functional
J
λ
∞
(
v
)
=
1
2
∫
ℝ
N
(
|
∇
v
|
2
+
V
∞
|
G
-
1
(
v
)
|
2
)
𝑑
x
-
λ
∫
ℝ
N
H
(
G
-
1
(
v
)
)
𝑑
x
.
Finally, choosing a sequence {(λm,vλm)}⊂[12,1]×H1(ℝN), with λm→1 as m→∞, we can prove that {vλm} is a bounded (PS) sequence for J=J1 due to the Pohozaev identity and Hardy’s inequality. By using the global compactness lemma again, we complete the proof of Theorem 1.1.
To apply the global compactness lemma, we first need to consider the existence of ground state solutions for the following equation:
(1.12)
-
Δ
v
+
V
∞
G
-
1
(
v
)
g
(
G
-
1
(
v
)
)
-
h
(
G
-
1
(
v
)
)
g
(
G
-
1
(
v
)
)
=
0
,
x
∈
ℝ
N
,
which can be viewed as the limiting equation of (1.11). We define the corresponding energy functional for the equation (1.12) by
(1.13)
J
∞
(
v
)
=
1
2
∫
ℝ
N
|
∇
v
|
2
d
x
+
1
2
∫
ℝ
N
V
∞
|
G
-
1
(
v
)
|
2
d
x
-
∫
ℝ
N
H
(
G
-
1
(
v
)
)
d
x
.
When h possesses general assumptions, it seems difficult to prove that J∞ possesses a bounded (PS) sequence by using the mountain pass lemma. To overcome this difficult, we need to use the technique developed in [15]. More precisely, by studying the behavior of J∞(u(e-θx)) for θ∈ℝ, we construct a (PS)c∞ sequence {vn} with the extra asymptotically property (see Proposition 3.2)
P
(
v
n
)
→
0
as
n
→
∞
,
where c∞ is the mountain pass level of J∞ and P(v)=0 is the Pohozaev identity related to problem (1.12). From this fact we can prove the boundedness of the (PS)c∞ sequence. Proceeding by the standard variational method, the existence of ground state solutions for equation (1.12) is established.
From the assumption of (A1), we can introduce an equivalent norm of H1(ℝN) defined as
∥
v
∥
=
(
∫
ℝ
N
(
|
∇
v
|
2
+
V
(
x
)
v
2
)
𝑑
x
)
1
2
.
And denote the usual Lebesgue space by Lq(ℝN) with norm ∥v∥q=(∫ℝN|v|qdx)1q, 1≤q<∞.
The outline of this paper is as follows. In Section 2, we provide some preliminary lemmas which will be used later. In Section 3, we analyze the limiting equation (1.12) and show the existence of positive ground state solutions. In Section 4, we employ the monotone method developed by Jeanjean in [17] to prove Theorem 1.1.
2 Some Preliminary Lemmas
In this section, we first give some properties for g and G. We can find the proof of the following lemma in [10].
Lemma 2.1.
The functions g(t) and G(t)=∫0tg(τ)𝑑τ enjoy the following properties under assumption (A1):
G
(
t
)
and
G
-
1
(
s
)
are odd.
For all
t
≥
0
, s≥0,
G
(
t
)
≤
g
(
t
)
t
,
G
-
1
(
s
)
≤
s
g
(
0
)
.
For all
s
≥
0
, G-1(s)s is nonincreasing and
lim
s
→
0
+
G
-
1
(
s
)
s
=
1
g
(
0
)
,
lim
s
→
+
∞
G
-
1
(
s
)
s
=
{
1
g
(
∞
)
if
g
is bounded
,
0
if
g
is unbounded.
For any
t
>
0
, we have
α
G
(
t
)
≥
g
(
t
)
t
𝑎𝑛𝑑
lim
t
→
+
∞
G
(
t
)
t
α
=
β
α
.
t
α
≤
β
α
G
(
t
)
for all
t
≥
0
, and there exist some constants
C
>
0
and
M
>
0
such that
0
≥
(
t
α
G
(
t
)
)
2
*
-
(
α
β
)
2
*
≥
-
C
G
(
t
)
-
(
1
-
γ
+
α
)
for
t
≥
M
,
where
γ
+
=
max
{
γ
,
0
}
.
For simplicity, we may assume that K=1 throughout this paper, where K is given in assumption (h3). Let
(2.1)
f
(
x
,
s
)
=
V
(
x
)
(
s
-
G
-
1
(
s
)
g
(
G
-
1
(
s
)
)
)
+
h
(
G
-
1
(
s
)
)
g
(
G
-
1
(
s
)
)
-
α
2
*
-
1
β
2
*
|
s
|
2
*
-
2
s
,
F
(
x
,
s
)
=
∫
0
s
f
(
x
,
τ
)
𝑑
τ
=
1
2
V
(
x
)
(
s
2
-
|
G
-
1
(
s
)
|
2
)
+
H
(
G
-
1
(
s
)
)
-
α
2
*
-
1
2
*
β
2
*
|
s
|
2
*
,
(2.2)
f
¯
(
s
)
=
V
∞
(
s
-
G
-
1
(
s
)
g
(
G
-
1
(
s
)
)
)
+
h
(
G
-
1
(
s
)
)
g
(
G
-
1
(
s
)
)
-
α
2
*
-
1
β
2
*
|
s
|
2
*
-
2
s
,
(2.3)
F
¯
(
s
)
=
∫
0
s
f
¯
(
τ
)
𝑑
τ
=
1
2
V
∞
(
s
2
-
|
G
-
1
(
s
)
|
2
)
+
H
(
G
-
1
(
s
)
)
-
α
2
*
-
1
2
*
β
2
*
|
s
|
2
*
.
Lemma 2.2.
The functions f(x,s), F(x,s), f¯(s) and F¯(s) enjoy the following properties under assumptions (A1), (A1) and (h1)–(h3):
lim
s
→
0
+
f
(
x
,
s
)
s
=
0
, lims→0+F(x,s)s2=0, lims→0+f¯(s)s=0 and lims→0+F¯(s)s2=0 uniformly in x∈ℝN.
lim
s
→
+
∞
f
(
x
,
s
)
s
2
*
-
1
=
0
, lims→+∞F(x,s)s2*=0, lims→+∞f¯(s)s2*-1=0 and lims→+∞F¯(s)s2*=0 uniformly in x∈ℝN.
Proof.
From (h2) and Lemma 2.1 (3), we get that
lim
s
→
0
+
f
(
x
,
s
)
s
=
V
(
x
)
(
1
-
1
g
2
(
0
)
)
+
lim
s
→
0
+
h
(
G
-
1
(
s
)
)
s
g
(
G
-
1
(
s
)
)
=
lim
s
→
0
+
h
(
G
-
1
(
s
)
)
G
-
1
(
s
)
⋅
G
-
1
(
s
)
s
g
(
G
-
1
(
s
)
)
=
0
,
by using the L’Hospital rule, it follows that lims→0+F(x,s)s2=0.
From (h3), Lemma 2.1 (4) and the fact that s=G(t), we have
lim
s
→
+
∞
h
(
G
-
1
(
s
)
)
s
2
*
-
1
g
(
G
-
1
(
s
)
)
=
lim
t
→
+
∞
h
(
t
)
g
(
t
)
G
(
t
)
2
*
-
1
=
lim
t
→
+
∞
h
(
t
)
t
α
2
*
-
1
⋅
t
α
2
*
-
1
g
(
t
)
G
(
t
)
2
*
-
1
=
α
2
*
-
1
β
2
*
,
which, together with (2.1), can verify that lims→+∞f(x,s)s2*-1=0. Furthermore, the L’Hospital rule gives that lims→+∞F(x,s)s2*=0.
By the same computation, we can deduce that f¯(s) and F¯(s) satisfy points (1) and (2). ∎
The following Pohozaev type identity can be deduced by a standard argument (see [8, Lemma 1.1]).
Lemma 2.3.
