Infinitely many solutions for two noncooperative
p(x)-Laplacian elliptic systems
∗
Wanmin Liu
†
Department of Mathematics, Lanzhou University, Lanzhou 730000, P. R. China
Abstract
The author deals with two noncooperative elliptic systems involving p(x)-
Laplacian in a smooth bounded domain and in R
N
respectively. With some
symmetry assumptions and growth conditions on nonlinearities, the existences
of infinitely many solutions are obtained by using a limit index theory devel-
oped by Li (Nonlinear Anal.: TMA, 25(1995) 1371) in variable exponent
Sobolev spaces.
Keywords: variable exponent Sobolev spaces; noncooperative elliptic sys-
tems; limit index theory; p(x)-Laplacian
Mathematics Subject Classification(2000): 35J50; 35B38
1 Introduction
The theory of variable exponent Lebesgue and Sobolev spaces has been developed
by several researchers in recent years. These spaces are natural generalization of the
classical Lebesgue space L
p
(Ω) and the Sobolev space W
k,p
(Ω). Although the study
of these spaces can go back to [21] and [20] as special cases of Musielak-Orlicz spaces,
the first paper systematically investing these spaces appeared in 1991 by Kov´aˇcik
and R´akosn´ık [17]. These spaces have been independently rediscovered by several
researchers based on different background. We refer to Samko [24], Fan and Zhao
[12], Acerbi and Mingione [1]. We also refer to three survey papers of these areas
by Harjulehto and H¨ast¨o [15], by Diening, H¨ast¨o and Nekvinda [4] and by Samko
[25]. Many applications have been found such as variational integrals with non-
standard growth conditions in nonlinear elasticity theory by Zhikov [31], models in
electrorheological fluids by R˚uˇziˇcka [23], and models in image restoration by Chen,
Levine and Rao [3].
∗
Project supported by National Natural Science Foundation of China (10671084)
†
Thesis(Master) – Lanzhou University (China), May 2007. Email: wanminliu@gmail.com
1
2
In this paper, we consider the following two noncooperative elliptic systems in-
volving p(x)-Laplacian in a smooth bounded domain Ω and in R
N
respectively:
4
p(x)
u = F
s
(x, u, v) in Ω,
−4
p(x)
v = F
t
(x, u, v) in Ω,
u|
∂Ω
= 0, v|
∂Ω
= 0;
(1.1)
(
4
p(x)
u − |u|
p(x)−2
u = G
s
(|x|, u, v) in R
N
,
−4
p(x)
v + |v|
p(x)−2
v = G
t
(|x|, u, v) in R
N
;
(1.2)
where 4
p(x)
u := div(|∇u|
p(x)−2
∇u) is called the p(x)-Laplacian, F (x, s, t) ∈ C
1
(
¯
Ω ×
R
2
, R), G(r, s, t) ∈ C
1
([0, ∞) × R
2
, R), F
s
=
∂F
∂s
and similar to F
t
, G
s
, G
t
.
Many results about p(x)-Laplacian equations with Dirichlet boundary conditions
([11, 5, 6]), Neumann boundary conditions ([19]) and in R
N
cases ([8, 28]) have
been obtained by variational approach and sub-supersolution method. Acerbi and
Mingione [1] have obtained the local C
1,α
regularity of minimizers of the integral
functional with p(x)-growth conditions under the assumption that p(x) is H¨older
continuous. The global regularity results have also been obtained by Fan [7]. There
are some results about elliptic systems. Hamidi [14] considered the following system
−4
p(x)
u = F
s
(x, u, v) in Ω,
−4
q(x)
v = F
t
(x, u, v) in Ω,
u|
∂Ω
= 0, v|
∂Ω
= 0;
(1.3)
and obtained the existence of solution since the integral functional of (1.3) is coercive
and satisfies mountain pass geometry under some assumptions on F . The author also
gave the multiplicity results by using the Fountain theorem when some symmetry
condition on F is assumed. Zhang [29] considered the system
−4
p(x)
u = f(v) in Ω,
−4
p(x)
v = g(u) in Ω,
u|
∂Ω
= 0, v|
∂Ω
= 0;
(1.4)
on a bounded radial symmetric domain with p(x) radial symmetric, and proved
the existence of a positive solution under some assumptions by sub-supersolution
method. He also considered the existence of solutions for weighted p(r)-Laplacian
system boundary value problems via Leray-Schauder degree in [30].
The main difficulties we meet here are that the corresponding integral functionals
of (1.1) and (1.2) are strongly indefinite. In addition to the nonlinearity of p(x)-
Laplacian operator, we also lose a compact embedding theorem in R
N
case. Thanks
to a limit index theory developed by Li [18] and the principle of symmetric criticality
3
due to Palais [22] in R
N
case, we can obtain the existence of infinitely many solutions
of problems (1.1) and (1.2) in the spaces W
1,p(x)
0
(Ω) and W
1,p(x)
(R
N
) respectively
under some nature assumptions on the nonlinearities.
Let us denote by c or c
i
some generic positive constants which may be different
throughout the paper.
Below are the assumptions.
(P1) p(x) ∈ C(
¯
Ω) and 1 < inf
Ω
p(x) := p
−
≤ p
+
:= sup
Ω
p(x) < ∞.
(P2) p(x) = p(|x|) := p(r) ∈ C
0,1
(R
N
) with 1 < inf
R
N
p(x) := p
−
≤ p
+
:=
sup
R
N
p(x) < N.
(F1) F ∈ C
1
(
¯
Ω × R
2
, R).
(F2) |F
s
(x, s, t)| + |F
t
(x, s, t)| ≤ c
1
+ c
2
(|s|
r(x)−1
+ |t|
r(x)−1
) where r(x) ∈ C(
¯
Ω),
2 ≤ r(x) < p
∗
(x).
p
∗
(x) :=
(
Np(x)
(N−p(x))
, if p(x) < N,
∞, if p(x) ≥ N.
(F3) ∃M > 0 and µ > p
+
such that
0 < µF (x, s, t) ≤ sF
s
(x, s, t) + tF
t
(x, s, t),
for all (x, s, t) ∈
¯
Ω×R
2
with s
2
+t
2
≥ M
2
. In this case, F (x, s, t) ≥ c
1
(|s|
µ
+|t|
µ
)−c
2
.
(F4) sF
s
(x, s, t) ≥ 0, for all (x, s, t) ∈
¯
Ω × R
2
.
