Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 170, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
NONLINEAR ELLIPTIC PROBLEM OF 2-q-LAPLACIAN TYPE
WITH ASYMMETRIC NONLINEARITIES
DANDAN YANG, CHUANZHI BAI
Abstract. In this article, we study the nonlinear elliptic problem of 2-qLaplacian type
−∆u − µ∆q u = −λ|u|r−2 u + au + b(u+ )θ−1
u=0
in Ω,
on ∂Ω,
where Ω ⊂ RN is a bounded domain. For a is between two eigenvalues, we
show the existence of three nontrivial solutions.
1. Introduction
In this article, we are interested in finding the multiple nontrivial weak solutions
to the nonlinear elliptic problem of 2-q-Laplacian type,
−∆u − µ∆q u = −λ|u|r−2 u + au + b(u+ )θ−1
u=0
in Ω,
on ∂Ω,
(1.1)
where Ω ⊂ RN is a bounded domain with samooth boundary ∂Ω, λ, µ > 0 are two
parameters, N > 2, 1 < min{q, r} ≤ max{q, r} < 2 < θ ≤ 2∗ = N2N
−2 , a ∈ R, b > 0,
and u+ = max{u, 0}. ∆q u = div(|∇u|q−2 ∇u) is the q-Laplacian of u.
Paiva and Presoto [12] studied the semilinear elliptic problem with asymmetric
nonlinearities,
−∆u = −λ|u|q−2 u + au + b(u+ )p−1
u=0
in Ω,
on ∂Ω.
(1.2)
Where N ≥ 3, 1 < q < 2 < p ≤ 2∗ , a ∈ R, b > 0 and λ is a positive parameter.
Problem (1.2) is also closely related to the class of superlinear Ambrosetti-Prodi
problems [6],
− ∆u = au + (u+ )p + f (x)
in Ω.
(1.3)
Further results for problem (1.3) can be found in [4, 5, 11, 14] and references cited
therein.
2000 Mathematics Subject Classification. 35J60, 35B38.
Key words and phrases. Quasilinear elliptic equations with q-Laplacian; critical exponent;
asymmetric nonlinearity; weak solution.
c
2014
Texas State University - San Marcos.
Submitted July 9, 2014. Published August 11, 2014.
1
2
D. YANG, C. BAI
EJDE-2014/170
Marano and Papageorgiou [10] obtained the existence of three solutions of the
(p, q)-Laplacian problem
−∆p u − µ∆q u = f (x, u)
u=0
in Ω,
on ∂Ω,
(1.4)
by using variational methods and truncation arguments. Nonlinear elliptic problems
involving the p-q-Laplacian operator is an active are of research; see [8, 9, 13, 15,
17, 18] and the references therein.
Motivated by the above works, we shall extend the results of problem (1.2) to
problem (1.1). By using variational methods, we obtain three solutions to (1.1).
We say that g is asymmetric when g satisfies the Ambrosetti-Prodi type condition
g− := lim
t→−∞
g(t)
g(t)
< λk < g+ := lim
.
t→+∞ t
t
Since problem (1.1) involves −∆ and −∆q , the arguments will be more complicated, and more analysis and estimates are needed.
The eigenvalue problem of the Laplacian, in Ω ⊂ RN , has the form
− ∆u = λu
in H01 (Ω).
(1.5)
By the Ljusternik-Schnirelman principle it is well known that there exists a nondecreasing sequence of nonnegative eigenvalues 0 < λ1 < λ2 ≤ · · · ≤ λj ≤ . . . and
a correspondent eigenfunctions ϕj . Also, the first eigenvalue λ1 is simple and the
eigenfunctions associated with λ1 do not change sign.
Now we are ready to state our main results.
Theorem 1.1. Let N ≥ 3, 1 < min{q, r} ≤ max{q, r} < 2 < θ < 2∗ and λk < a <
λk+1 . Then, for λ > 0 and µ > 0 small enough, problem (1.1) has at least three
nontrivial solutions.
Theorem 1.2. Let N ≥ 4, 1 < min{q, r} ≤ max{q, r} < 2 < θ = 2∗ and λk < a <
λk+1 . Then, for λ > 0 and µ > 0 small enough, problem (1.1) has at least three
nontrivial solutions.
This article is organized as follows. In Section 2, we show some geometric conditions to establish the Mountain-Pass levels and give a technical lemma which is
crucial in the proof of our main results. In Section 3, we establish the existence of
three nontrivial solutions for the nonlinear elliptic problem (1.1).
2. Preliminaries
In this article, k·kp and |·|p denote the norms on W01,p (Ω) and Lp (Ω), respectively;
Z
1/p
Z
1/p
kukp =
|∇u|p dx
, |u|p =
|u|p dx
.