Assume (A1), (A1)–(A2) and (h1)–(h4) hold. Let v be a weak solution of problem (1.11). Then we have the following Pohozaev identity:
N
-
2
2
∫
ℝ
N
|
∇
v
|
2
d
x
+
N
2
∫
ℝ
N
V
(
x
)
|
G
-
1
(
v
)
|
2
d
x
+
1
2
∫
ℝ
N
(
∇
V
(
x
)
⋅
x
)
|
G
-
1
(
v
)
|
2
d
x
=
N
∫
ℝ
N
H
(
G
-
1
(
v
)
)
d
x
.
Using the Brezis–Lieb lemma in [3], we can prove the following lemma.
Lemma 2.4 (See [36, Lemma 2.2]).
Suppose that p∈C(RN×R) and there exists a constant T<+∞ such that
lim
s
→
0
|
p
(
x
,
s
)
s
|
≤
T
𝑎𝑛𝑑
lim
s
→
∞
p
(
x
,
s
)
|
s
|
2
*
-
1
=
0
uniformly in
x
∈
ℝ
N
.
Let {un}⊂H1(RN) be a bounded sequence and u∈H1(RN) with un⇀u in H1(RN). Then
lim
n
→
∞
[
∫
ℝ
N
P
(
x
,
u
n
)
𝑑
x
-
∫
ℝ
N
P
(
x
,
u
)
𝑑
x
-
∫
ℝ
N
P
(
x
,
u
n
-
u
)
𝑑
x
]
=
0
,
where P(x,s)=∫0sp(x,t)𝑑t.
The following general minimax principle is due to Willem [34].
Lemma 2.5 ([34, Theorem 2.8]).
Let X be a Banach space. Let M0 be a closed subspace of the metric space M and Γ0⊂C(M0,X). Define
Γ
≜
{
η
∈
C
(
M
,
X
)
:
η
|
M
0
∈
Γ
0
}
.
If φ∈C1(X,R) satisfies
∞
>
c
≜
inf
η
∈
Γ
sup
u
∈
M
φ
(
η
(
u
)
)
>
a
≜
sup
η
0
∈
Γ
0
sup
u
∈
M
0
φ
(
η
0
(
u
)
)
,
then, for every ε∈(0,(c-a)/2), δ>0 and η∈Γ such that supMφ∘η≤c+ε, there exists u∈X such that
c
-
2
ε
≤
φ
(
u
)
≤
c
+
2
ε
,
dist
(
u
,
η
(
M
)
)
≤
2
δ
,
∥
φ
′
(
u
)
∥
≤
8
ε
/
δ
.
On the existence of bounded (PS) sequences, we shall introduce the following abstract result developed by Jeanjean [17].
Proposition 2.6.
Let X be a Banach space equipped ∥⋅∥, and let L⊂R+ be an interval. We consider a family (Iλ)λ∈L of C1-functionals on X of the form
I
λ
(
u
)
=
A
(
u
)
-
λ
B
(
u
)
for all
λ
∈
L
,
where B(u)≥0 for all u∈X, and such that either A(u)→+∞ or B(u)→+∞ as ∥u∥→∞. We assume that there are two points (v1,v2) in X such that, by setting
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
X
)
:
γ
(
0
)
=
v
1
,
γ
(
1
)
=
v
2
}
,
we have, for all λ∈L,
c
λ
:=
inf
γ
∈
Γ
max
t
∈
[
0
,
1
]
I
λ
(
γ
(
t
)
)
>
max
{
I
λ
(
v
1
)
,
I
λ
(
v
2
)
}
.
Then, for almost every λ∈L, there exists a bounded (PS)cλ sequence in X. Moreover, the map λ↦cλ is continuous from the left.
3 The Limiting Problem
In this section, we consider the limiting equation (1.12). The norm on the H1(ℝN) is taken as
∥
v
∥
=
(
∫
ℝ
N
(
|
∇
v
|
2
+
V
∞
v
2
)
𝑑
x
)
1
2
.
From (2.2) and (2.3), we find that equation (1.12) can be rewritten as
(3.1)
-
Δ
v
+
V
∞
v
=
f
¯
(
v
)
+
α
2
*
-
1
β
2
*
|
v
|
2
*
-
2
v
,
x
∈
ℝ
N
,
and the functional J∞ defined in (1.13) can be rewritten as
J
∞
(
v
)
=
1
2
∫
ℝ
N
|
∇
v
|
2
d
x
+
1
2
∫
ℝ
N
V
∞
v
2
d
x
-
∫
ℝ
N
F
¯
(
v
)
d
x
-
α
2
*
-
1
2
*
β
2
*
∫
ℝ
N
|
v
|
2
*
d
x
.
In view of Lemma 2.3, if v∈H1(ℝN) is a weak solution of equation (1.12) or (3.1), the Pohozaev identity can be rewritten as follows:
P
(
v
)
≜
N
-
2
2
∫
ℝ
N
|
∇
v
|
2
d
x
+
N
2
∫
ℝ
N
V
∞
|
G
-
1
(
v
)
|
2
d
x
-
N
∫
ℝ
N
H
(
G
-
1
(
v
)
)
d
x
(3.2)
=
N
-
2
2
∫
ℝ
N
|
∇
v
|
2
d
x
+
N
2
∫
ℝ
N
V
∞
v
2
d
x
-
N
∫
ℝ
N
F
¯
(
v
)
d
x
-
N
α
2
*
-
1
2
*
β
2
*
∫
ℝ
N
|
v
|
2
*
d
x
=
0
.
We verify that the functional J∞ exhibits the mountain pass geometry.
Lemma 3.1.
The functional J∞ satisfies the following:
there exist
α
,
ρ
>
0
such that
J
∞
(
v
)
≥
α
for all
∥
v
∥
=
ρ
,
there exists
w
∈
H
1
(
ℝ
N
)
such that
∥
w
∥
>
ρ
and
J
∞
(
w
)
<
0
.
Proof.
From Lemma 2.2, we have that for any ε>0, there exists a Cε>0 such that F¯(v)≤εv2+Cε|v|2*. By choosing ε sufficiently small, the Sobolev inequality gives
J
∞
(
v
)
=
1
2
∫
ℝ
N
(
|
∇
v
|
2
+
V
∞
v
2
)
d
x
-
∫
ℝ
N
F
¯
(
v
)
d
x
-
α
2
*
-
1
2
*
β
2
*
∫
ℝ
N
|
v
|
2
*
d
x
≥
1
2
∫
ℝ
N
(
|
∇
v
|
2
+
V
∞
v
2
)
d
x
-
∫
ℝ
N
(
ε
v
2
+
C
ε
|
v
|
2
*
)
d
x
-
α
2
*
-
1
2
*
β
2
*
∫
ℝ
N
|
v
|
2
*
d
x
≥
C
∥
v
∥
2
-
C
∥
v
∥
2
*
.
If we choose ∥v∥=ρ small, then J∞(v)≥α>0, and we can get (i).
On the other hand, for given φ∈C0∞(ℝN,[0,1]) with supp(φ)=B¯1, from (h4) and Lemma 2.1 (2) and (5), we have
J
∞
(
t
φ
)
=
1
2
t
2
∫
ℝ
N
|
∇
φ
|
2
d
x
+
1
2
∫
ℝ
N
V
∞
|
G
-
1
(
t
φ
)
|
2
d
x
-
∫
ℝ
N
H
(
G
-
1
(
t
φ
)
)
d
x
≤
1
2
t
2
∫
ℝ
N
|
∇
φ
|
2
d
x
+
1
2
t
2
∫
ℝ
N
V
∞
φ
2
d
x
-
D
p
∫
ℝ
N
|
G
-
1
(
t
φ
)
|
p
d
x
-
1
α
2
*
∫
{
x
:
|
G
-
1
(
t
φ
)
|
≥
M
}
|
G
-
1
(
t
φ
)
|
α
2
*
d
x
≤
1
2
t
2
∫
ℝ
N
|
∇
φ
|
2
d
x
+
1
2
t
2
∫
ℝ
N
V
∞
φ
2
d
x
-
α
2
*
-
1
2
*
β
2
*
t
2
*
∫
{
x
:
|
G
-
1
(
t
φ
)
|
≥
M
}
|
φ
|
2
*
d
x
+
C
t
2
*
-
1
+
γ
+
α
+
∫
{
x
:
|
G
-
1
(
t
φ
)
|
≥
M
}
|
φ
|
2
*
-
1
+
γ
+
α
d
x
→
-
∞
as
t
→
+
∞
,
which gives (ii) if we take w=tφ with t sufficiently large. ∎
In view of Lemma 3.1, we can define the mountain pass level of J∞(v) by
(3.3)
c
∞
≜
inf
η
∈
Γ
∞
max
t
∈
[
0
,
1
]
J
∞
(
η
(
t
)
)
,
where
Γ
∞
=
{
η
∈
C
(
[
0
,
1
]
,
H
1
(
ℝ
N
)
)
:
η
(
0
)
=
0
,
η
(
1
)
=
w
}
.