(F5) F (x, −s, −t) = F (x, s, t), for all (x, s, t) ∈
¯
Ω × R
2
.
(G1) G ∈ C
1
([0, ∞) × R
2
, R).
(G2) For some p(x) q(x) p
∗
(x),
|G
s
(|x|, s, t)| + |G
t
(|x|, s, t)| ≤ c
1
(|s|
p(x)−1
+ |t|
p(x)−1
) + c
2
(|s|
q(x)−1
+ |t|
q(x)−1
).
The symbol α(x) β(x) means inf
¯
Ω
(β(x) − α(x)) > 0.
(G3) ∃M > 0 and µ > p
+
such that
0 < µG(r, s, t) ≤ sG
s
(r, s, t) + tG
t
(r, s, t),
for all (r, s, t) ∈ [0, ∞) × R
2
with s
2
+ t
2
≥ M
2
.
(G4) |G
s
(|x|, s, t)| + |G
t
(|x|, s, t)| = o(|s|
p(x)−1
) + o(|t|
p(x)−1
) uniformly on R
N
as
s
2
+ t
2
→ 0.
(G5) sG
s
(r, s, t) ≥ 0, for all (r, s, t) ∈ [0, ∞) × R
2
.
(G6) G(r, −s, −t) = G(r, s, t), for all (r, s, t) ∈ [0, ∞) × R
2
.
The following are the main results.
Theorem 1.1 Suppose that (P1) and (F1)-(F5) are satisfied. Let Φ(u, v) be the in-
tegral functional of (1.1). Then problem (1.1) possesses a sequence of weak solutions
{±(u
n
, v
n
)} in W
1,p(x)
0
(Ω) × W
1,p(x)
0
(Ω) such that Φ(u
n
, v
n
) → +∞ as n → ∞.
4
Theorem 1.2 Suppose that (P2) and (G1)-(G6) are satisfied. Let Ψ(u, v) be the
integral functional of (1.2). Then problem (1.2) possesses a sequence of radial weak
solutions {±(u
n
, v
n
)} in W
1,p(x)
(R
N
) × W
1,p(x)
(R
N
) such that Ψ(u
n
, v
n
) → +∞ as
n → ∞. In addition, if N = 4 or N ≥ 6, the problem (1.2) possesses infinitely
many nonradial weak solutions.
Remark 1.3 The definitions of weak solution of (1.1) and (1.2) are given in Defi-
nition 3.1 and Definition 4.1 respectively.
Remark 1.4 When p(x) ≡ p (a constant), the corresponding results have been ob-
tained by Li in [18] and by Huang and Li in [16]. The aim of the present paper is
to generalize their results to general cases.
The paper is organized as follows. In Section 2.1 we do some preliminaries of the
space W
1,p(x)
0
(Ω) and W
1,p(x)
(R
N
), review some basic properties of p(x)-Laplacian
operator. In Section 2.2 we recall a limit index theory due to Li. In Section 3, we
prove Theorem 1.1. In Section 4, we prove Theorem 1.2.
2 Preliminaries
2.1 Variable exponent Sobolev spaces and p(x)-Laplacian
operator
Let Ω be an open subset of R
N
. In this subsection, without further assumption, Ω
could be R
N
. On the basic properties of the space W
1,p(x)
(Ω) we refer to [17, 12].
In the following we display some facts which we will use later.
Denote by S(Ω) the set of all measurable real functions defined on Ω, and el-
ements in S(Ω) that equal to each other almost everywhere are considered as one
element. Denote L
∞
+
(Ω) = {p ∈ L
∞
(Ω) : ess inf
Ω
p(x) := p
−
≥ 1}.
For p ∈ L
∞
+
(Ω), define
L
p(x)
(Ω) = {u ∈ S(Ω) :
Z
Ω
|u|
p(x)
dx < ∞},
with the norm
|u|
L
p(x)
(Ω
= |u|
p(x)
= inf{λ > 0 :
Z
Ω
|u/λ|
p(x)
dx ≤ 1};
and define
W
1,p(x)
(Ω) = {u ∈ L
p(x)
(Ω) : |∇u| ∈ L
p(x)
(Ω)},
with the norm
kuk
W
1,p(x)
(Ω)
= |u|
L
p(x)
(Ω)
+ |∇u|
L
p(x)
(Ω)
.
5
We denote by W
1,p(x)
0
(Ω) the closure of C
∞
0
(Ω) in W
1,p(x)
(Ω). Define
W
1,p(x)
r
(R
N
) = {u ∈ W
1,p(x)
(R
N
) : u is radially symmetric}.
Hereafter, we always assume that p(x) is continuous and p
−
> 1.
Proposition 2.1 ([17, 12]) The spaces L
p(x)
(Ω), W
1,p(x)
(Ω), W
1,p(x)
0
(Ω) and
W
1,p(x)
r
(R
N
) all are separable and reflexive Banach spaces.
Proposition 2.2 ([17, 12, 9]) The conjugate space of L
p(x)
(Ω) is L
p
o
(x)
(Ω), where
1
p(x)
+
1
p
o
(x)
= 1. For any u ∈ L
p(x)
(Ω) and v ∈ L
p
o
(x)
(Ω), the H¨older inequality holds:
Z
Ω
|uv|dx ≤ 2|u|
p(x)
|v|
p
o
(x)
. (2.1)
Remark 2.3 In the right of (2.1), the constant 2 is suitable, but not the best. The
best constant is given in [9] denoted by d
(p
−
,p
+
)
which only depends on p
−
and p
+
when p(x) is given and d
(p
−
,p
+
)
is smaller than
1
p
−
+
1
p
+
.
Proposition 2.4 ([12] Theorem 2.7) Suppose that Ω is a bounded domain. In
W
1,p(x)
0
(Ω) the Poincar´e inequality holds, that is, there exists a positive constant
c such that
|u|
p(x)
≤ c|∇u|
p(x)
, for all u ∈ W
1,p(x)
0
(Ω).
So |∇u|
p(x)
is an equivalent norm in W
1,p(x)
0
(Ω).