Ω
Ω
For convenience, we substitute k · k for k · k2 . The best Sobolev constant S of the
∗
embedding H01 (Ω) ,→ L2 (Ω) is denoted by
S=
kuk2
2 .
u∈H0 (Ω)\{0} |u|2∗
inf
1
EJDE-2014/170
NONLINEAR ELLIPTIC PROBLEM OF 2-q-LAPLACIAN TYPE
3
It is known that S is independent of Ω and is never achieved except when Ω = RN
(see [16]). Consider the energy functional Iλ,µ defined on H01 (Ω) given by
Z
Z
Z
1
µ
λ
a
b
Iλ,µ (u) = kuk2 + kukqq +
|u|r dx −
|u|2 dx −
(u+ )θ dx.
(2.1)
2
q
r Ω
2 Ω
θ Ω
It is easy to know that Iλ,µ is of class C 2 and there exists a one to one correspondence between the weak solutions of (1.1) and the critical points of Iλ,µ on H01 (Ω).
By a weak solution of (1.1) we mean that u ∈ H01 (Ω) satisfying
Z
Z
0
hIλ,µ
(u), vi = [∇u∇v + µ|∇u|q−2 ∇u∇v]dx + λ
|u|r−2 uvdx
Ω
Ω
Z
Z
+ θ−1
−a
uvdx − b (u ) vdx = 0
Ω
Ω
H01 (Ω).
for all v ∈
Denote by ϕi a normalized eigenvector relative to eigenvalue λi of (1.5). Let
Vk = hϕ1 , . . . , ϕk i and Wk = Vk⊥ . Without loss of generality, we suppose 0 ∈ Ω,
and m ∈ N large enough so that B2/m ⊂ Ω, where B2/m denotes the ball of radius
2/m with center in 0. Consider the functions introduced in [7],


if x ∈ B1/m ,
0
ζm (x) = m|x| − 1 if x ∈ Am = B2/m \ B1/m ,


1
if x ∈ Ω \ B2/m .
Set ϕm
i = ζm ϕi ,
m
m
Vkm = hϕm
1 , ϕ2 , . . . , ϕk i
and Wkm = (Vkm )⊥ . For each m ∈ N, define a positive cut-off function η ∈
Cc∞ (B1/m ) such that η ≡ 1 in B1/2m , η ≤ 1 in B1/m and k∇ηk∞ ≤ 4m; take
ϕm
k+1 = ηϕk+1 . Then
supp u ∩ supp ϕm
(2.2)
k+1 = ∅
m
whenever u ∈ Vk . By [7], it is easy to check the following Lemma.
Lemma 2.1. As m → ∞ we have
ϕm
i → ϕi
in H01 (Ω)
and
max
R
u∈Vkm :
Ω
|u|2 =1
kuk2 ≤ λk + ck m2−N .
Corollary 2.2. For m large enough
Vkm ⊕ Wk = H01 .
(2.3)
As an easy consequence of Lemma 2.1 we have the following decomposition of
H01 .
Lemma 2.3. Assume λ1 < a, 1 < min{q, r} ≤ max{q, r} < 2 < θ ≤ 2∗ and
λ, µ > 0. Then every (PS) sequence of Iλ,µ is bounded.
Proof. Suppose {un } ⊂ H01 (Ω) is a (PS) sequence of Iλ,µ ; i.e., it satisfies
Z
Z
Z
1
µ
λ
a
b
2
q
r
2
θ
ku
k
+
ku
k
+
|u
|
dx
−
|u
|
dx
−
(u+
n
n q
n
n
n ) dx ≤ C,
2
q
r Ω
2 Ω
θ Ω
Z
Z
|un |r−2 un vdx
[∇un ∇v + µ|∇un |q−2 ∇un ∇v]dx + λ
Ω
Ω
Z
Z
θ−1
−a
un vdx − b (u+
vdx ≤ n kvk, ∀v ∈ H01 (Ω),
n)
Ω
Ω
(2.4)
(2.5)
4
D. YANG, C. BAI
EJDE-2014/170
where n → 0 as n → ∞. By (2.4) and (2.5), we obtain
C + n kun k
1
≥ Iλ (un ) − hIλ0 (un ), un i
2
Z
Z
µ µ
λ λ
b
b
θ
kun kqq +
−
−
|un |r dx +
−
(u+
)
dx
=
q
2
r
2 Ω
2 θ Ω n
Z
b
b
≥
−
(u+ )θ dx.
2 θ Ω n
(2.6)
Thus, we have
Z
θ
(u+
n ) dx ≤ C + n kun k.
(2.7)
Ω
Moreover, by Hölder inequality, we have
Z
Z
2/θ
θ−2
θ
+ 2
θ
.
(u+
(un ) dx ≤ |Ω|
n ) dx
(2.8)
Ω
Ω
On the other hand, by (2.5) we have
− 2
− q
0
− r
− 2
−
|hIλ,µ
(un ), u−
n i| = kun k + µkun kq + λ|un |r − a|un |2 ≤ n kun k,
with u− = max{−u, 0}. It follows from (2.4), (2.7), (2.8) and (2.9) that
Z
1 + 2
µ µ − q
λ λ
ku k ≤
−
kun kq +
−
|un |r
2 n
2
q
2
r Ω
Z
Z
b
1 0
a
2
(u+
)
dx
+
(u+ )θ dx + |hIλ,µ
(un ), u−
+
n i| + C
2 Ω n
θ Ω n
2
Z
Z
a
b
2
≤
(u+
)
dx
+
(u+ )θ dx + n ku−
nk+C
2 Ω n
θ Ω n
≤ n kun k + n ku−
n k + C.