Now, we will construct a (PS) sequence {vn}⊂H1(ℝN) for J∞(v) at the level c∞ with P(vn)→0 as n→∞.
Proposition 3.2.
There exists a sequence {vn}⊂H1(RN) such that
J
∞
(
v
n
)
→
c
∞
,
(
J
∞
)
′
(
v
n
)
→
0
𝑎𝑛𝑑
P
(
v
n
)
→
0
as
n
→
∞
.
Proof.
Following the idea in [15, 16], we define the map Φ:ℝ×H1(ℝN)→H1(ℝN), for θ∈ℝ, u∈H1(ℝN) and x∈ℝN, by Φ(θ,u)=u(e-θx). For every θ∈ℝ, u∈H1(ℝN), the functional J∞∘Φ is computed as
J
∞
∘
Φ
(
θ
,
u
)
=
1
2
e
(
N
-
2
)
θ
∫
ℝ
N
|
∇
u
|
2
d
x
+
1
2
e
N
θ
∫
ℝ
N
V
∞
|
G
-
1
(
u
)
|
2
d
x
-
e
N
θ
∫
ℝ
N
H
(
G
-
1
(
u
)
)
d
x
.
By Lemma 3.1, we can easily check that J∞∘Φ(θ,u)>0 for all (θ,u), with |θ|, ∥u∥H1(ℝN) being small and J∞∘Φ(0,w)<0, where w is given in Lemma 3.1. Hence, J∞∘Φ possesses the mountain pass geometry in ℝ×H1(ℝN). As a result, we can define the mountain pass level of J∞∘Φ by
c
~
∞
≜
inf
η
~
∈
Γ
~
∞
max
t
∈
[
0
,
1
]
J
∞
∘
Φ
(
η
~
(
t
)
)
,
where
Γ
~
∞
=
{
η
~
∈
C
(
[
0
,
1
]
,
ℝ
×
H
1
(
ℝ
N
)
)
:
η
~
(
0
)
=
(
0
,
0
)
,
η
~
(
1
)
=
(
0
,
w
)
}
.
As Γ∞={Φ∘η~:η~∈Γ~∞}, the mountain pass levels of J∞ and J∞∘Φ coincide, i.e., c∞=c~∞. By Lemma 2.5, we see that there exists a sequence {(θn,un)} in ℝ×H1(ℝN) such that, as n→∞,
(3.4)
J
∞
∘
Φ
(
θ
n
,
u
n
)
→
c
∞
,
(3.5)
(
J
∞
∘
Φ
)
′
(
θ
n
,
u
n
)
→
0
in
(
ℝ
×
H
1
(
ℝ
N
)
)
-
1
,
(3.6)
θ
n
→
0
.
Indeed, set ε=εn=1n2, δ=δn=1n in Lemma 2.5. Then (3.4) and (3.5) are direct conclusions of (a) and (c) of Lemma 2.5; we just need to verify (3.6). In view of the definition of c∞ in (3.3), for ε=εn=1n2, there exists ηn∈Γ∞ such that
max
t
∈
[
0
,
1
]
J
∞
(
η
n
(
t
)
)
≤
c
∞
+
1
n
2
.
Set η~n(t)=(0,ηn(t)). Then
max
t
∈
[
0
,
1
]
J
∞
∘
Φ
(
η
~
n
(
t
)
)
=
max
t
∈
[
0
,
1
]
J
∞
(
η
n
(
t
)
)
≤
c
∞
+
1
n
2
.
By Lemma 2.5 (b), there exists (θn,un)∈ℝ×H1(ℝN) such that dist((θn,un),(0,ηn(t)))≤2n. Then (3.6) holds.
Set vn=Φ(θn,un). We have
J
∞
(
v
n
)
→
c
∞
as
n
→
∞
.
Since, for any (h,w¯)∈ℝ×H1(ℝN),
(3.7)
〈
(
J
∞
∘
Φ
)
′
(
θ
n
,
u
n
)
,
(
h
,
w
¯
)
〉
=
〈
(
J
∞
)
′
(
v
n
)
,
Φ
(
θ
n
,
w
¯
)
〉
+
P
(
v
n
)
h
→
0
as
n
→
∞
,
taking (h,w¯)=(1,0) in (3.7), we have
P
(
v
n
)
→
0
as
n
→
∞
.
For any u∈H1(ℝN), set w¯(x)=u(eθnx), h=0 in (3.7). Then we get
〈
(
J
∞
)
′
(
v
n
)
,
u
〉
=
o
n
(
1
)
∥
u
(
e
θ
n
x
)
∥
=
o
n
(
1
)
∥
u
∥
for θn→0 as n→∞, where on(1)→0 as n→∞. Thus, we have
(
J
∞
)
′
(
v
n
)
→
0
in
H
-
1
(
ℝ
N
)
as
n
→
∞
.
Therefore, the lemma is proved. ∎
Lemma 3.3.
The sequence {vn}⊂H1(RN) given in Proposition 3.2 is bounded.
Proof.
To show the boundedness of {vn}, we see that
c
∞
+
o
n
(
1
)
=
J
∞
(
v
n
)
-
1
N
P
(
v
n
)
=
1
N
∫
ℝ
N
|
∇
v
n
|
2
d
x
,
which implies that ∫ℝN|∇vn|2dx≤C. Next, we only need to show the boundedness of ∫ℝNvn2𝑑x. In fact, recall that (J∞)′(vn)→0, and by Lemma 2.2, we have, for any ε>0, that there exists Cε>0 such that
∫
ℝ
N
|
∇
v
n
|
2
d
x
+
∫
ℝ
N
V
∞
v
n
2
d
x
=
∫
ℝ
N
f
¯
(
v
n
)
v
n
d
x
+
α
2
*
-
1
β
2
*
∫
ℝ
N
|
v
n
|
2
*
d
x
+
〈
(
J
∞
)
′
(
v
n
)
,
v
n
〉
≤
ε
2
∫
ℝ
N
v
n
2
d
x
+
C
ε
∫
ℝ
N
|
v
n
|
2
*
d
x
+
∥
(
J
∞
)
′
(
v
n
)
∥
H
-
1
(
ℝ
N
)
∥
v
n
∥
≤
ε
2
∫
ℝ
N
(
|
∇
v
n
|
2
+
v
n
2
)
d
x
+
C
ε
∫
ℝ
N
|
∇
v
n
|
2
d
x
+
ε
2
∥
v
n
∥
2
+
C
ε
∥
(
J
∞
)
′
(
v
n
)
∥
H
-
1
(
ℝ
N
)
2
.
By choosing ε>0 small enough, we obtain that {vn} is bounded in H1(ℝN). ∎
Now, we will give an appropriate estimate on the mountain pass value c∞. To this end, we introduce a well-known fact that the minimization problem S=inf{∥∇v∥22:v∈D1,2(ℝN),∥v∥2*=1} has a solution given by
w
ε
(
x
)
=
(
N
(
N
-
2
)
ε
)
N
-
2
4
(
ε
+
|
x
|
2
)
N
-
2
2
and
∥
∇
w
ε
∥
2
2
=
∥
w
ε
∥
2
*
2
*
=
S
N
2
.
Let φ∈C0∞(ℝN,[0,1]) be a radial cut-off function such that φ(x)=1 if |x|≤1, φ(x)=0 if |x|≥2 and |∇φ|≤2. Set ψε(x)=φ(x)wε(x). Then we get the following estimations (see [10]).
Lemma 3.4.