Remark 2.5 When Ω is a bounded domain, we denote by kuk := |∇u|
p(x)
as the
equivalent norm in W
1,p(x)
0
(Ω) in Section 3. In Section 4 we will use the following
equivalent norm on W
1,p(x)
(R
N
) also with the symbol kuk:
kuk := inf{λ > 0 :
Z
Ω
(|∇u|
p(x)
+ |u|
p(x)
)/λ
p(x)
dx ≤ 1}. (2.2)
Proposition 2.6 ([12] Theorem 1.3) Set ρ(u) =
R
Ω
|u(x)|
p(x)
dx. For u, u
k
in the
space L
p(x)
(Ω), we have
(1) |u|
p(x)
< 1 (= 1; > 1) ⇐⇒ ρ(u) < 1 (= 1; > 1);
(2) |u|
p(x)
> 1 ⇒ |u|
p
−
p(x)
≤ ρ(u) ≤ |u|
p
+
p(x)
; |u|
p(x)
< 1 ⇒ |u|
p
+
p(x)
≤ ρ(u) ≤ |u|
p
−
p(x)
;
(3) lim
k→∞
|u
k
|
p(x)
= 0 (= ∞) ⇐⇒ lim
k→∞
ρ(u
k
) = 0 (= ∞).
Proposition 2.7 Let X be the space W
1,p(x)
0
(Ω) or the space W
1,p(x)
(R
N
) with the
norm k · k as Remark 2.5. Set I(u) =
R
Ω
|∇u(x)|
p(x)
dx when Ω is bounded or
I(u) =
R
R
N
(|∇u(x)|
p(x)
+ |u(x)|
p(x)
)dx in the R
N
case respectively. If u, u
k
∈ X then
the similar conclusions of Proposition 2.6 hold for k · k and I(·).
6
Proposition 2.8 ([17, 12]) Let F : Ω × R → R satisfies Carath´eodory conditions,
and
|F (x, t)| ≤ a(x) + b|t|
p
1
(x)
p
2
(x)
, for all (x, t) ∈ Ω × R,
where a ∈ L
p
2
(x)
(Ω), b is a positive constant, p
1
, p
2
∈ L
∞
+
(Ω). Denote by N
F
the
Nemytsky operator defined by F , i.e.
(N
F
(u))(x) = F (x, u(x)),
then N
F
: L
p
1
(x)
(Ω) → L
p
2
(x)
(Ω) is a continuous and bounded map.
Proposition 2.9 ([10] Theorem 1.1.) If p: Ω → R is Lipschitz continuous and
p
+
< N, then for q ∈ L
∞
+
(Ω) with p(x) ≤ q(x) ≤ p
∗
(x), there is a continuous
embedding W
1,p(x)
(Ω) → L
q(x)
(Ω).
Proposition 2.10 ([5] Proposition 2.4.) Assume that the boundary of Ω possesses
the cone property and p ∈ C(
¯
Ω). If q ∈ C(
¯
Ω) and 1 ≤ q(x) < p
∗
(x) for x ∈
¯
Ω, then
there is a compact embedding W
1,p(x)
(Ω) → L
q(x)
(Ω).
Proposition 2.11 ([13] Theorem 3.1.) Suppose that p : R
N
→ R is a uniformly
continuous and radially symmetric function satisfying 1 < p
−
≤ p
+
< N. Then, for
any measurable function q : R
N
→ R with
p(x) q(x) p
∗
(x), for all x ∈ R
N
,
there is a compact embedding
W
1,p(x)
r
(R
N
) → L
q(x)
(R
N
).
Definition 2.12 On the space L
p(x)
(Ω) ∩ L
q(x)
(Ω), we define the norm |u|
p(x)∧q(x)
=
|u|
p(x)
+ |u|
q(x)
. On the space (L
p(x)
(Ω))
2
:= L
p(x)
(Ω) × L
p(x)
(Ω), we define the
norm |(u, v)|
p(x)
= inf{λ > 0 :
R
Ω
(|u|
p(x)
+ |v|
p(x)
)/λ
p(x)
dx ≤ 1}. On the space
(L
p(x)
(Ω))
2
∩(L
q(x)
(Ω))
2
, we define the norm |(u, v)|
p(x)∧q(x)
= |(u, v)|
p(x)
+|(u, v)|
q(x)
.
On the space L
p(x)
(Ω)+L
q(x)
(Ω), we define the norm |u|
p(x)∨q(x)
= inf{|v|
p(x)
+|w|
q(x)
:
v ∈ L
p(x)
(Ω), w ∈ L
p(x)
(Ω), u = v + w}.
Similar to Proposition 2.8, we have the following proposition.
Proposition 2.13 (1) Assume 1 ≤ p(x), r(x) < ∞, f ∈ C(Ω × R
2
) and
f(x, s, t) ≤ c
1
(|s|
p(x)
r(x)
+ |t|
p(x)
r(x)
).
Then for every (u, v) ∈ (L
p(x)
(Ω))
2
, f(·, u, v) ∈ L
r(x)
(Ω) and the operator
T
1
: (L
p(x)
(Ω))
2
→ L
r(x)
(Ω) : (u, v) 7→ f(x, u, v)
7
is continuous.
(2) Assume 1 ≤ p(x), r(x), q(x), s(x) < ∞, f ∈ C(Ω × R
2
) and
f(x, s, t) ≤ c
2
(|s|
p(x)
r(x)
+ |t|
p(x)
r(x)
) + c
3
(|s|
q(x)
s(x)
+ |t|
q(x)
s(x)
).
Then for every (u, v) ∈ (L
p(x)
(Ω))
2
∩ (L
q(x)
(Ω))
2
, f (·, u, v) ∈ L
r(x)
(Ω) + L
s(x)
(Ω) and
the operator
T
2
: (L
p(x)
(Ω))
2
∩ (L
q(x)
(Ω))
2
→ L
r(x)
(Ω) + L
s(x)
(Ω) : (u, v) 7→ f(x, u, v)
is continuous.
Now we display some basic properties of p(x)-Laplacian operators. Let X be
W
1,p(x)
0
(Ω) or W
1,p(x)
(R
N
). Consider the following two functionals:
J
1
(u) =
Z
Ω
1
p(x)
|∇u|
p(x)
dx, for all u ∈ X = W
1,p(x)
0
(Ω);
J
2
(u) =
Z
R
N
1
p(x)
(|∇u|
p(x)
+ |u|
p(x)
)dx, for all u ∈ X = W
1,p(x)
(R
N
).
We know that J
1
, J
2
∈ C
1
(X, R), and the p(x)-Laplacian operator is the derivative
operator of J
1
in the weak sense. We denote L = J
0
1
: X → X
∗
and T = J
0
2
:
X → X
∗
, then
hLu, ˜ui =
Z
Ω
|∇u|
p(x)−2
∇u∇˜udx, for all u, ˜u ∈ X, (2.3)
hT u, ˜ui =
Z
R
N
(|∇u|
p(x)−2
∇u∇˜u + |u|
p(x)−2
u˜u)dx, for all u, ˜u ∈ X. (2.4)
Proposition 2.14 ([11, 8])
(1) L, T : X → X
∗
are two continuous, bounded and strictly monotone operators.