(2.9)
(2.10)
1
Firstly, we show that (u+
n ) is bounded in H0 (Ω). Suppose by contradiction that
−
→ ∞, by (2.10), we know that (un ) is also unbounded. Let wn = un /kun k.
Since {wn } is bounded in H01 (Ω), there exists w ∈ H01 (Ω) such that
ku+
nk
wn * w
wn → w
in H01 (Ω),
in Ls , ∀1 ≤ s < 2∗ ,
wn → w
a.e. in Ω.
From (2.10), there exists σ > 0 satisfying
+ 2
ku−
n k ≥ σkun k
whenever n is large. Notice that
wn+ =
u+
u+
u+
n
n
n
=
,
+ 2
− 2 1/2 ≤
+ 2
4 1/2
kun k
(kun k + kun k )
(kun k + σ 2 ku+
nk )
which implies that w ≤ 0. Furthermore, by
wn− =
u−
u−
u−
ku−
n
n
n
nk
=
+ 2
− 2 1/2 =
− ·
+ 2
2 1/2
kun k
(kun k + kun k )
kun k (kun k + ku−
nk )
(2.11)
EJDE-2014/170
NONLINEAR ELLIPTIC PROBLEM OF 2-q-LAPLACIAN TYPE
5
and (2.11), we obtain kwn− k → 1. Hence, by (2.9),
−λ
q
r
2
ku−
|u−
|u−
n kq
n |r
n|
+
µ
− 2
− 2 +a
− 2 → 1.
kun k
kun k
kun k
(2.12)
Recalling that q, r < 2, we obtain
q
ku−
n kq
2−q
2 ku− kq−2 → 0,
n
− 2 ≤ |Ω|
kun k
r
2∗ −r
r
|u−
n |r
− 22r∗
−2
2∗ S
2∗
ku−
→
nk
− 2 ≤ |Ω|
kun k
(2.13)
0.
(2.14)
Moreover, by (2.11) and kwn− k → 1, we have
u−
u−
u−
kun k
n
n
n
−
=
−
− −1 →0
kun k kun k
kun k kun k
in H01 (Ω).
Thus we may exchange ku−
n k√for kun k in (2.12), and substituting (2.13) and (2.14)
into it, we obtain |wn− | → 1/ a, then w 6= 0. Taking v = ϕ1 in (2.5), one has
Z
Z
kun kq−1
q
[∇wn ∇ϕ1 dx + µ
|∇wn |q−2 ∇wn ∇ϕ1 dx
ku
k
n
Ω
Ω
Z
Z
Z
λ
b
r−2
+
|un | un ϕ1 dx − a
wn ϕ1 dx −
(u+ )θ−1 ϕ1 dx → 0;
kun k Ω
kun k Ω n
Ω
that is,
Z
kun kqq−1
|∇wn |q−2 ∇wn ∇ϕ1 dx
(λ1 − a)
wn ϕ1 dx + µ
kun k
Ω
Ω
(2.15)
Z
Z
λ
b
θ−1
+
|un |r−2 un ϕ1 dx −
(u+
)
ϕ
dx
→
0.
1
kun k Ω
kun k Ω n
Since the second, the third and the fourth term above approach zero, it follows that
Z
(λ1 − a)
wϕ1 dx = 0,
Z
Ω
which is a contradiction, as w ≤ 0, w 6= 0 and λ1 < a, so that (u+
n ) is bounded.
Finally, assume that kun k → ∞ and ku+
k
≤
C
for
all
n
∈
N.
Taking
v = wn in
n
(2.5), by (2.13) and
Z
1
(u+ )θ dx → 0,
kun k Ω n
for θ ≤ 2∗ , we obtain a|wn |22 → 1, so that wn → w in L2 (Ω) with w 6= 0. Then by
(2.5) we obtain
Z
Z
∇w∇vdx − a
wvdx = 0 for all v ∈ H01 (Ω),
Ω
Ω
with w 6= 0 and w ≤ 0, which is a contradiction, as a is not the first eigenvalue.
Hence, we conclude that {un } must be bounded in H01 (Ω).
In the subcritical case, 1 ≤ θ < 2∗ , we can easily know according to the lemma
above, Iλ,µ satisfies the (PS) condition at every level.
Lemma 2.4. Let λ1 < a and θ = 2∗ . For each λ, µ > 0, Iλ,µ satisfies the (PS)
2−N
condition at level c with c < N1 b 2 S N/2 .
