Infinitely many solutions for elliptic problems in R N involving the p ( x ) -Laplacian
Qing-Mei Zhou
B1and Ke-Qi Wang
21Library, Northeast Forestry University, Harbin, 150040, P.R. China
2College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin, 150040, P.R. China
Received 5 September 2015, appeared 28 December 2015 Communicated by Dimitri Mugnai
Abstract. We consider thep(x)-Laplacian equations inRN. The potential function does not satisfy the coercive condition. We obtain the existence of infinitely many solutions of the equations, improving a recent result of Duan–Huang [L. Duan, L. H. Huang, Electron. J. Qual. Theory Differ. Equ.2014, No. 28, 1–13].
Keywords: p(x)-Laplacian, variational method, variant fountain theorem, critical point.
2010 Mathematics Subject Classification: 35J35, 35J60, 35J70.
1 Introduction and main results
Let us consider the following nonlinear elliptic problem:
− 4p(x)u+V(x)|u|p(x)−2u= f(x,u), inRN, (P) where p :RN →R is Lipschitz continuous and 1< p− :=infRN p(x) ≤supRN p(x):= p+ <
N,V is the new potential function, and the nonlinear term f is sublinear with some precise assumptions that we state below.
We emphasize that the operator−∆p(x)u=div(|∇u|p(x)−2∇u)is said to bep(x)-Laplacian, which becomes p-Laplacian when p(x)≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian, for example, it is inhomogeneous and in general, it does not have the first eigenvalue. The study of various mathematical problems with variable exponent growth condition has received considerable attention in recent years.
These problems appear in a lot of applications, such as image processing models (see e.g.
[6,21]), stationary thermorheological viscous flows (see [2]) and the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium (see [3]). We refer to [4,14–18,23–25] for the study of the p(x)-Laplacian equations and the corresponding variational problems.
BCorresponding author. Email: zhouqingmei2008@163.com
It is well known, the main difficulty in treating problem (P) inRN arises from the lack of compactness of the Sobolev embeddings, which prevents from checking directly that the en- ergy functional associated with (P) satisfies theC-condition. To overcome the difficulty of the noncompact embedding, Dai in [7,8] proves that the subspace of radially symmetric functions of W1,p(x)(RN), denoted further by Wr1,p(x)(RN), can be embedded compactly into L∞(RN) whenever 2 ≤ N < p− ≤ p+ < +∞. Later, Alves–Liu [1] (who use the conditions (V1)and (V2)), Ge–Zhou–Xue [19,20] (who assume that conditions (V1) and(V3) hold) also establish new compact embedding theorems for the subspace ofW1,p(x)(RN)when 1< p− ≤ p+< N.
Furthermore, the authors make the following assumptions on the potential functionV.
(V1) V∈C(RN)and infV>0.
(V2) For anyM >0,{x ∈RN :V(x)≤ M}is a bounded set.
(V3) There existsr>0 such that for anyb>0
|ylim|→∞µ {x ∈RN :V(x)≤b} ∩Br(y)) =0, whereµis the Lebesgue measure onRN.
We emphasize that in our approach, no coerciveness hypothesis (V2) and not necessarily radial symmetry will be required on the potential V. To the best of our knowledge, there are only a few works concerning on this case up to now. Inspired by the above facts and the aforementioned papers, the main purpose of this paper is to study the existence of infinitely many solutions for problem (P) when F(x,u)is sublinear inu at infinity. Our tool used here is a variational method combined with the theory of variable exponent Sobolev spaces.
We are now in the position to state our main results.
Theorem 1.1. Suppose that(V1)and the following condition H(f)holds, H(f) f(x,u) = b(x)
q(x)|u|q(x)−2u, b: RN → R+ is a positive continuous function such that b∈L s
(x)
s(x)−q(x)(RN)and1<q−≤q+ < p−, where p(x)≤s(x) p∗(x), p∗(x) = N p(x)
N−p(x), and s(x) p∗(x)means thatx∈RN(p∗(x)−s(x))>0.