ψ
ε
(
x
)
satisfies the following estimations as ε→0:
∫
ℝ
N
|
∇
ψ
ε
|
2
d
x
=
S
N
2
+
O
(
ε
N
-
2
2
)
,
∫
ℝ
N
|
ψ
ε
|
2
*
d
x
=
S
N
2
+
O
(
ε
N
2
)
,
∫
ℝ
N
|
ψ
ε
|
d
x
≤
C
ε
N
-
2
4
,
∫
ℝ
N
|
ψ
ε
|
2
*
-
1
d
x
≤
C
ε
N
-
2
4
,
∫
ℝ
N
|
∇
ψ
ε
|
d
x
≤
C
ε
N
-
2
4
,
∫
ℝ
N
|
ψ
ε
|
2
d
x
=
{
C
ε
+
O
(
ε
N
-
2
2
)
for
N
≥
5
,
C
ε
|
ln
ε
|
+
O
(
ε
N
-
2
2
)
for
N
=
4
,
O
(
ε
1
2
)
for
N
=
3
,
∫
ℝ
N
|
ψ
ε
|
t
d
x
=
{
C
ε
N
2
-
t
(
N
-
2
)
4
for
t
>
N
N
-
2
,
C
ε
t
(
N
-
2
)
4
|
ln
ε
|
for
t
=
N
N
-
2
,
O
(
ε
t
(
N
-
2
)
4
)
for
t
<
N
N
-
2
.
Now we are ready to prove the following result.
Lemma 3.5.
Assume (A1) and (h1)–(h4) hold. Then, for all D>0, the level
c
∞
<
β
N
N
α
N
+
2
2
S
N
2
holds if either p>α(N+2)N-2+γ+ for 3≤N<6 or p>max{α(N+2)N-2+γ+,2α} for N≥6. Moreover, the same conclusion holds provided that 2<p<α2* for N≥3 and D>0 sufficiently large, where D is given in assumption (h4).
Proof.
Since J∞(0)=0 and limt→+∞J∞(tψε)=-∞, we can prove (see [9]) that for ε>0 small enough, there exists a constant tε>0 such that
J
∞
(
t
ε
ψ
ε
)
=
max
t
≥
0
J
∞
(
t
ψ
ε
)
and
(3.8)
0
<
A
1
<
t
ε
<
A
2
<
+
∞
,
where A1 and A2 are positive constants independent of ε.
Now, we are going to estimate J∞(tεψε). From (h4) and Lemma 2.1 (4) and (5), we have
J
∞
(
t
ε
ψ
ε
)
=
t
ε
2
2
∫
ℝ
N
|
∇
ψ
ε
|
2
d
x
+
1
2
∫
ℝ
N
V
∞
|
G
-
1
(
t
ε
ψ
ε
)
|
2
d
x
-
∫
ℝ
N
H
(
G
-
1
(
t
ε
ψ
ε
)
)
d
x
≤
t
ε
2
2
∫
ℝ
N
|
∇
ψ
ε
|
2
d
x
+
C
t
ε
2
∫
ℝ
N
ψ
ε
2
d
x
-
D
p
∫
{
x
:
|
G
-
1
(
t
ε
ψ
ε
)
|
≥
M
}
|
G
-
1
(
t
ε
ψ
ε
)
|
p
d
x
-
1
α
2
*
∫
{
x
:
|
G
-
1
(
t
ε
ψ
ε
)
|
≥
M
}
|
G
-
1
(
t
ε
ψ
ε
)
|
α
2
*
d
x
≤
t
ε
2
2
∫
ℝ
N
|
∇
ψ
ε
|
2
d
x
+
C
t
ε
2
∫
ℝ
N
ψ
ε
2
d
x
-
C
⋅
D
t
ε
p
α
∫
{
x
:
|
G
-
1
(
t
ε
ψ
ε
)
|
≥
M
}
|
ψ
ε
|
p
α
d
x
-
α
2
*
-
1
t
ε
2
*
2
*
β
2
*
∫
{
x
:
|
G
-
1
(
t
ε
ψ
ε
)
|
≥
M
}
|
ψ
ε
|
2
*
d
x
+
C
t
ε
2
*
-
1
+
γ
+
α
∫
{
x
:
|
G
-
1
(
t
ε
ψ
ε
)
|
≥
M
}
|
ψ
ε
|
2
*
-
1
+
γ
+
α
d
x
.
By a direct calculation, we have
(3.9)
∫
{
x
:
|
G
-
1
(
t
ε
ψ
ε
)
|
≥
M
}
|
ψ
ε
|
2
*
-
1
+
γ
+
α
d
x
≤
∫
ℝ
N
|
ψ
ε
|
2
*
-
1
+
γ
+
α
d
x
≤
C
ε
N
-
2
4
(
1
-
γ
+
α
)
.
By choosing ε>0 small enough such that, for |x|≤ε, ψε=wε≥Cε-N-24 and |G-1(tεψε)|≥M, one has
(3.10)
∫
{
x
:
|
G
-
1
(
t
ε
ψ
ε
)
|
≥
M
}
|
ψ
ε
|
p
α
d
x
≥
∫
B
ε
(
0
)
|
w
ε
|
p
α
d
x
=
O
(
ε
N
2
-
N
-
2
4
α
p
)
.
By choosing ε>0 small enough and fixing a small constant ε0>0 such that |G-1(tεψε)|≥M for |x|≤ε0ε4, we have that
(3.11)
∫
{
x
:
|
G
-
1
(
t
ε
ψ
ε
)
|
≥
M
}
|
ψ
ε
|
2
*
d
x
≥
∫
B
ε
0
ε
4
(
0
)
|
w
ε
|
2
*
d
x
=
S
N
2
+
O
(
ε
N
4
)
.
It follows from Lemma 3.4 and (3.8)–(3.11) that
J
∞
(
t
ε
ψ
ε
)
≤
(
t
ε
2
2
-
α
2
*
-
1
t
ε
2
*
2
*
β
2
*
)
S
N
2
+
C
ε
N
-
2
4
(
1
-
γ
+
α
)
-
C
⋅
D
ε
N
2
-
N
-
2
4
α
p
+
C
{
ε
+
O
(
ε
N
-
2
2
)
if
N
≥
5
,
ε
|
ln
ε
|
+
O
(
ε
)
if
N
=
4
,
O
(
ε
1
2
)
if
N
=
3
,
≤
β
N
N
α
N
+
2
2
S
N
2
+
C
ε
N
-
2
4
(
1
-
γ
+
α
)
-
C
⋅
D
ε
N
2
-
N
-
2
4
α
p
+
C
{
ε
+
O
(
ε
N
-
2
2
)
if
N
≥
5
,
ε
|
ln
ε
|
+
O
(
ε
)
if
N
=
4
,
O
(
ε
1
2
)
if
N
=
3
,
≜
β
N
N
α
N
+
2
2
S
N
2
+
I
.
In the following, we only need to prove that I<0 for small ε. By a simple computation, we have some cases as follows:
For the case when 3≤N<6, p>α(N+2)N-2+γ+ or N≥6, p>max{α(N+2)N-2+γ+,2α}, we can verify that I<0 for all D>0 as ε>0 is small.
For the case when 2<p<α2*, we can take D=ε-τ, which again gives that I<0 as ε>0 is small, where τ=N+24+(N-2)(γ+-2)4α for 3≤N<6 and τ=max{N+24+(N-2)(γ+-2)4α,N-22-N-22α} for N≥6.
As a result, the proof is completed. ∎
Proposition 3.6.
Assume (A1) and (h1)–(h4) hold. Then the limiting equation (1.12) has a nontrivial solution v~∈H1(RN).
Proof.
From Proposition 3.2 and Lemma 3.3, we deduce a bounded (PS)c∞ sequence {vn} for J∞, with P(vn)→0. Let
δ
=
lim sup
n
→
∞
sup
y
∈
ℝ
N
∫
B
1
(
y
)
|
v
n
|
2
d
x
.
If δ=0, then, by Lions’ compactness lemma [22],
(3.12)
v
n
→
0
in
L
q
(
ℝ
N
)
for
2
<
q
<
2
*
.