6
D. YANG, C. BAI
EJDE-2014/170
Proof. Let {un } ⊂ H01 (Ω) be a sequence satisfying
0
Iλ,µ (un ) → c and |hIλ,µ
(un ), vi| ≤ n kvkp ,
∀v ∈ H01 (Ω),
(2.16)
with n → 0 as n → ∞. By Lemma 2.3 we obtain that {un } is bounded. Thus, by
passing to a subsequence, we have
un * u in H01 (Ω),
un → u in Ls , ∀1 ≤ s < 2∗ ,
(2.17)
un → u a.e. in Ω.
1
Since {u+
the Gagliardo-Nirenberg inequality it follows
n } is bounded in H0 (Ω), from
∗
+
that {un } is also bounded in L2 . By passing to a subsequence again, we have
+
2∗
u+
n * u in L . Hence, we obtain by [11, Lemma 2.3] that
−∆u − µ∆q u = −λ|u|r−2 u + au + b(u+ )2
u=0
∗
−1
,
in Ω
on ∂Ω,
(2.18)
Thus, by (2.18) we have
λ λ
µ µ
−
kukqq +
−
Iλ,µ (u) =
q
2
r
2
Z
b
b
|u| dx +
− ∗
2
2
Ω
r
Z
∗
(u+ )2 dx ≥ 0. (2.19)
Ω
Set wn = un − u. It is easy to check that
+ s
+ s
+ s
|u+
n − u |s ≤ |(un − u) |s = |wn |s ,
1 ≤ s ≤ 2∗ .
(2.20)
By (2.16) and the Brezis-Lieb Lemma, we have
∗
+ 2
kwn k2 + µkwn kqq + λ|wn |rr − a|wn |22 − b|u+
n − u |2∗
= kun k2 − kuk2 + µ(kun kqq − kukqq ) + λ(|un |rr − |u|rr )
2∗
+ 2∗
− a(|un |22 − |u|22 ) − b |u+
n |2∗ − |u |2∗ + on (1)
0
0
= hIλ,µ
(un ), un i − hIλ,µ
(u), ui + o(1),
which implies that
+ 2∗
lim kwn k2 + µkwn kqq + λ|wn |rr − a|wn |22 − b|u+
n − u |2∗ = 0.
n→∞
(2.21)
Moreover, by (2.17) we have wn → 0 in Lr and L2 . Thus, we have from (2.20) and
(2.21) that
∗
∗
+ 2
+ 2
kwn k2 + µkwn kqq = b|u+
n − u |2∗ + o(1) ≤ b|wn |2∗ + o(1).
(2.22)
Without loss of generality, we assume that
kwn k2 = d + o(1),
kwn kqq = h + o(1).
By (2.22), (2.23) and Sobolev inequality, we obtain
d + µh 2/2∗
∗
∗
≥ Sb−2/2 d2/2 .
d≥S
b
(2.23)
(2.24)
If d = 0, then we complete the proof. Otherwise, (2.24) implies that
d ≥ S N/2 b
2−N
2
.
(2.25)
EJDE-2014/170
NONLINEAR ELLIPTIC PROBLEM OF 2-q-LAPLACIAN TYPE
7
Then by (2.16), (2.19) and the Brezis-Lieb Lemma, we conclude
c ≥ c − Iλ,µ (u) = Iλ,µ (un ) − Iλ,µ (u) + o(1)
µ
1
=
kun k2 − kuk2 +
kun kqq − kukqq
2
q
λ
a
b
+ 2∗
2∗
+ (|un |rr − |u|rr ) −
|un |22 − |u|22 − ∗ |u+
n |2∗ − |u |2∗ + o(1)
r
2
2
1
µ
λ
a
b
+ 2∗
2
q
r
2
= kwn k + kwn kq + |wn |r − |wn |2 − ∗ |u+
n − u |2∗ + o(1).
2
q
r
2
2
(2.26)
Let n → ∞ in (2.26), we obtain by (2.22), (2.23), (2.25) and wn → 0 in Lr and L2
that
d µh d + µh
c≥ +
−
2
q
2∗
1
1
µ
µ
=
− ∗ d+
− ∗ h
2 2
q
2
1
1
−
d
≥
2 2∗
2−N
1
≥ S N/2 b 2 ,
N
which is a contradiction.
3. Main result
Firstly, we consider the existence of the nonnegative solution of (1.1) . Define
+
the functional Iλ,µ
: H01 (Ω) → R as follows
Z
Z
Z
µ
λ
a
b
1
+
(u+ )r dx −
(u+ )2 dx −
(u+ )θ dx. (3.1)
Iλ,µ
(u) = kuk2 + kukqq +
2
q
r Ω
2 Ω
θ Ω
+
+
It follows that Iλ,µ
∈ C 1 and the critical points u+ of Iλ,µ
satisfy u+ ≥ 0 and
+ 0
so are critical points of Iλ,µ as well, actually, (Iλ,µ ) (u+ )[(u+ )− ] = −k(u+ )− k2 −
µk(u+ )− kqq = 0.
+
Similar to the proofs of Lemma 2.3 and Lemma 2.4, we can show that Iλ,µ
satisfies the (PS) condition.