Then problem(P)possesses infinitely many solutions{uk}satisfying Z
RN
1 p(x)
|∇uk|p(x)+V(x)|uk|p(x)dx−
Z
RNF(x,uk)dx→0−, ask→∞, where F(x,uk) =Ruk
0 f(x,t)dt.
The rest of this paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the nonsmooth critical point theory of the locally Lipschitz functionals. In Section 3, the proof of the main results is given.
2 Preliminaries
In order to discuss problem (P), we need some facts on space W1,p(x)(RN) which are called variable exponent Sobolev spaces. For a deeper treatment on these spaces, we refer to [10–12, 22]. Write
C+(RN) =np ∈C(RN): p(x)>1 for anyx ∈RNo,
p−= inf
x∈RNp(x), p+= sup
x∈RN
p(x)for any p∈ C+(RN).
Denote by S(RN) the set of all measurable real-valued functions defined on RN. Note that two measurable functions in S(RN)are considered as the same element of S(RN)when they are equal almost everywhere.
Let p∈C+(RN). The variable exponent Lebesgue space Lp(x)(RN)is defined by Lp(x)(RN) =
u∈S(RN): Z
RN|u|p(x)dx< +∞
, endowed with the norm
|u|Lp(x)(RN)=|u|p(x)=inf (
λ>0 : Z
RN
u(x) λ
p(x)
dx≤1 )
. Then we define the variable exponent Sobolev space
W1,p(x)(RN) =nu∈Lp(x)(RN):|∇u| ∈Lp(x)(RN)o with the norm
kuk=kukW1,p(x)(RN)=|u|p(x)+|∇u|p(x).
With these norms, the spaces Lp(x)(RN) and W1,p(x)(RN) are separable reflexive Banach spaces; see [12] for the details.
Proposition 2.1 ([13]). Setρ(u) =R
RN|u|p(x)dx.For u∈ Lp(x)(RN), we have (i) for u6=0,|u |p(x)=λ⇔ρ(uλ) =1;
(ii) |u|p(x)<1(=1;>1)⇔ρ(u)<1(=1;>1); (iii) if|u|p(x) ≥1, then|u|p−
p(x)≤ ρ(u)≤ |u|p+
p(x); (iv) if|u|p(x) ≤1, then|u|p+
p(x)≤ ρ(u)≤ |u|p−
p(x).
Next, we consider the case thatV satisfies(V1). On the linear subspace E=
u∈W1,p(x)(RN): Z
RN
|∇u|p(x)+V(x)|u|p(x)dx<+∞
, we equip with the norm
kukE=inf
λ>0 : Z
RN
∇u λ
p(x)
+V(x) u λ
p(x) dx≤1
.
Then (E,k · kE)is continuously embedded intoW1,p(x)(RN)as a closed subspace. Therefore, (E,k · kE)is also a separable reflexive Banach space. It is easy to see that with the normk · kE, Proposition2.1remains valid, that is,
Proposition 2.2. Set I(u) =R
RN(|∇u|p(x)+V(x)|u|p(x))dx. If u∈W1,p(x)(RN), then (i) for u6=0,kukE =λ⇔ I(u
λ) =1;
(ii) kukE <1(=1;>1)⇔ I(u)<1(=1;>1); (iii) ifkukE ≥1, thenkukpE− ≤ I(u)≤ kukpE+; (iv) ifkukE ≤1, thenkukpE+ ≤ I(u)≤ kukpE−.
Proposition 2.3([13]). The conjugate space of Lp(x)(RN)is Lq(x)(RN), where p(1x)+ q(1x) =1. For any u∈ Lp(x)(RN)and v∈ Lq(x)(RN), we have
Z
RNuvdx
≤2|u|p(x)|v|q(x).
Proposition 2.4 ([11]). Let p : RN → R be Lipschitz continuous and satisfy1 < p− ≤ p+ < N, and q:RN →Rbe a measurable function.