Since 〈(J∞)′(vn),vn〉=on(1), it follows form (3.12) that
∫
ℝ
N
|
∇
v
n
|
2
d
x
+
∫
ℝ
N
V
∞
v
n
2
d
x
-
α
2
*
-
1
β
2
*
∫
ℝ
N
|
v
n
|
2
*
d
x
=
o
n
(
1
)
.
Recalling that {vn}⊂H1(ℝN) is bounded, up to a subsequence, we obtain
∫
ℝ
N
|
∇
v
n
|
2
d
x
+
∫
ℝ
N
V
∞
v
n
2
d
x
→
α
2
*
-
1
β
2
*
l
≥
0
as
n
→
∞
,
where
l
=
lim
n
→
∞
∫
ℝ
N
|
v
n
|
2
*
d
x
.
It is easy to check that l>0. Otherwise, ∥vn∥→0 as n→∞, which contradicts c∞>0. From the definition of S, we have that
∫
ℝ
N
|
∇
v
n
|
2
d
x
+
∫
ℝ
N
V
∞
v
n
2
d
x
≥
∫
ℝ
N
|
∇
v
n
|
2
d
x
≥
S
(
∫
ℝ
N
|
v
n
|
2
*
d
x
)
2
2
*
.
Taking the limit as n→∞, we obtain l≥(β2*α2*-1S)N2. On the other hand, since J∞(vn)→c∞, we get
c
∞
=
1
2
∫
ℝ
N
|
∇
v
n
|
2
d
x
+
1
2
∫
ℝ
N
V
∞
v
n
2
d
x
-
α
2
*
-
1
2
*
β
2
*
∫
ℝ
N
|
v
n
|
2
*
d
x
+
o
n
(
1
)
=
(
1
2
-
1
2
*
)
α
2
*
-
1
β
2
*
l
≥
β
N
N
α
N
+
2
2
S
N
2
,
which contradicts Lemma 3.5. So δ>0 and there exists a sequence {yn}⊂ℝN such that
(3.13)
∫
B
1
(
y
n
)
v
n
2
𝑑
x
≥
δ
2
>
0
.
Let v~n(x)=vn(x+yn). Then {v~n} is still a bounded (PS)c∞ sequence for J∞ with P(v~n)→0. By (3.13), up to a subsequence, we may assume that there exists v~∈H1(ℝN)∖{0} such that
v
~
n
⇀
v
~
in
H
1
(
ℝ
N
)
,
v
~
n
→
v
~
in
L
loc
q
(
ℝ
N
)
,
1
≤
q
<
2
*
,
v
~
n
→
v
~
a.e. in
ℝ
N
.
Hence, v~ is a nontrivial solution of equation (1.12). ∎
Motivated by the idea in [18], we have the following lemma.
Lemma 3.7.
Let u be a nontrivial solution for equation (1.12). Then there exists a path η∈C([0,1],H1(RN)) such that η(0)=0, J∞(η(1))<0, u∈η([0,1]) and maxt∈[0,1]J∞(η(t))=J∞(u). Moreover, 0∉η((0,1]).
Proof.
For a nontrivial solution u∈H1(ℝN) for equation (1.12), we set
u
t
(
x
)
=
u
(
x
t
)
for
t
>
0
.
By the Pohozaev identity (3.2), we have
J
∞
(
u
t
)
=
1
2
t
N
-
2
∫
ℝ
N
|
∇
u
|
2
d
x
+
1
2
t
N
∫
ℝ
N
V
∞
|
G
-
1
(
u
)
|
2
d
x
-
t
N
∫
ℝ
N
H
(
G
-
1
(
u
)
)
d
x
=
(
1
2
t
N
-
2
-
1
2
*
t
N
)
∫
ℝ
N
|
∇
u
|
2
d
x
.
We can see easily that
max
t
>
0
J
∞
(
u
t
)
=
J
∞
(
u
)
,
J
∞
(
u
t
)
→
-
∞
as t→+∞,
∥
u
t
∥
2
=
t
N
-
2
∫
ℝ
N
|
∇
u
|
2
d
x
+
t
N
∫
ℝ
N
V
∞
u
2
d
x
→
0
as t→0.
We choose L>1 such that J∞(uL)<0 and set η(t)=uLt for t∈(0,1], η(0)=0. This is the desired path. ∎
We end this section by showing the existence of ground state solutions for the limiting equation (1.12). We say that ω(x) is a ground state solution of (1.12) if J∞(ω)=m∞, where
m
∞
=
inf
{
J
∞
(
u
)
:
u
∈
H
1
(
ℝ
N
)
∖
{
0
}
is a solution of (1.12)
}
.
Theorem 3.8.
The nontrivial solution v~∈H1(RN) given in Proposition 3.6 is a positive ground state solution of (1.12).
Proof.
Let v~ be the solution of (1.12) obtained in Proposition 3.6. By Proposition 3.2 and Fatou’s lemma, we have that
J
∞
(
v
~
)
=
J
∞
(
v
~
)
-
1
N
P
(
v
~
)
=
1
N
∫
ℝ
N
|
∇
v
~
|
2
d
x
≤
lim inf
n
→
∞
1
N
∫
ℝ
N
|
∇
v
~
n
|
2
d
x
=
lim inf
n
→
∞
[
J
∞
(
v
~
n
)
-
1
N
P
(
v
~
n
)
]
=
c
∞
.
On the other hand, as a consequence of Lemma 3.7, we know that J∞(v~)≥c∞. Hence, J∞(v~)=c∞.
Now we claim that v~ is the ground state solution of (1.12). Otherwise, there exists a nontrivial solution w of (1.12) satisfying J∞(w)<J∞(v~). By Lemma 3.7, there exists a path η∈C([0,1],H1(ℝN)) such that η(0)=0, J∞(η(1))<0, w∈η([0,1]) and maxt∈[0,1]J∞(η(t))=J∞(w). Moreover, 0∉η((0,1]). It follows from the definition of c∞ that J∞(w)≥c∞=J∞(v~), a contradiction. Hence, J∞(v~)=c∞=m∞. By the standard regularity argument and the strong maximum principle, we can show that v~ is positive. ∎
4 Proof of Main Result
This section is devoted to the case when the potential V(x) is not constant. Set L=[12,1]. We consider a family of C1-functionals on H1(ℝN)
J
λ
(
v
)
=
1
2
∫
ℝ
N
(
|
∇
v
|
2
+
V
(
x
)
|
G
-
1
(
v
)
|
2
)
𝑑
x
-
λ
∫
ℝ
N
H
(
G
-
1
(
v
)
)
𝑑
x
for all
λ
∈
[
1
2
,
1
]
.
Set
A
(
v
)
=
1
2
∫
ℝ
N
(
|
∇
v
|
2
+
V
(
x
)
|
G
-
1
(
v
)
|
2
)
𝑑
x
,
B
(
v
)
=
∫
ℝ
N
H
(
G
-
1
(
v
)
)
𝑑
x
.
We have Jλ(v)=A(v)-λB(v) and it is easy to see that B(v)≥0. Also, if ∥v∥=(∫ℝN(|∇v|2+V(x)v2)𝑑x)12→+∞, then either ∫ℝN|∇v|2dx→+∞ or ∫ℝNV(x)v2𝑑x→+∞. The former implies A(v)→+∞ directly. On the other hand, by Lemma 2.1 (2) and the Sobolev inequality, we have
∫
ℝ
N
V
(
x
)
v
2
𝑑
x
=
∫
{
x
:
|
v
(
x
)
|
≤
1
}
V
(
x
)
v
2
𝑑
x
+
∫
{
x
:
|
v
(
x
)
|
>
1
}
V
(
x
)
v
2
𝑑
x
≤
g
2
(
1
)
∫
{
x
:
|
v
(
x
)
|
≤
1
}
V
(
x
)
|
G
-
1
(
v
)
|
2
𝑑
x
+
∫
{
x
:
|
v
(
x
)
|
>
1
}
V
(
x
)
|
v
|
2
*
𝑑
x
≤
g
2
(
1
)
∫
{
x
:
|
v
(
x
)
|
≤
1
}
V
(
x
)
|
G
-
1
(
v
)
|
2
d
x
+
C
(
∫
ℝ
N
|
∇
v
|
2
d
x
)
2
*
2
,
which implies that the latter gives A(v)→+∞ as well.