+
Lemma 3.1. Let 2 < θ ≤ 2∗ . If λ, µ > 0, then Iλ,µ
satisfies the (PS) condition at
level c with c <
1 N/2 2−N
b 2
NS
.
+
Lemma 3.2. The trivial solution u ≡ 0 is a local minimizer for Iλ,µ
, for all
λ, µ > 0.
+
Proof. It suffices to show that 0 is a local minimizer of Iλ,µ
in the topology (see
1
[3]). For u ∈ C0 (Ω), we have
Z
Z
Z
1
µ
λ
a
b
+
Iλ,µ
(u) = kuk2 + kukqq +
(u+ )r dx −
(u+ )2 dx −
(u+ )θ dx
2
q
r Ω
2 Ω
θ Ω
Z
Z
Z
λ
a
b
≥
(u+ )r dx −
(u+ )2 dx −
(u+ )θ dx
r Ω
2 Ω
θ Ω
Z
λ a
b θ−r 2−r
≥
− |u|C 0 − |u|C 0
(u+ )r dx ≥ 0
r
2
θ
Ω
8
D. YANG, C. BAI
EJDE-2014/170
whenever
a 2−r b θ−r
λ
|u|C 0 + |u|C 0 ≤ .
2
θ
r
+
Lemma 3.3. There exists t0 > 0 such that Iλ,µ
(t0 ϕ1 ) ≤ 0, for all λ, µ in a bounded
set.
Proof. Let ϕ1 be the positive eigenfunction associated to λ1 , for t > 0, we have
Z
Z
Z
tq µ
tr λ
t2 a
tθ b
t2
+
2
q
r
2
kϕ1 kq +
ϕ dx −
ϕ dx −
ϕθ dx
Iλ,µ (tϕ1 ) = kϕ1 k +
2
q
r Ω 1
2 Ω 1
θ Ω 1
Z
Z
Z
t2
tq µ
tr λ
tθ b
2
q
r
= (λ1 − a)
ϕ1 dx +
kϕ1 kq +
ϕ dx −
ϕθ dx
2
q
r Ω 1
θ Ω 1
Ω
+
Since λ1 < a and q, r < 2 < θ, there exists a choice of t0 > 0 such that Iλ,µ
(t0 ϕ1 ) ≤
0 for λ, µ in a bounded set.
Define
c+
λ,µ = inf
+
sup Iλ,µ
(γ(t)),
γ∈Γ+ t∈[0,1]
where
Γ+ = {γ ∈ C([0, 1], γ(0) = 0, γ(1) = t0 ϕ1 }.
On the other hand, by the proof of Lemma 3.3, we obtain
Z
tq µ
tr λ
+
q
Iλ,µ (tϕ1 ) ≤
kϕ1 kq +
ϕr dx.
q
r Ω 1
2−N
1 N/2
Then, if λ and µ are small enough, c+
b 2 , consequently, by means
λ,µ < N S
+
+
of the Mountain Pass Theorem, cλ,µ is a critical value of Iλ,µ
. Thus, we have the
following result.
Lemma 3.4. Let N > 2, 1 < min{q, r} ≤ max{q, r} < 2 < θ ≤ 2∗ and λ1 < a. If
λ, µ are small enough, then (1.1) has at least a nontrivial positive solution.
−
To obtain the negative solution, consider the functional Iλ,µ
: H01 (Ω) → R given
by
Z
Z
1
µ
λ
a
kuk2 + kukqq +
(u− )r dx −
(u− )2 dx.
(3.2)
2
q
r Ω
2 Ω
−
−
satisfy u− ≤ 0 and so are
Again, Iλ,µ
∈ C 1 and the critical points u− of Iλ,µ
critical points of Iλ,µ as well. We will apply once again the mountain pass theorem
−
to obtain a critical point of Iλ,µ
.
−
Iλ,µ
(u) =
−
Lemma 3.5. The trivial solution u ≡ 0 is a local minimizer for Iλ,µ
, for all
λ, µ > 0.
−
Proof. It suffices to show that 0 is a local minimizer of Iλ,µ
in the topology. For
u ∈ C01 (Ω), we have
Z
Z
1
µ
λ
a
−
Iλ,µ
(u) = kuk2 + kukqq +
(u− )r dx −
(u− )2 dx
2
q
r Ω
2 Ω
Z
Z
λ
a
− r
(u ) dx −
(u− )2 dx
≥
r Ω
2 Ω
EJDE-2014/170
NONLINEAR ELLIPTIC PROBLEM OF 2-q-LAPLACIAN TYPE
≥
whenever
2−r
a
2 |u|C 0
λ a 2−r − |u|C 0
r
2
Z
9
(u− )r dx ≥ 0
Ω
≤ λ/r.
−
Lemma 3.6. There exists t0 > 0 such that Iλ,µ
(−t0 ϕ1 ) ≤ 0, for all λ, µ in a
bounded set.
Proof. For t > 0, we have
Z
Z
tq µ
tr λ
t2
t2 a
2
q
r
kϕ1 kq +
ϕ dx −
ϕ2 dx
= kϕ1 k +
2
q
r Ω 1
2 Ω 1
Z
Z
tr λ
t2
tq µ
q
2
kϕ1 kq +
ϕr dx.