(i) If p≤ q≤ p∗, then there is a continuous embedding W1,p(x)(RN),→Lq(x)(RN). (ii) If p≤ q p∗, then there is a compact embedding W1,p(x)(RN),→Lqloc(x)(RN).
Proposition 2.5 ([10]). Let p(x)and q(x)be measurable functions such that p(x) ∈ L∞(RN)and 1≤ p(x)q(x)≤∞almost everywhere inRN. Let u ∈ Lq(x)(RN), u6=0. Then
|u|p(x)q(x) ≥1⇒ |u|pp−(x)q(x)≤|u|p(x)
q(x)≤ |u|pp+(x)q(x),
|u|p(x)q(x) ≤1⇒ |u|p+
p(x)q(x)≤|u|p(x)
q(x)≤ |u|p−
p(x)q(x). In particular, if p(x) = p is a constant, then
|u|p
q(x)= |u|ppq(x). Set
I(u) =
Z
RN
1
p(x)|∇u|p(x)dx+
Z
RN
V(x)
p(x)|u|p(x)dx.
We know that (see [5]), I ∈ C1(E,R)and p(x)-Laplacian operator −∆p(x)u is the derivative operator of J in the weak sense. We denoteL= I0 :E→E∗, then
hLu,vi=
Z
RN(|∇u(x)|p(x)−2∇u· ∇v+V(x)|u|p(x)−2uv)dx, ∀u,v∈E.
Proposition 2.6([13]). Set E andLas above, then
(i) L: E→E∗ is a continuous, bounded and strictly monotone operator;
(ii) Lis a mapping of type(S+), if un *u (weak) in E andlim supn→∞(L(un)− L(u),un−u)≤ 0, then un→u in E;
(iii) L: E→E∗ is a homeomorphism.
In order to assure the existence of infinitely many solutions for the problem (P), our main tool will be the variant fountain theorem [26, Theorem 2.2], which will be used in our proof.
Let X be a Banach space with the normk · k and X = L∞i∈NXi with dimXi < ∞ for any i∈N. Set
Yk =
k
M
i=0
Xi, Zk =
M∞ i=k
Xi. (2.1)
Consider the followingC1-functional ϕλ :E→Rdefined by ϕλ(u) =A(u)−λB(u), λ∈[1, 2], where A,B:X→Rare two functionals.
Lemma 2.7. Suppose that the functionalϕλ defined above, and satisfies the following conditions.
(1) ϕλ maps bounded sets to bounded sets uniformly forλ∈[1, 2]. Furthermore, ϕλ(−u) = ϕλ(u) for all(λ,u)∈ [1, 2]×X.
(2) B(u)≥0; B(u)→∞askuk →∞on any finite dimensional subspace of X.
(3) There existρk >rk >0such that ak(λ):= inf
u∈Zk,kuk=ρk
ϕλ(u)≥0>bk(λ):= max
u∈Yk,kuk=rkϕλ(u) for allλ∈[1, 2], dk(λ):= inf
u∈Zk,kuk≤ρk
ϕλ(u)→0as k→∞uniformly forλ∈ [1, 2]. Then there existλn →1, u(λn)∈Ynsuch that
ϕ0λn|Yn(u(λn)) =0, ϕλn(u(λn))→ck ∈[dk(2),bk(1)] asn→∞.
Particularly, if {u(λn)}has a convergent subsequence for every k, then ϕ1 has infinitely many non- trivial critical points{uk} ∈X\{0}satisfyingϕ1(uk)→0−as k→∞.
In order to discuss the problem (P), we need to consider the energy functional ϕ: E→R defined by
ϕ(u) =
Z
RN
1 p(x)
h|∇u|p(x)+V(x)|u|p(x)idx−
Z
RNF(x,u)dx.