(2) L, T : X → X
∗
are two mappings of type (S
+
). Here a map L is called of type
(S
+
) if we have the property that u
n
u in X and lim sup
n→∞
hL(u
n
) − L(u), u
n
−
ui ≤ 0, then u
n
→ u in X.
(3) L, T : X → X
∗
are two homeomorphisms.
2.2 A limit index theory due to Li
In this section, we will recall a limit index theory developed by Li [18]. Suppose Z
is a G -Banach space, where G is a topological group. For the definition of index i
we refer to [26] Definition 5.9.
8
Definition 2.15 An index i is said to satisfy the d-dimension property if there is a
positive integer d such that
i(V
dk
∩ S
1
) = k
for all dk-dimensional subspaces V
dk
∈ Σ := {A ⊂ Z : A is closed and gA =
A for all g ∈ G } such that V
dk
∩ Fix G = {0}, where S
1
is the unit sphere in Z.
Proposition 2.16 ([18] Lemma 2.3) Suppose Z = W
1
⊕ W
2
and dim W
1
= kd,
where W
j
is a G -invariant subspace, j = 1, 2. Let i be an index satisfying the
d-dimension property. If W
1
∩ Fix G = {0}, A ∈ Σ and i(A) > k, then A ∩ W
2
6= ∅.
Suppose U and V are G -invariant closed subspaces of Z such that Z = U ⊕ V ,
where V is infinite dimension and V = ∪
∞
j=1
V
j
. Here V
j
is a dn
j
-dimensional G -
invariant subspace of V , and V
1
⊂ V
2
⊂ . . . for j = 1, 2, . . . . Let Z
j
= U ⊕ V
j
, and
let A
j
= A ∩ Z
j
for all A ∈ Σ.
Definition 2.17 ([18] Definition 2.4) Let i be an index satisfying the d-dimension
property. A limit index i
∞
with respect to {Z
j
} induced by i is a mapping
i
∞
: Σ → Z ∪ {−∞, +∞}
given by
i
∞
(A) = lim sup
j→∞
(i(A
j
) − n
j
).
Proposition 2.18 ([18] Proposition 2.5) Let A, B ∈ Σ. Then i
∞
satisfies:
(1) A = ∅ ⇐⇒ i
∞
(A) = −∞;
(2) (Monotonicity) A ⊂ B ⇒ i
∞
(A) ≤ i
∞
(B);
(3) (Subadditivity) i
∞
(A ∪ B) ≤ i
∞
(A) + i(B);
(4) If V ∩ Fix G = {0}, then i
∞
(S
ρ
∩ V ) = 0, where S
ρ
= {z ∈ Z, kzk = ρ};
(5) If Y
0
and
˜
Y
0
are G -invariant closed subspaces of V such that V = Y
0
⊕
˜
Y
0
,
˜
Y
0
⊂ V
j
0
for some j
0
and dim
˜
Y
0
= dm, then i
∞
(S
ρ
∩ Y
0
) ≥ −m.
Definition 2.19 Let Z be a Banach space which has a decomposition Z = ∪
∞
j=1
Z
j
where Z
1
⊂ Z
2
· · · , dim Z
j
= dn
j
. A functional f ∈ C
1
(Z, R) is said to satisfy
the (P S)
∗
c
condition with respect to {Z
n
} at the level c ∈ R if any sequence {z
n
k
},
z
n
k
∈ Z
n
k
such that
f(z
n
k
) → c and k(f
n
k
)
0
(z
n
k
)k → 0 as n
k
→ ∞
possesses a subsequence which converges in Z to a critical point of f, where f
n
k
:=
f|
Z
n
k
.
9
Theorem 2.20 ([18] Corollary 4.4, [16] Theorem 2.7) Assume that
(B1) f ∈ C
1
(Z, R) is G -invariant;
(B2) there are G -invariant closed subspaces U and V such that V is infinite dimen-
sion and
Z = U ⊕ V ;
(B3) there is a sequence of G -invariant finite-dimensional subspaces
V
1
⊂ V
2
· · · ⊂ V
j
⊂ · · · , dim V
j
= dn
j
,
such that V = ∪
∞
j=1
V
j
;
(B4) there is an index i on Z satisfying the d-dimension property;
(B5) there are G -invariant subspaces Y
0
,
˜
Y
0
, Y
1
of V such that V = Y
0
⊕
˜
Y
0
, Y
1
,
˜
Y
0
⊂ V
j
0
for some
j
0
and dim
˜
Y
0
= dm ≤ dk = dim Y
1
;
(B6) there are α and β, α < β such that f satisfies (PS)
∗
c
with respect to Z
n
:=
U ⊕ V
n
,
for all c ∈ [α, β];
(B7)
(a) either Fix G ⊂ U ⊕ Y
1
or Fix G ∩ V = {0},
(b) there is ρ > 0 such that f(z) ≥ α, for all z ∈ Y
0
∩ S
ρ
,
(c) f(z) ≤ β, for all z ∈ U ⊕ Y
1
.
If i
∞
is the limit index induced by i, then the numbers
d
j
= sup
i
∞
(A)≥j
inf
z∈A
f(z)
are critical values of f and α ≤ d
−m
≤ d
−m−1
≤ · · · ≤ d
−k+1
≤ β. Moreover, if d =
d
l
= · · · = d
l+r
, r > 0, then i(K
c
) ≥ r + 1, where K
c
= {z ∈ Z; f
0
(z) = 0, f(z) = d}.
Proof. By Proposition 2.18(5), i
∞
(S
ρ
∩ Y
0
) ≥ −m thus α ≤ d
−m
. It is obvious that
d
−m
≤ d
−m−1
≤ · · · ≤ d
−k+1
. Let us turn to prove d
−k+1
≤ β. Let V
j
Y
1
be a fixed
G -invariant complementary subspace of Y
1
in V
j
, j ≥ j
0
. It is easy to obtain that
(V
j
Y
1
) ∩ Fix G = {0} since of (B7)(a). Suppose A ∈ Σ and i
∞
(A) ≥ −k + 1, there
must be some j such that i(A
j
)−n
j
> −k, that is i(A
j
) > n
j
−k. On the other hand,
we have dim(V
j
Y
1
) = d(n
j
− k). By Proposition 2.16 we get A
j
∩ (U ⊕ Y
1
) 6= ∅.