For λ∈[12,1], let
f
λ
(
x
,
s
)
=
V
(
x
)
(
s
-
G
-
1
(
s
)
g
(
G
-
1
(
s
)
)
)
+
λ
h
(
G
-
1
(
s
)
)
g
(
G
-
1
(
s
)
)
-
λ
α
2
*
-
1
β
2
*
|
s
|
2
*
-
2
s
and
F
λ
(
x
,
s
)
=
∫
0
s
f
λ
(
x
,
t
)
𝑑
t
=
1
2
V
(
x
)
(
s
2
-
|
G
-
1
(
s
)
|
2
)
+
λ
H
(
G
-
1
(
s
)
)
-
λ
α
2
*
-
1
2
*
β
2
*
|
s
|
2
*
.
Hence, the functional Jλ can be rewritten as
J
λ
(
v
)
=
1
2
∫
ℝ
N
(
|
∇
v
|
2
+
V
(
x
)
v
2
)
d
x
-
∫
ℝ
N
F
λ
(
x
,
v
)
d
x
-
λ
α
2
*
-
1
2
*
β
2
*
∫
ℝ
N
|
v
|
2
*
d
x
.
Moreover, by Lemma 2.2, it is easy to verify that the functions fλ(x,s) and Fλ(x,s) satisfy
(4.1)
lim
s
→
0
+
f
λ
(
x
,
s
)
s
=
0
,
lim
s
→
0
+
F
λ
(
x
,
s
)
s
2
=
0
,
lim
s
→
+
∞
f
λ
(
x
,
s
)
s
2
*
-
1
=
0
and
lim
s
→
+
∞
F
λ
(
x
,
s
)
s
2
*
=
0
uniformly in x∈ℝN and λ∈[12,1].
By the standard argument, as in the proof of Lemma 3.1, we can verify that the functional Jλ has a mountain pass geometry.
Lemma 4.1.
Assume (A1), (A1) and (h1)–(h4) hold. Then, for all fixed λ∈[12,1], the functional Jλ satisfies the following:
there exists
w
∈
H
1
(
ℝ
N
)
∖
{
0
}
such that
J
λ
(
w
)
≤
0
,
c
λ
=
inf
η
∈
Γ
max
t
∈
[
0
,
1
]
J
λ
(
η
(
t
)
)
>
max
{
J
λ
(
0
)
,
J
λ
(
w
)
}
, where
Γ
=
{
η
∈
C
(
[
0
,
1
]
,
H
1
(
ℝ
N
)
)
:
η
(
0
)
=
0
,
η
(
1
)
=
w
}
.
From Theorem 3.8, we conclude that for any λ∈[12,1], the associated limiting equation of Jλ given as
(4.2)
-
Δ
v
V
∞
G
-
1
(
v
)
g
(
G
-
1
(
v
)
)
-
λ
h
(
G
-
1
(
v
)
)
g
(
G
-
1
(
v
)
)
=
0
,
x
∈
ℝ
N
,
has a positive ground state solution in H1(ℝN), that is, there exists uλ∞(x)>0, for all x∈ℝN, such that (Jλ∞)′(uλ∞)=0 and Jλ∞(uλ∞)=mλ∞, where
m
λ
∞
=
inf
{
J
λ
∞
(
u
)
:
u
∈
H
1
(
ℝ
N
)
∖
{
0
}
is a solution of (4.2)
}
and
J
λ
∞
(
v
)
=
1
2
∫
ℝ
N
(
|
∇
v
|
2
+
V
∞
|
G
-
1
(
v
)
|
2
)
𝑑
x
-
λ
∫
ℝ
N
H
(
G
-
1
(
v
)
)
𝑑
x
.
Motivated by the ideas in [19] or [13, 34], we can establish a version of the global compactness lemma related to the functional Jλ and its limiting functional Jλ∞.
Lemma 4.2.
Assume (A1), (A1)–(A2) and (h1)–(h4) hold. Let λ∈[12,1] be fixed and let {vn}⊂H1(RN) be a bounded (PS)cλ sequence for Jλ with cλ∈(0,λ1-N2βNSN2/(NαN+22)). Then there exist a subsequence of {vn}, still denoted by {vn}, an integer l∈N∪{0}, a sequence {ynk}⊂RN and wk∈H1(RN) for 1≤k≤l such that
v
n
⇀
v
in
H
1
(
ℝ
N
)
with
J
λ
′
(
v
)
=
0
,
|
y
n
k
|
→
+
∞
and
|
y
n
k
-
y
n
k
′
|
→
+
∞
for
k
≠
k
′
,
w
k
≠
0
and
(
J
λ
∞
)
′
(
w
k
)
=
0
for
1
≤
k
≤
l
,
∥
v
n
-
v
-
∑
k
=
1
l
w
k
(
⋅
-
y
n
k
)
∥
→
0
,
J
λ
(
v
n
)
→
J
λ
(
v
)
+
∑
k
=
1
l
J
λ
∞
(
w
k
)
,
where we agree that in the case l=0, the above hold without wk and {ynk}.
Proof.
Since {vn}⊂H1(ℝN) is bounded, we may assume, up to a subsequence, vn⇀v in H1(ℝN). So Jλ′(vn)→0 implies that for any ψ∈C0∞(ℝN),
〈
J
λ
′
(
v
n
)
,
ψ
〉
=
∫
ℝ
N
∇
v
n
∇
ψ
d
x
+
∫
ℝ
N
V
(
x
)
G
-
1
(
v
n
)
g
(
G
-
1
(
v
n
)
)
ψ
𝑑
x
-
λ
∫
ℝ
N
h
(
G
-
1
(
v
n
)
)
g
(
G
-
1
(
v
n
)
)
ψ
𝑑
x
=
o
n
(
1
)
∥
ψ
∥
as n→∞. By using the Lebesgue dominated theorem, we can obtain that Jλ′(v)=0. Thus, (i) holds.
By Lemma 2.3, the Pohozaev identity gives that
J
λ
(
v
)
=
1
N
∫
ℝ
N
|
∇
v
|
2
d
x
-
1
2
N
∫
ℝ
N
(
∇
V
(
x
)
⋅
x
)
|
G
-
1
(
v
)
|
2
d
x
.
From (A2) and Hardy’s inequality (see [14])
∫
ℝ
N
|
∇
v
|
2
d
x
≥
(
N
-
2
)
2
4
∫
ℝ
N
v
2
|
x
|
2
d
x
for all
v
∈
H
1
(
ℝ
N
)
,
we conclude that
1
N
∫
ℝ
N
|
∇
v
|
2
d
x
≥
(
N
-
2
)
2
4
N
∫
ℝ
N
v
2
|
x
|
2
d
x
≥
(
N
-
2
)
2
4
N
C
0
∫
ℝ
N
(
∇
V
(
x
)
⋅
x
)
v
2
d
x
>
1
2
N
∫
ℝ
N
(
∇
V
(
x
)
⋅
x
)
v
2
d
x
.
Therefore, from Lemma 2.1 (2), we obtain that
(4.3)
J
λ
(
v
)
≥
C
∫
ℝ
N
|
∇
v
|
2
d
x
≥
0
.
Step 1. Set vn1=vn-v. Then we have that
∥
v
n
1
∥
2
=
∥
v
n
∥
2
-
∥
v
∥
2
+
o
n
(
1
)
,
∥
v
n
1
∥
2
*
2
*
=
∥
v
n
∥
2
*
2
*
-
∥
v
∥
2
*
2
*
+
o
n
(
1
)
,
J
λ
(
v
n
)
-
J
λ
(
v
)
=
J
λ
∞
(
v
n
1
)
+
o
n
(
1
)
,
(
J
λ
∞
)
′
(
v
n
1
)
→
0
in H-1(ℝN).
The proof of (1a) and (1b) are standard and follow from the Brezis–Lieb lemma. By (4.1) and Lemma 2.4, we can check that
(4.4)
∫
ℝ
N
F
λ
(
x
,
v
n
1
)
𝑑
x
=
∫
ℝ
N
F
λ
(
x
,
v
n
)
𝑑
x
-
∫
ℝ
N
F
λ
(
x
,
v
)
𝑑
x
+
o
n
(
1
)
.