= (λ1 − a)
ϕ1 dx +
2
q
r Ω 1
Ω
Since λ1 < a and r, q < 2, there exists a choice of t0 > 0 which proves the lemma.
−
Iλ,µ
(−tϕ1 )
As in the nonnegative solution case, we obtain a critical value
c−
λ,µ = inf
−
sup Iλ,µ
(γ(t)),
γ∈Γ− t∈[0,1]
where
Γ− = {γ ∈ C([0, 1] : γ(0) = 0, γ(1) = −t0 ϕ1 }.
Similar to the proof of Lemma 3.5, we obtain the estimate
Z
tr0 λ
tq0 µ
−
q
kϕ
k
+
ϕr dx,
c−
≤
max
I
(−st
ϕ
)
≤
1 q
0 1
λ,µ
λ,µ
q
r Ω 1
s∈[0,1]
which implies that if λ, µ are small enough, then we obtain the estimate c−
λ,µ <
1 N/2 2−N
b 2 , consequently, by the Mountain
NS
−
of Iλ,µ . Hence, we obtain another important
Pass Theorem, c−
λ,µ is a critical value
result.
Lemma 3.7. Let N > 2, 1 < min{q, r} ≤ max{q, r} < 2 < θ ≤ 2∗ and λ1 < a. If
λ, µ small enough, then (1.1) has at least a nontrivial negative solution.
For Wk and Vkm are as in Section 2, we now consider the existence of the third
solution.
Lemma 3.8. There exist α > 0 and ρ > 0 such that
Iλ,µ (u) ≥ α
whenever u ∈ Wk and kuk = ρ.
Proof. If u ∈ Wk , then
Z
Z
Z
1
µ
λ
a
b
Iλ,µ (u) = kuk2 + kukqq +
|u|r dx −
|u|2 dx −
(u+ )θ dx
2
q
r Ω
2 Ω
θ Ω
Z
Z
1
a
b
≥ kuk2 −
|u|2 dx −
(u+ )θ dx
2
2 Ω
θ Ω
a b
1
≥
−
kuk2 − |u|θθ
2 2λk+1
θ
≥ kuk2 A − Bkukθ−2 ,
1
with A, B > 0. Then it suffices to take ρ < (A/B) θ−2 .
10
D. YANG, C. BAI
EJDE-2014/170
Lemma 3.9. Given λ0 > 0 and µ0 > 0, there exist m0 ∈ N and R > ρ such that
Z
µ
λ
Iλ,µ (u) ≤ kukqq +
|u|r dx,
q
r Ω
whenever u ∈ ∂Qm , where Qm = (BR ∩ Vkm ) ⊕ [0, Rϕm
k+1 ], m ≥ m0 , λ ≤ λ0 and
µ ≤ µ0 . Henceforth ∂ means the boundary relative to subspace Vkm .
Proof. Let m be large enough and ak < a such that
λk + ck m2−N ≤ ak < a.
For u ∈
(3.3)
Vkm ,
by Lemma 2.1 and (3.3) one can obtain
Z
Z
Z
µ
λ
a
b
1
|u|r dx −
|u|2 dx −
(u+ )θ dx
Iλ,µ (u) = kuk2 + kukqq +
2
q
r Ω
2 Ω
θ Ω
Z
Z
a λ
1
µ
b
≤
−
kuk2 + kukqq +
|u|r dx −
(u+ )θ dx
2 2ak
q
r Ω
θ Ω
Z
λ
µ
q
|u|r dx,
≤ kukq +
q
r Ω
(3.4)
and
Iλ,µ (ξϕm
k+1 )
Z
ξ 2 m 2 µξ q m q λξ r
kϕk+1 k +
kϕk+1 kq +
|ϕm |r dx
2
q
r Ω k+1
Z
Z
(3.5)
aξ 2
bξ θ
2
+ θ
−
|ϕm
|
dx
−
((ϕm
k+1
k+1 ) ) dx
2 Ω
θ Ω
Z
Z
2
q
ξ
µ0 ξ
λ0 ξ r
bξ θ
2
m
q
m
r
≤ kϕm
k
+
kϕ
k
+
|ϕ
|
dx
−
((ϕm )+ )θ dx.
k+1 q
k+1
2 k+1
q
r
θ Ω k+1
Ω
=
1,2
Since ϕm
k+1 → ϕk+1 in W0 (Ω) as m → ∞, ϕk+1 changes of sign, and θ > 2, q, r,
there exist m0 ∈ N and R > 0 such that
Iλ,µ (Rϕm
k+1 ) ≤ 0
∀m ≥ m0 .