Then A ∩ (U ⊕ Y
1
) 6= ∅. By the definition of d
−k+1
and (B7)(c), we get d
−k+1
≤ β.
The proof that d
j
are critical values of f is the Theorem 4.1 in [18].
Remark 2.21 In [18] Corollary 4.4 and [16] Theorem 2.7, this theorem is stated
incorrectly , but the proof they gave there is essentially correct.
10
3 The bounded case
In this section, we always assume that (P1) is satisfied. Denote by X the space
W
1,p(x)
0
(Ω) with the norm kuk = |∇u|
p(x)
as in Remark 2.5. The integral functional
of (1.1) is
Φ(u, v) = −
Z
Ω
1
p(x)
|∇u|
p(x)
dx +
Z
Ω
1
p(x)
|∇v|
p(x)
dx − F(u, v),
where
F(u, v) :=
Z
Ω
F (x, u, v)dx, u, v ∈ X.
Definition 3.1 The pair (u, v) ∈ X × X is called a weak solution of (1.1) if
−
Z
Ω
|∇u|
p(x)−2
∇u∇˜udx +
Z
Ω
|∇v|
p(x)−2
∇v∇˜vdx
=
Z
Ω
F
s
(x, u, v)˜udx +
Z
Ω
F
t
(x, u, v)˜vdx, for all (˜u, ˜v) ∈ X × X. (3.1)
For simplicity, using the operator L defined in (2.3), we rewrite (3.1) as
h(−Lu, Lv), (˜u, ˜v)i = hF
0
(u, v), (˜u, ˜v)i, for all (˜u, ˜v) ∈ X × X,
where
h(−Lu, Lv), (˜u, ˜v)i := h−Lu, ˜ui + hLv, ˜vi,
and
hF
0
(u, v), (˜u, ˜v)i :=
Z
Ω
F
s
(x, u, v)˜udx +
Z
Ω
F
t
(x, u, v)˜vdx.
Lemma 3.2 Suppose F satisfies (F1) and (F2), then
(1) Φ, F ∈ C
1
(X × X, R) and
hΦ
0
(u, v), (˜u, ˜v)i = h(−Lu, Lv), (˜u, ˜v)i − hF
0
(u, v), (˜u, ˜v)i. (3.2)
In particular, each critical point of Φ is a weak solution of (1.1).
(2) F
0
: X × X → X
∗
× X
∗
is completely continuous.
Proof . The proof of (1) is routine. The proof of (2) relies on Proposition 2.10 and
we omit it.
As X is a separable and reflexive Banach space, there exist {e
j
}
∞
j=1
⊂ X and
{f
i
}
∞
i=1
⊂ X
∗
such that
X = span{e
j
|j = 1, 2, · · · }, X
∗
= span{f
i
|i = 1, 2, · · · }
W
∗
, and hf
i
, e
j
i = δ
ij
.
11
For convenience, we write X
n
= span{e
1
, · · · , e
n
}, X
⊥
n
= span{e
n+1
, · · · }. Now
set E = X × X, E
n
= X
n
× X
n
. Define a group of G = {ι, τ}
∼
=
Z
2
by setting
τ(u, v) = (−u, −v), ι(u, v) = (u, v). (3.3)
Let
Σ = {A ⊂ E : A is closed and (u, v) ∈ A ⇒ (−u, −v) ∈ A}. (3.4)
An index γ on Σ is defined by
γ(A) =
0 if A = ∅,
min{m ∈ Z
+
: ∃h ∈ C(A, R
m
\{0}) such that h(−u, −v) = −h(u, v)} ,
+∞ if such h dose not exist.
(3.5)
Then γ is an index satisfying 1-dimension property by Borsuk-Ulam Theorem (see
[26] Proposition II 5.2.). We can obtain a limit index γ
∞
with respect to {E
n
} from
γ.
Lemma 3.3 Assume that F satisfies (F1) and (F2). Then any bounded sequence
{(u
n
k
, v
n
k
)} such that
(u
n
k
, v
n
k
) ∈ E
n
k
, Φ(u
n
k
, v
n
k
) → c, k(Φ
n
k
)
0
(u
n
k
, v
n
k
)k → 0 as n
k
→ ∞ (3.6)
possesses a subsequence which converges in E to a critical point of Φ, where Φ
n
k
:=
Φ|
E
n
k
.
Proof . Since E is reflexive, going if necessary to a subsequence, we can assume that
u
n
k
u and v
n
k
v. Observing that E = ∪
∞
n=1
E
n
, we can choose (¯u
n
k
, ¯v
n
k
) ∈ E
n
k
such that ¯u
n
k
→ u and ¯v
n
k
→ v. Hence
lim
n
k
→∞
hΦ
0
(u
n
k
, v
n
k
), (u
n
k
− u, 0)i
= lim
n
k
→∞
hΦ
0
(u
n
k
, v
n
k
), (u
n
k
− ¯u
n
k
, 0)i + lim
n
k
→∞
hΦ
0
(u
n
k
, v
n
k
), (¯u
n
k
− u, 0)i
= lim
n
k
→∞
h(Φ
n
k
)
0
(u
n
k
, v
n
k
), (u
n
k
− ¯u
n
k
, 0)i = 0. (3.7)
Substituting (3.2) into (3.7) and noticing that F
0
is completely continuous, we obtain
lim
n
k
→∞
hLu
n
k
, u
n
k
− ui = 0. (3.8)
By computing the limit of hΦ
0
(u
n
k
, v
n
k
), (0, v
n
k
− v)i in the similar way using ¯v
n
k
,
we obtain
lim
n
k
→∞
hLv
n
k
, v
n
k
− vi = 0. (3.9)
12
From (3.8) and (3.9), we conclude that u
n
k
→ u and v
n
k
→ v since L is of type (S
+
).
It remains to show that (u, v) is a critical point of Φ. Taking arbitrarily (¯u
j
, ¯v
j
) ∈
E
j
, then for n
k
≥ j we have
hΦ
0
(u, v), (¯u
j
, ¯v
j
)i = hΦ
0
(u, v) − Φ
0
(u
n
k
, v
n
k
), (¯u
j
, ¯v
j
)i + h(Φ
n
k
)
0
(u
n
k
, v
n
k
), (¯u
j
, ¯v
j
)i.