Combining (1a), (1b), (4.4) and the fact that vn1⇀0 in H1(ℝN), we deduce that
J
λ
(
v
n
)
-
J
λ
(
v
)
=
J
λ
(
v
n
1
)
+
o
n
(
1
)
=
J
λ
∞
(
v
n
1
)
+
o
n
(
1
)
,
which gives item (1c). Finally, we prove item (1d). By the elliptic estimate, we have v∈L∞(ℝN). From [34, Lemma 8.9], one has
(4.5)
|
∫
ℝ
N
(
|
v
n
|
2
*
-
2
v
n
-
|
v
|
2
*
-
2
v
-
|
v
n
-
v
|
2
*
-
2
(
v
n
-
v
)
)
ψ
𝑑
x
|
=
o
n
(
1
)
∥
ψ
∥
for all
ψ
∈
C
0
∞
(
ℝ
N
)
.
By a similar argument as that used in the proof of [34, Lemma 8.1], we also have
(4.6)
|
∫
ℝ
N
(
f
λ
(
x
,
v
n
)
-
f
λ
(
x
,
v
)
-
f
λ
(
x
,
v
n
1
)
)
ψ
𝑑
x
|
=
o
n
(
1
)
∥
ψ
∥
for all
ψ
∈
C
0
∞
(
ℝ
N
)
.
On the other hand, a direct computation shows that
〈
J
λ
′
(
v
n
)
-
J
λ
′
(
v
)
,
ψ
〉
=
∫
ℝ
N
(
(
∇
v
n
-
∇
v
)
∇
ψ
+
V
(
x
)
(
v
n
-
v
)
ψ
)
𝑑
x
-
∫
ℝ
N
(
f
λ
(
x
,
v
n
)
-
f
λ
(
x
,
v
)
)
ψ
𝑑
x
(4.7)
-
λ
α
2
*
-
1
β
2
*
∫
ℝ
N
(
|
v
n
|
2
*
-
2
v
n
-
|
v
|
2
*
-
2
v
)
ψ
𝑑
x
for all
ψ
∈
C
0
∞
(
ℝ
N
)
.
Combining (4.5)–(4.7) and the fact that vn1⇀0 in H1(ℝN), we have
〈
(
J
λ
∞
)
′
(
v
n
1
)
,
ψ
〉
=
〈
J
λ
′
(
v
n
1
)
,
ψ
〉
+
o
n
(
1
)
=
〈
J
λ
′
(
v
n
)
-
J
λ
′
(
v
)
,
ψ
〉
+
o
n
(
1
)
=
o
n
(
1
)
.
Hence, {vn1} is a (PS) sequence of Jλ∞.
Let
σ
1
=
lim sup
n
→
∞
sup
y
∈
ℝ
N
∫
B
1
(
y
)
|
v
n
1
|
2
d
x
.
Vanishing. If σ1=0, by Lions’ compactness lemma [22],
(4.8)
v
n
1
→
0
in
L
q
(
ℝ
N
)
for
2
<
q
<
2
*
.
Combining (1c), (1d) and (4.8), we deduce that
J
λ
(
v
n
)
-
J
λ
(
v
)
=
1
2
∥
v
n
1
∥
2
-
λ
α
2
*
-
1
2
*
β
2
*
∫
ℝ
N
|
v
n
1
|
2
*
d
x
+
o
n
(
1
)
and
∥
v
n
1
∥
2
=
λ
α
2
*
-
1
β
2
*
∫
ℝ
N
|
v
n
1
|
2
*
d
x
+
o
n
(
1
)
.
Without loss of generality, we may assume that
∥
v
n
1
∥
2
→
λ
α
2
*
-
1
β
2
*
ξ
≥
0
and
∫
ℝ
N
|
v
n
1
|
2
*
d
x
→
ξ
≥
0
.
By the Sobolev inequality, we have
∥
v
n
1
∥
2
≥
∫
ℝ
N
|
∇
v
n
1
|
2
d
x
≥
S
(
∫
ℝ
N
|
v
n
1
|
2
*
d
x
)
2
2
*
.
If ξ>0, then, by taking the limit as n→∞, we obtain ξ≥(β2*λα2*-1S)N2. From (4.3), we have that
c
λ
≥
J
λ
(
v
n
)
-
J
λ
(
v
)
+
o
n
(
1
)
=
(
1
2
-
1
2
*
)
λ
α
2
*
-
1
β
2
*
ξ
≥
λ
1
-
N
2
β
N
N
α
N
+
2
2
S
N
2
,
which is a contradiction. Hence, ξ=0. Then vn→v in H1(ℝN) and Lemma 4.2 holds with l=0. Non-vanishing. If σ1>0, there exists a sequence {yn1}⊂ℝN such that
∫
B
1
(
y
n
1
)
|
v
n
1
|
2
d
x
≥
σ
1
2
>
0
.
Set wn1=vn1(⋅+yn1). So {wn1} is bounded in H1(ℝN) and we may assume that wn1⇀w1 in H1(ℝN). Since
∫
B
1
(
0
)
|
w
n
1
|
2
d
x
≥
σ
1
2
,
we see that w1≠0. Moreover, vn1⇀0 in H1(ℝN) implies that {yn1} is unbounded. Hence, we may assume that |yn1|→+∞. We see that it is not difficult to verify that (Jλ∞)′(w1)=0. Step 2. Set vn2=vn-v-w1(⋅-yn1). We can similarly check that
∥
v
n
2
∥
2
=
∥
v
n
∥
2
-
∥
v
∥
2
-
∥
w
1
∥
2
+
o
n
(
1
)
,
∥
v
n
2
∥
2
*
2
*
=
∥
v
n
∥
2
*
2
*
-
∥
v
∥
2
*
2
*
-
∥
w
1
∥
2
*
2
*
+
o
n
(
1
)
,
J
λ
(
v
n
)
-
J
λ
(
v
)
-
J
λ
∞
(
w
1
)
=
J
λ
∞
(
v
n
2
)
+
o
n
(
1
)
,
(
J
λ
∞
)
′
(
v
n
2
)
→
0
in H-1(ℝN).
Similar to the argument made in step 1, let
σ
2
=
lim sup
n
→
∞
sup
y
∈
ℝ
N
∫
B
1
(
y
)
|
v
n
2
|
2
d
x
.
If vanishing occurs, by using the fact that Jλ∞(w1)>0 and the similar argument of the vanishing case in step 1, we know that ∥vn2∥→0, i.e., vn-v-w1(⋅-yn1)→0 in H1(ℝN). Moreover, by (2c), we see that Jλ(vn)+on(1)=Jλ(v)+Jλ∞(w1), and Lemma 4.2 holds with l=1.
If non-vanishing occurs, there exist a sequence {yn2}⊂ℝN and a nontrivial w2∈H1(ℝN) such that wn2=vn2(⋅+yn2)⇀w2 in H1(ℝN). It follows form (2d) that (Jλ∞)′(w2)=0. Furthermore, vn2⇀0 in H1(ℝN) implies that |yn2|→+∞ and |yn2-yn1|→+∞.
Finally, we can continue this procedure and obtain vnk=vnk-1-wk-1(⋅-ynk-1), with k≥2, such that vnk⇀0 in H1(ℝN), (Jλ∞)′(wk)=0. Furthermore, we have sequences {ynk}⊂ℝN such that |ynk|→+∞ and |ynk-ynk′|→+∞ for k≠k′. By the properties of the weak convergence, one has
(4.9)
∥
v
n
∥
2
-
∥
v
∥
2
-
∥
∑
k
=
1
l
w
k
(
⋅
-
y
n
k
)
∥
2
=
∥
v
n
-
v
-
∑
k
=
1
l
w
k
(
⋅
-
y
n
k
)
∥
2
+
o
n
(
1
)
and
J
λ
(
v
n
)
-
J
λ
(
v
)
-
∑
k
=
1
l
J
λ
∞
(
w
k
)
-
J
λ
∞
(
v
n
l
+
1
)
=
o
n
(
1
)
.