(3.6)
Then combining (2.2), (3.4) and (3.6) leads to
Iλ,µ (u) ≤
λ
µ
kukqq +
q
r
Z
|u|r dx,
(3.7)
Ω
whenever u ∈ Vkm ∪ (Vkm ⊕ Rϕm
k+1 ). By (3.5), there exists β > 0 satisfying
Iλ,µ (ξϕm
k+1 ) ≤ β,
(3.8)
for all ξ ≥ 0 and m ≥ m0 . Since a > λk , we may take R > 0 such that
Z
1
a µ
λ
−
kuk2 + kukqq +
|u|r dx
Iλ,µ (u) ≤
2 2λk
q
r Ω
Z
µ
λ
≤ −β + kukqq +
|u|r dx.
q
r Ω
(3.9)
Hence, by (2.2), (3.8) and (3.9) we obtain
m
Iλ,µ (u + ξϕm
k+1 ) = Iλ,µ (u) + Iλ,µ (ξϕk+1 ) ≤
µ
λ
kukqq +
q
r
Z
|u|r dx
(3.10)
Ω
for all m ≥ m0 and u ∈ ∂(BR ∩ Vkm ). Thus, by (3.7) and (3.10), we complete the
proof.
EJDE-2014/170
NONLINEAR ELLIPTIC PROBLEM OF 2-q-LAPLACIAN TYPE
11
Proof of Theorem 1.1. For the subcritical case, if θ < 2∗ , α is given by Lemma 3.8.
Take λ and µ small enough in order that
Z
µ
λ
kukqq +
|u|r dx < α
q
r Ω
for all u ∈ ∂Qm . Then by Lemma 3.9 we have
Iλ,µ (u) < α
whenever u ∈ ∂Qm and m ≥ m0 . Applying the Linking Theorem, Iλ,µ possesses a
critical point u at level cλ,µ , where
cλ,µ = inf max Iλ,µ (η(u)),
Γ u∈Qm
Γ = {η ∈ C(Qm , W01,p (Ω)); η = Id
on ∂Qm },
Finally, since cλ,µ ≥ α, Iλ,µ (u) ≥ α > 0 and c±
λ,µ → 0 as λ, µ → 0. Hence, if λ, µ are
small enough c±
<
α
≤
c
,
and
we
know
that u may be neither of the critical
λ,µ
λ,µ
+
−
points found above for Iλ,µ and Iλ,µ ; that is, u is the third solution of (1.1). Thus,
combining Lemmas 3.4 and 3.7, we conclude that (1.1) has at least three nontrivial
solutions.
Proof of Theorem 1.2. For the critical case, θ = 2∗ . Consider the family of functions taken from [1]:
CN (N −2)/2
u = 2
, > 0,
( + |x|2 )(N −2)/2
where
CN = (N (N − 2))(N −2)/4 .
m
m
Let um
= ηu , where η is given as section 2, and Qm = (BR ∩ Vk ) ⊕ [0, Ru ].
m
m
Replacing u by ϕk+1 in Lemma 3.7, we obtain
Z
µ
λ
q
Iλ,µ (u) ≤ kukq +
|u|r dx, ∀u ∈ ∂Qm
q
r Ω
whenever m is large. Hence, to conclude the proof of Theorem 1.2, it remains to
show that
2−N
1
sup Iλ,µ (u) < S N/2 b 2
(3.11)
N
u∈Qm
for all , λ and µ small enough. Let
Z
Z
∗
a
b
1
2
2
|u| dx − ∗
J(u) = kuk −
(u+ )2 dx.
2
2 Ω
2 Ω
Then, we have
Z
λ
µ
Iλ,µ (u) = J(u) +
|u|r dx + kukqq .
r Ω
q
It is sufficient to prove that there exist m0 > 0 and 0 > 0 such that
2−N
1
sup J(u) < S N/2 b 2
N
u∈Qm
for all m ≥ m0 and < 0 . It is not difficult to obtain the following expressions [2]:
Z
2
N/2
|∇um
+ O(N −2 ),
(3.12)
| dx = S
Ω
12
D. YANG, C. BAI
Z
EJDE-2014/170
∗
2
N/2
|um
+ O(N ).
| dx = S
(3.13)
Ω
Moreover, we obtain
Z
Z
2
|um
|
dx
=
Ω
|u |2 dx + O(N −2 )
B(0,1/m)
Z
≥
B(0,)
2 N −2
CN
+
[22 ]N −2
Z
<|x|<1/m
2 N −2
CN
+ O(N −2 )
[2|x|2 ]N −2
(3.14)
(
d2 | ln | + O(2 ), if N = 4,
=
d2 + O(N −2 ),
if N ≥ 5,
where d is a positive constant. If N = 4, according (3.12), (3.13) and (3.14), one
has
2
m 2
S 2 − ad2 | ln | + O(2 )
kum
k − a|u |
≤
2
|um
(S 2 + O(4 ))1/2
|2∗
= S − ad2 | ln |S −1 + O(2 ) < S,
for > 0 sufficiently small. And similarly, if N ≥ 5, we obtain
2
m 2
S N/2 − ad2 + O(N −2 )
kum
k − a|u |
≤
2
|um
(S N/2 + O(N ))2/2∗
|2∗
= S − ad2 S (2−N )/2 + O(N −2 ) < S,
for > 0 sufficiently small. Let u = v + tum
∈ Qm . By simple computation, we
obtain
2−N
2
m 2 N/2
2−N
b 2 kum
1
k − a|u |
m
max J(tu ) =
(3.15)
< S N/2 b 2 .