(3.10)
Taking n
k
→ ∞ in the right side of (3.10), we obtain hΦ
0
(u, v), (¯u
j
, ¯v
j
)i = 0. Hence
Φ
0
(u, v) = 0.
Lemma 3.4 Suppose that F satisfies (F1)-(F4). Then the functional Φ satisfies
(P S)
∗
c
with respect to {E
n
} for each c.
Proof . By Lemma 3.3, we only need to prove that each sequence satisfying (3.6)
is bounded. We can assume that ku
n
k
k ≥ 1 and kv
n
k
k ≥ 1. From Proposition 2.7
and (F4), we have
ku
n
k
k ≥ h−(Φ
n
k
)
0
(u
n
k
, v
n
k
), (u
n
k
, 0)i
= hLu
n
k
, u
n
k
i +
Z
Ω
F
s
(x, u
n
k
, v
n
k
)u
n
k
dx ≥ ku
n
k
k
p
−
. (3.11)
So ku
n
k
k is bounded. On the other hand, from (F3), Proposition 2.7 and H¨older
inequality, we have
c
1
≥ Φ(u
n
k
, v
n
k
)
= −
Z
Ω
1
p(x)
|∇u
n
k
|
p(x)
dx +
Z
Ω
1
p(x)
|∇v
n
k
|
p(x)
dx − F(u
n
k
, v
n
k
)
≥ −
Z
Ω
1
p(x)
|∇u
n
k
|
p(x)
dx +
Z
Ω
1
p(x)
|∇v
n
k
|
p(x)
dx
−
1
µ
Z
Ω
(u
n
k
F
s
(x, u
n
k
, v
n
k
) + v
n
k
F
t
(x, u
n
k
, v
n
k
))dx
= −
Z
Ω
1
p(x)
|∇u
n
k
|
p(x)
dx +
Z
Ω
1
p(x)
|∇v
n
k
|
p(x)
dx −
1
µ
hF
0
(u
n
k
, v
n
k
), (u
n
k
, v
n
k
)i
= −
Z
Ω
1
p(x)
−
1
µ
|∇u
n
k
|
p(x)
dx +
Z
Ω
1
p(x)
−
1
µ
|∇v
n
k
|
p(x)
dx
+
1
µ
h(Φ
n
k
)
0
(u
n
k
, v
n
k
), (u
n
k
, v
n
k
)i
≥ −
1
p
−
−
1
µ
ku
n
k
k
p
+
+
1
p
+
−
1
µ
kv
n
k
k
p
−
−
2
µ
k(Φ
n
k
)
0
(u
n
k
, v
n
k
)k (ku
n
k
k + kv
n
k
k) . (3.12)
So kv
n
k
k is bounded. Thus {(u
n
k
, v
n
k
)} is a bounded sequence in E.
13
Proposition 3.5 ([8] Lemma 3.3) Assume that X = span{e
j
|j = 1, 2, · · · }, X
⊥
m
=
span{e
m+1
, · · · }, f : X → R is a weakly-strongly continuous and f(0) = 0. Then
δ
m
:= sup
u∈X
⊥
m
, kuk=1
|f(u)| → 0, as m → ∞.
Proof of Theorem 1.1. Note that Φ is invariant with respect to the action of G .
We shall verify that Φ satisfies the hypotheses of Theorem 2.20. Set E = U ⊕ V ,
where U = X ×{0} and V = {0} × X. Set Y
0
= {0} ×X
⊥
m
and Y
1
= {0} ×X
k
where
m and k are to be determined. Then Y
0
and Y
1
are G -invariant and codim
V
Y
0
= m,
dim Y
1
= k, Fix G = {(0, 0)}. So Fix G ∩ V = {(0, 0)} and (B7)(a) of Theorem 2.20
is satisfied. It remains to verify (b) and (c) of (B7).
First, we verify (b) of (B7). By (F3), we have
Φ(u, 0) = −
Z
Ω
1
p(x)
|∇u|
p(x)
dx − F(u, 0) ≤ −
1
p
+
Z
Ω
|∇u|
p(x)
dx − c
1
Z
Ω
|u|
µ
dx + c
2
.
Therefore sup
u∈X
Φ(u, 0) < +∞. Choose α such that α > sup
u∈X
Φ(u, 0).
If (0, v) ∈ Y
0
∩ S
ρ
(where ρ > 1 is to be determined), we have v ∈ X
⊥
m
and
kvk = ρ. Define f : X → R, f(v) = |v|
r(x)
. Since the embedding X → L
r(x)
(Ω) is
compact by Proposition 2.10, f is weakly-strongly continuous. By Proposition 3.5,
we have δ
m
→ 0 as m → ∞. By (F2) we obtain
Φ(0, v) =
Z
Ω
1
p(x)
|∇v|
p(x)
dx − F(0, v)
≥
1
p
+
Z
Ω
|∇v|
p(x)
dx − c
3
Z
Ω
|v|
r(x)
dx − c
4
≥
1
p
+
kvk
p
−
− c
3
|v|
r
+
r(x)
− c
4
≥
1
p
+
ρ
p
−
− c
3
δ
r
+
m
ρ
r
+
− c
4
.
Setting ρ = (
c
3
p
+
r
+
δ
r
+
m
p
−
)
1
p
−
−r
+
, we have
Φ|
Y
0
∩S
ρ
≥ (r
+
− p
−
)(p
+
r
+
)
r
+
p
−
−r
+
c
3
p
−
p
−
p
−
−r
+
δ
p
−
r
+
p
−
−r
+
m
− c
4
→ +∞ as m → ∞.
Next, we verify (c) of (B7). For each (u, v) ∈ U ⊕ Y
1
and kuk > 1, kvk > 1,
Φ(u, v) ≤ −
1
p
+
kuk
p
−
+
1
p
−
kvk
p
+
− c
5
Z
Ω
(|u|
µ
+ |v|
µ
)dx + c
6
≤
1
p
−
kvk
p
+
− c
5
Z
Ω
|v|
µ
dx + c
6
.
14
Since all norms are equivalent in the finite dimension space Y
1
, we get
Φ(u, v) ≤
1
p
−
kvk
p
+
− c
7
kvk
µ
+ c
8
.
Then we have sup Φ|
U⊕Y
1
< +∞ since µ > p
+
. Thus we can choose k > m and
β > α such that Φ|
U⊕Y
1
≤ β.