Since {vn}⊂H1(ℝN) is bounded and there exists ρ>0 such that ∥w∥>ρ for any nontrivial critical point of Jλ∞, (4.9) implies that the iteration stops at some finite l+1∈ℕ. The proof of this lemma is complete. ∎
On the convergence of bounded (PS) sequence {vn} for Jλ, we can establish the following result.
Lemma 4.3.
Assume (A1), (A1)–(A2) and (h1)–(h4) hold. Let λ∈[12,1] be fixed and {vn}⊂H1(RN) be a bounded (PS)cλ sequence for Jλ with cλ∈(0,λ1-N2βNSN2/(NαN+22)). Then there exist subsequence of {vn}, still denoted by {vn} and 0≠vλ∈H1(RN), such that
v
n
→
v
λ
in
H
1
(
ℝ
N
)
.
Proof.
By Lemma 4.2, for λ∈[12,1], there exists vλ∈H1(ℝN) such that vn⇀vλ in H1(ℝN) and Jλ′(vλ)=0. Moreover,
∥
v
n
-
v
λ
-
∑
k
=
1
l
w
k
(
⋅
-
y
n
k
)
∥
→
0
and
J
λ
(
v
n
)
→
J
λ
(
v
λ
)
+
∑
k
=
1
l
J
λ
∞
(
w
k
)
,
l
≥
0
,
where wk (1≤k≤l) are nontrivial critical points of Jλ∞.
Thus, to prove that vn→vλ in H1(ℝN), it suffices to show that l=0. Indeed, suppose by contradiction that l>0. Then, by (4.3), we deduce that
(4.10)
c
λ
=
lim
n
→
∞
J
λ
(
v
n
)
=
J
λ
(
v
λ
)
+
∑
k
=
1
l
J
λ
∞
(
w
k
)
≥
l
m
λ
∞
≥
m
λ
∞
.
On the other hand, let ωλ be a ground state solution of (4.2). Arguing as in Lemma 3.7, we can find a path η∈C([0,1],H1(ℝN)) such that η(0)=0, Jλ∞(η(1))<0, ωλ∈η([0,1]) and
max
t
∈
[
0
,
1
]
J
λ
∞
(
η
(
t
)
)
=
J
λ
∞
(
ω
λ
)
.
Moreover, 0∉η((0,1]). Since V(x)≤V∞ and V(x)≢V∞, it follows from the definition of cλ that
c
λ
≤
max
t
∈
[
0
,
1
]
J
λ
(
η
(
t
)
)
<
max
t
∈
[
0
,
1
]
J
λ
∞
(
η
(
t
)
)
=
J
λ
∞
(
ω
λ
)
=
m
λ
∞
,
which contradicts (4.10). Hence, l=0, i.e., vn→vλ in H1(ℝN) and vλ is a nontrivial critical point for Jλ with Jλ(vλ)=cλ. ∎
Combining Proposition 2.6 and Lemma 4.1, we deduce that for a.e. λ∈[12,1], there exists a bounded (PS) sequence {vn}⊂H1(ℝN) such that Jλ(vn)→cλ<λ1-N2βNSN2/(NαN+22). By Lemma 4.3, we deduce that Jλ has a nontrivial critical point vλ∈H1(ℝN) with Jλ(vλ)=cλ for a.e. λ∈[12,1]. As a special case, we obtain the existence of a sequence {(λm,vλm)}⊂[12,1]×H1(ℝN) with λm→1 as m→∞ and vλm≠0 satisfying
(4.11)
J
λ
m
′
(
v
λ
m
)
=
0
and
J
λ
m
(
v
λ
m
)
=
c
λ
m
,
where cλm∈(0,λm1-N2βNSN2/(NαN+22)). In order to prove Theorem 1.1, we need to show that the critical point sequence {vλm} obtained in (4.11) is bounded and that is a (PS) sequence for J=J1 satisfying limm→∞J(vλm)=c1, where J is given by (1.9). By applying Lemma 4.3 again, we obtain a nontrivial critical point of J and the proof is completed.
Proof of Theorem 1.1.
First, we show that the sequence {vλm}⊂H1(ℝN) obtained in (4.11) is bounded. Since Jλm(vλm)=cλm≤c12, from Lemma 2.3, we deduce that
c
1
2
≥
J
λ
m
(
v
λ
m
)
=
1
N
∫
ℝ
N
|
∇
v
λ
m
|
2
d
x
-
1
2
N
∫
ℝ
N
(
∇
V
(
x
)
⋅
x
)
|
G
-
1
(
v
λ
m
)
|
2
d
x
.
By a similar argument as that in the proof of (4.3), we can obtain ∫ℝN|∇vλm|2dx≤C. Next, we only need to show the boundedness of ∫ℝNvλm2𝑑x. In fact, recalling that Jλm′(vλm)=0 and (4.1), we have that for any ε>0, there exists Cε>0 such that
∫
ℝ
N
|
∇
v
λ
m
|
2
d
x
+
∫
ℝ
N
V
(
x
)
v
λ
m
2
d
x
=
∫
ℝ
N
f
λ
m
(
x
,
v
λ
m
)
v
λ
m
d
x
+
λ
m
α
2
*
-
1
β
2
*
∫
ℝ
N
|
v
λ
m
|
2
*
d
x
≤
ε
∫
ℝ
N
v
λ
m
2
d
x
+
C
ε
∫
ℝ
N
|
v
λ
m
|
2
*
d
x
,
and by choosing ε>0 small enough, we obtain that {vλm} is bounded in H1(ℝN). Thus, for any ψ∈C0∞(ℝN), we have
|
〈
J
λ
m
′
(
v
λ
m
)
,
ψ
〉
-
〈
J
′
(
v
λ
m
)
,
ψ
〉
|
=
|
(
λ
m
-
1
)
∫
ℝ
N
h
(
G
-
1
(
v
λ
m
)
)
g
(
G
-
1
(
v
λ
m
)
)
ψ
𝑑
x
|
→
0
as m→∞. On the other hand,
lim
m
→
∞
J
(
v
λ
m
)
=
lim
m
→
∞
[
J
λ
m
(
v
λ
m
)
+
(
λ
m
-
1
)
∫
ℝ
N
H
(
G
-
1
(
v
λ
m
)
)
𝑑
x
]
=
lim
m
→
∞
c
λ
m
=
c
1
,
where we used the fact that the map λ↦cλ is continuous from the left. Thus, {vλm} is a bounded (PS) sequence for J satisfying limm→∞J(vλm)=c1<βNSN2/(NαN+22). By applying Lemma 4.3 again, we obtain a nontrivial critical point v0∈H1(ℝN) for J and J(v0)=c1.
Finally, we end this proof by showing the existence of a ground state solution for problem (1.11), which is equivalent to (1.1). Let
d
=
inf
{
J
(
v
)
:
v
∈
H
1
(
ℝ
N
)
∖
{
0
}
is a solution of (1.11)
}
,
we can deduce that 0≤d≤J(v0)≤J∞(v0)<βNSN2/(NαN+22). In fact, for any v satisfying J′(v)=0, by a standard argument, we see ∥v∥≥ρ for some positive constant ρ. Similar to the proof of (4.3), we infer
(4.12)
J
(
v
)
≥
C
∫
ℝ
N
|
∇
v
|
2
d
x
.
Therefore, d≥0. In the following, we rule out d=0. Suppose by contradiction that {vn} is a critical point sequence of J satisfying limn→∞J(vn)=0. From (4.12), we have limn→∞∫ℝN|∇vn|2dx=0. Form this, together with 〈J′(vn),vn〉=0, we can verify that limn→∞∫ℝNvn2𝑑x=0. Therefore, we obtain limn→∞∥vn∥=0, which is a contradiction with ∥vn∥≥ρ>0 for all n∈ℕ.
Let {vn}⊂H1(ℝN) be a sequence of nontrivial critical point of J satisfying J(vn)→d<βNSN2/(NαN+22). Similarly, we can deduce that {vn} is bounded in H1(ℝN), i.e., {vn} is a bounded (PS)d sequence for J. Similar to the argument in the proof of Lemma 4.3, there exists a nontrivial w∈H1(ℝN) such that J(w)=d and J′(w)=0. Moreover, by using the strong maximum principle and standard regularity arguments, we show for the ground state solution w>0. ∎
,
Wentao Huang