2
t≥0
N
|um
|
N
2∗
Fix m0 > 0 such that λk + ck m2−N
≤ σ < a. Then, for m ≥ m0 , we obtain
0
Z
Z
∗
1
a
b
2
2
|v| dx − ∗
(v + )2 dx
J(v) = kvk −
2
2 Ω
2 Ω
(3.16)
1
a 2
σ 2 a 2
2
≤ kvk − |v| ≤ |v| − |v| ≤ 0.
2
2
2
2
From (3.15) and (3.16), we obtain
m
m
J(u) = J(v + tum
) = J(v) + J(tu ) ≤ J(tu ) <
So, (3.11) holds.
1 N/2 2−N
S
b 2 .
N
Letting µ → 0 in Theorem 1.1 and Theorem 1.2, we easily show that Theorems
1.1 and 1.2 extend the main results in Paiva and Presoto [12].
Acknowledgments. The authors want to thank the anonymous referees for their
valuable comments and suggestions. This work is supported by Natural Science
Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of
China (11271364 and 10771212).
EJDE-2014/170
NONLINEAR ELLIPTIC PROBLEM OF 2-q-LAPLACIAN TYPE
13
References
[1] C. O. Alves, Y. Ding; Multiplicity of positive solutions to a p-Laplacian equation involving
critical nonlinearity, J. Math. Anal. Appl. 279 (2003), 508-521.
[2] J.G. Azorero, I. P. Alonso; Multiplicity of solutions for elliptic problems with critical exponent
or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 877-895.
[3] H. Brezis, L. Nirenberg; H 1 Versus C 1 local minimizers, C. R. Acad. Sci., Paris 1317 (1993),
465-472.
[4] M. Calanchi, B. Ruf; Elliptic equations with one-sided critical growth, Electron. J. Differential
Equations 89 (2002), 21 (electronic).
[5] M. Cuesta, D. G. de Figueiredo, P. N. Srikanth; On a resonant-superlinear elliptic problem,
Calc. Var. Partial Differential Equations 17 (2003), 221-233.
[6] D. G. De Figueiredo, J. Yang; Critical superlinear Ambrosetti-Prodi problems, Topol. Methods Nonlinear Anal. 14 (1999), 59-80.
[7] F. Gazzola, B. Ruf; Lower-order perturbations of critical growth nonlinearities in semilinear
elliptic equations, Adv. Differential Equations 2 (1997), 555-572.
[8] C. He, G. Li; The existence of nontrivial solution to the p-q-Laplacian problem with nonlinearlity asymptotic to up−1 at infinity in RN , Nonlinear Anal. 68 (2008), 1100-1119.
[9] G. B. Li, G. Zhang; Multiple solutions for the p-q-Laplacian problem with critical exponent,
Acta Math. Sci. Ser. B 29 (4) (2009), 903-918.
[10] S. A. Maranoa, N. S. Papageorgiou; Constant-sign and nodal solutions of coercive (p, q)Laplacian problems, Nonlinear Anal. 77 (2013), 118-129.
[11] D. C. de Morais Filho, F. R. Pereira; Critical Ambrosetti-Prodi type problems for systems of
elliptic equations, Nonlinear Anal. 68 (2008), 194-207.
[12] F. O. de Paiva, A. E. Presoto; Semilinear elliptic problems with asymmetric nonlinearities,
J. Math. Anal. Appl. 409 (2014), 254-262.
[13] N. S. Papageorgiou, V. D. Rǎdulescu; Qualitative phenomena for some classes of quasilinear
elliptic equations with multiple resonance, Applied Mathematics Optimization, 69 (2014),
393-430.
[14] B. Ribeiro; The Ambrosetti-Prodi problem for gradient elliptic systems with critical homogeneous nonlinearity, J. Math. Anal. Appl. 363 (2010) 606-617.
[15] M. Sun; Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance,
J.Math. Anal. Appl. 386 (2012), 661-668.
[16] G. Talenti; Best constant in Sobolev inequality, Ann. Math. 110 (1976), 353-372.
[17] H. Yin, Z. Yang; A Class of p-q-Laplacian type equation with concave-convex nonlinearities
in bounded domain, J. Math. Anal. Appl. 382 (2011), 843-855.
[18] H. Yin, Z. Yang; Multiplicity of positive solutions to a p-q-Laplacian equation involving
critical nonlinearity, Nonlinear Anal. 75 (2012), 3021-3035.
Dandan Yang
School of Mathematical Science, Huaiyin Normal University, Huaian, Jiangsu 223300,
China
E-mail address: ydd423@sohu.com
Chuanzhi Bai
School of Mathematical Science, Huaiyin Normal University, Huaian, Jiangsu 223300,
China
E-mail address: czbai@hytc.edu.cn