So
d
j
= sup
γ
∞
(A)≥j
inf
z∈A
Φ(z), −k + 1 ≤ j ≤ −m,
are critical values of Φ and α ≤ d
j
≤ β. Since α can be chosen arbitrarily large, Φ
has a sequence of critical values d
n
→ +∞.
4 The R
N
case
In this section, we always assume that (P2) is satisfied and denote X by W
1,p(x)
(R
N
)
with the norm kuk defined by (2.2) and denote X
r
by W
1,p(x)
r
(R
N
) with the same
norm. The integral functional of (1.2) is
Ψ(u, v) = −
Z
R
N
1
p(x)
(|∇u|
p(x)
+ |u|
p(x)
)dx +
Z
R
N
1
p(x)
(|∇v|
p(x)
+ |v|
p(x)
)dx − G(u, v),
where
G(u, v) :=
Z
R
N
G(|x|, u, v)dx, u, v ∈ X.
Definition 4.1 (u, v) ∈ X × X is called a weak solution of (1.2) if
−
Z
R
N
(|∇u|
p(x)−2
∇u∇˜u + |u|
p(x)−2
u˜u)dx
+
Z
R
N
(|∇v|
p(x)−2
∇v∇˜v + |v|
p(x)−2
v˜v)dx
=
Z
R
N
G
s
(|x|, u, v)˜udx +
Z
R
N
G
t
(|x|, u, v)˜vdx, (4.1)
for all (˜u, ˜v) ∈ X × X.
Denote
h(T u, T v), (˜u, ˜v)i := hT u, ˜ui + hT v, ˜vi,
where T is defined as (2.4) and denote
hG
0
(u, v), (˜u, ˜v)i :=
Z
R
N
G
s
(|x|, u, v)˜udx +
Z
R
N
G
t
(|x|, u, v)˜vdx.
Then (4.1) can be rewritten as
h(−T u, T v), (˜u, ˜v)i = hG
0
(u, v), (˜u, ˜v)i, for all (˜u, ˜v) ∈ X × X.
15
Proposition 4.2 ([22] Principle of symmetric criticality) If u is a critical point
of Ψ|
X
r
×X
r
, then u is also a critical point of Ψ|
X×X
and thus a radially symmetric
solution of problem (1.2).
By the principle of symmetric criticality, to solve problem (1.2), we shall to find
the critical points of Ψ restricted on X
r
× X
r
using the limit index theory.
Lemma 4.3 Suppose G satisfies (G1)-(G4). Then
(1) Ψ, G ∈ C
1
(X
r
× X
r
, R) and
hΨ
0
(u, v), (˜u, ˜v)i = h(−T u, T v), (˜u, ˜v)i−hG
0
(u, v), (˜u, ˜v)i, for all (˜u, ˜v) ∈ X
r
×X
r
.
(4.2)
In particular, each critical point of Ψ is a weak solution of the problem (1.2).
(2) G
0
: X
r
× X
r
→ X
∗
r
× X
∗
r
is completely continuous.
Proof . (1) is obvious. Now we shall prove G
0
is continuous. Suppose (u
n
, v
n
) →
(u, v) ∈ X
r
× X
r
. By Proposition 2.9, we have (u
n
, v
n
) → (u, v) ∈ (L
p(x)
(R
N
))
2
∩
(L
q(x)
(R
N
))
2
. It follows from (G2) and Proposition 2.13(2) that
G
s
(|x|, u
n
, v
n
) → G
s
(|x|, u, v) in L
p
o
(x)
(R
N
) + L
q
o
(x)
(R
N
),
G
t
(|x|, u
n
, v
n
) → G
t
(|x|, u, v) in L
p
o
(x)
(R
N
) + L
q
o
(x)
(R
N
).
For all (˜u, ˜v) ∈ X
r
× X
r
, we obtain, by H¨older inequality (2.1),
|hG
0
(u
n
, v
n
), (˜u, ˜v)i − hG
0
(u, v), (˜u, ˜v)i|
≤
Z
R
N
|G
s
(|x|, u
n
, v
n
) − G
s
(|x|, u, v)||˜u|dx
+
Z
R
N
|G
t
(|x|, u
n
, v
n
) − G
t
(|x|, u, v)||˜v|dx
≤ 2|G
s
(|x|, u
n
, v
n
) − G
s
(|x|, u, v)|
p
o
(x)∨q
o
(x)
|˜u|
p(x)∧q(x)
+2|G
t
(|x|, u
n
, v
n
) − G
t
(|x|, u, v)|
p
o
(x)∨q
o
(x)
|˜v|
p(x)∧q(x)
,
where 1/p(x) + 1/p
o
(x) = 1, 1/q(x) + 1/q
o
(x) = 1. Thus
kG
0
(u
n
, v
n
) − G
0
(u, v)k
X
∗
r
×X
∗
r
→ 0
as n → ∞.
Now let us prove that G
0
is completely continuous. For any ε > 0, using (G2)
and (G4), we obtain C
ε
> 0 such that
|G
s
(|x|, s, t)| + |G
t
(|x|, s, t)| ≤ ε(|s|
p(x)−1
+ |t|
p(x)−1
) + C
ε
(|s|
q(x)−1
+ |t|
q(x)−1
).
16
Assume that (u
n
, v
n
) (u, v) in X
r
× X
r
. Since X
r
→ L
q(x)
(R
N
) is compact by
Proposition 2.11, we have (u
n
, v
n
) → (u, v) in (L
q(x)
(R
N
))
2
. By Proposition 2.13(1)
we have
G
s
(|x|, u
n
, v
n
) − ε(|u
n
|
p(x)−1
+ |v
n
|
p(x)−1
) → G
s
(|x|, u, v) − ε(|u|
p(x)−1
+ |v|
p(x)−1
)
in (L
q
o
(x)
(R
N
))
2
, and
G
t
(|x|, u
n
, v
n
) − ε(|u
n
|
p(x)−1
+ |v
n
|
p(x)−1
) → G
t
(|x|, u, v) − ε(|u|
p(x)−1
+ |v|
p(x)−1
)
in (L
q
o
(x)
(R
N
))
2
.
So we obtain
kG
s
(|x|, u
n
, v
n
) − G
s
(|x|, u, v)k + kG
t
(|x|, u
n
, v
n
) − G
t
(|x|, u, v)k
= sup
k˜uk≤1
Z
R
N
|G
s
(|x|, u
n
, v
n
) − G
s
(|x|, u, v)||˜u|dx
+ sup
k˜vk≤1
Z
R
N
|G
t
(|x|, u
n
, v
n
) − G
t
(|x|, u, v)||˜v|dx < cε.
Therefore G
0
is completely continuous.
Since X
r
is a separable and reflexive Banach space, there exist {e
j
}
∞
j=1
⊂ X
r
such
that (X
r
)
n
:= span{e
1
, · · · , e
n
} and (X
r
)
⊥
n
= span{e
n+1
, · · · }. Now set E = X
r
×X
r
and E
n
= (X
r
)
n
× (X
r
)
n
. As we have done in (3.3), (3.4) and (3.5), we can obtain
a limit index γ
∞
with respect to {E
n
}.
Lemma 4.4 Suppose that G satisfied (G1)-(G5). Then Ψ satisfies (P S)
∗
c
condition
with respect to {E
n
} for each c.
Proof . Lemma 3.3 is also suitable here if we replace Φ and L by Ψ and T respec-
tively. Thus we only need to prove each sequence satisfying
{(u
n
k
, v
n
k
)} ∈ E
n
k
, Ψ(u
n
k
, v
n
k
) → c, k(Ψ
n
k
)
0
(u
n
k
, v
n
k
)k → 0 as n
k
→ ∞,
is bounded where Ψ
n
k
:= Ψ|
E
n
k
. By (G5) and Proposition 2.7 similar to (3.11), we
have
ku
n
k
k ≥ h−(Ψ
n
k
)
0
(u
n
k
, v
n
k
), (u
n
k
, 0)i ≥ ku
n
k
k
p
−
.
So ku
n
k
k is bounded in X
r
. On the other hand, by (G3), similar to (3.12), we have
c
1
≥ −
1
p
−
−
1
µ
ku
n
k
k
p
+
+
1
p
+
−
1
µ
kv
n
k
k
p
−
−
2
µ
k(Ψ
n
k
)
0
(u
n
k
, v
n
k
)k (ku
n
k
k + kv
n
k
k) .
17
So kv
n
k
k is bounded in X
r
. Thus {(u
n
k
, v
n
k
)} is a bounded sequence in E.
Proof of Theorem 1.2. We shall find the critical points of Ψ in E by using
Theorem 2.20. By the assumption (G6), Ψ is invariant with respect to G . Set
E = U ⊕ V , where U = X
r
× {0} and V = {0} × X
r
. Set Y
0
= {0} × (X
r
)
⊥
m
and
Y
1
= {0} × (X
r
)
k
where m and k are to be determined. Then Y
0
and Y
1
are G -
invariant and codim
V
Y
0
= m, dim Y
1
= k, Fix G = {(0, 0)}. So Fix G ∩ V = {(0, 0)}
and (B7)(a) of Theorem 2.20 is satisfied. It remains to verify (b) and (c) of (B7).
First, we verify (b) of (B7). After integrating, we obtain from (G2)-(G4) the
existence of two positive constants c
1
and c
2
< 1/p
+
such that
G(|x|, s, 0) ≥ c
1
|s|
µ
− c
2
|s|
p(x)
, for all x ∈ R
N
, s ∈ R.
Hence, for all u ∈ X
r
, we have
Ψ(u, 0) = −
Z
R
N
1
p(x)
(|∇u|
p(x)
+ |u|
p(x)
)dx − G(u, 0)
≤ −
Z
R
N
1
p(x)
(|∇u|
p(x)
+ |u|
p(x)
)dx − c
1
Z
R
N
|u|
µ
dx + c
2
Z
R
N
|u|
p(x)
dx
< ∞.
Then we can choose α such that α > sup
u∈X
r
Ψ(u, 0).
If (0, v) ∈ Y
0
∩ S
ρ
(where ρ > 1 is to be determined), we have v ∈ (X
r
)
⊥
m
and kvk = ρ. Define f : X
r
→ R, f(v) = |v|
q(x)
. Since the compact embedding
X
r
→ L
q(x)
(Ω), f is weakly-strongly continuous. By Proposition 3.5, δ
m
→ 0 as
m → ∞. Then by (G2), (G3),
Ψ(0, v) =
Z
R
N
1
p(x)
(|∇v|
p(x)
+ |v|
p(x)
)dx − G(0, v)
≥
1
p
+
Z
R
N
(|∇v|
p(x)
+ |v|
p(x)
)dx − c
3
Z
R
N
|v|
q(x)
dx − c
4
≥
1
p
+
kvk
p
−
− c
3
|v|
q
+
q(x)
− c
4
≥
1
p
+
ρ
p
−
− c
3
δ
q
+
m
ρ
q
+
− c
4
.
Setting ρ = (
c
3
p
+
q
+
δ
q
+
m
p
−
)
1
p
−
−q
+
, we have
Ψ|
Y
0
∩S
ρ
≥ (q
+
− p
−
)(p
+
q
+
)
q
+
p
−
−q
+
c
3
p
−
p
−
p
−
−q
+
δ
p
−
q
+
p
−
−q
+
m
− c
4
→ +∞ as m → ∞.
18
Next, we verify (c) of (B7). For each (u, v) ∈ U ⊕ Y
1
, and kuk > 1, kvk > 1,
Ψ(u, v) ≤ −
1
p
+
kuk
p
−
+
1
p
−
kvk
p
+
− c
5
Z
Ω
(|u|
µ
+ |v|
µ
)dx + c
6
≤
1
p
−
kvk
p
+
− c
5
Z
Ω
|v|
µ
dx + c
6
.
Since all norms are equivalent in the finite dimension space Y
1
, we get
Ψ(u, v) ≤
1
p
−
kvk
p
+
− c
7
kvk
µ
+ c
8
.
Then we have sup Ψ|
U⊕Y
1
< +∞ since µ > p
+
. Thus we can choose k > m and
β > α such that Ψ|
U⊕Y
1
≤ β.
So
d
j
= sup
γ
∞
(A)≥j
inf
z∈A
Ψ(z), −k + 1 ≤ j ≤ −m,
are critical values of Ψ and α ≤ d
j
≤ β. Since α can be chosen arbitrarily large, Ψ
has a sequence of critical values d
n
→ +∞.
If N = 4 or N ≥ 6, using the Bartsch-Willem’s famous nonradial solutions result
in [2] (see also [27] Theorem 1.31), the problem (1.2) possesses infinitely many
nonradial solutions.
Acknowledgement.
The author would like to thank Professor Xian-Ling Fan for his valuable suggestions
and comments.
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