Under our hypotheses, it follows from a Hölder-type inequality and Sobolev’s embedding theorem that the energy functional ϕis well-defined. It is well known that ϕ∈ C1(E,R)and its derivative is given by
hϕ0(u),vi=
Z
RN
|∇u|p(x)−2∇u∇v+V(x)|u|p(x)−2uv dx−
Z
RN f(x,u)vdx
for v ∈ E. It is standard to verify that the weak solutions of problem (P) correspond to the critical points of the functional ϕ.
3 Proof of the main results
In this section, for the notation in Lemma 2.7, the space X = E, and related functionals onE are
A(u) =
Z
RN
1 p(x)
h|∇u|p(x)+V(x)|u|p(x)idx, B(u) =
Z
RN F(x,u)dx. (3.1) So the perturbed functional which we will study is
ϕλ(u) =
Z
RN
1 p(x)
h|∇u|p(x)+V(x)|u|p(x)idx−λ Z
RNF(x,u)dx.
Clearly,ϕλ ∈ C1(E,R)for allλ∈[1, 2]. We choose a completely orthogonal basis{ei}ofEand define Xi :=Rei, andZk,Yk defined as (2.1). We shall prove that ϕλ satisfies the conditions of Lemma2.7. Following along the same lines as in [9], we can obtain that
• B(u) ≥ 0, and B(u) → ∞ as kuk → ∞ on any finite dimensional subspace of E (see [9, Lemma 3.1]).
• There exists a sequenceρk →0+ask →∞such that ak(λ):= inf
u∈Zk,kuk=ρk
ϕλ(u)≥0 and
dk(λ):= inf
u∈Zk,kuk≤ρk
ϕλ(u)→0
as k → ∞ uniformly for λ ∈ [1, 2]. For further details, we refer the reader to Duan–
Huang [9, Lemma 3.2],
• There exists a sequence{rk}with 0<rk <ρk for allk ∈Nsuch that bk(λ):= inf
u∈Yk,kuk=rkϕλ(u)<0, ∀λ∈ [1, 2], for details, see [9, Lemma 3.3].
Obviously, condition (1) in Lemma 2.7 have been satisfied. Thus, conditions (1), (2) and (3) in Lemma 2.7 have been verified. Therefore, we know from Lemma 2.7 that there exist λn→1,u(λn)∈Yn such that
ϕ0λn|Yn(u(λn)) =0, ϕλn(u(λn))→ck ∈[dk(2),bk(1)] asn→∞. (3.2) For simplicity, we denoteu(λn)byunfor all n∈N.
Claim: The sequence{un}is bounded inE.
By virtue of hypothesisH(f), Proposition2.2, Proposition2.3and Proposition2.5, we have 1
p+min{kunkp+,kunkp−} ≤
Z
RN
1 p(x)
h|∇un|p(x)+V(x)|un|p(x)idx
= ϕλn(un) +λn Z
RNF(x,un)dx
= ϕλn(un) +λn Z
RNb(x)|un|q(x)dx
≤ M0+2|b| s(x) s(x)−q(x)
|un|q(x)s(x) q(x)
≤
M0+2|b| s(x) s(x)−q(x)
|un|qs(+x), if|un|s(x) ≥1, M0+2|b| s(x)
s(x)−q(x)
|un|q−
s(x), if|un|s(x) ≤1,
(3.3)
for some M0 > 0. Since 1 < q− ≤ q+ < p−, (3.3) implies that {un} is bounded inE. Next, we show that there is a strongly convergent subsequence of{un}in E. Indeed, in view of the boundedness of{un}, passing to a subsequence if necessary, still denoted by {un}, we may assume thatun *uweakly inE.
From the choice of the function b ∈ L
s(x)
s(x)−q(x)(RN), for any given number ε > 0, we can chooseRε >0 such that
Z
|x|>Rε
|b(x)|s(xs)−(xq)(x)dx <ε
s− −q+
s− . (3.4)
Since the embedding E ,→ Llocs(x)(RN) is compact, un * u0 in E implies un → u0 in Lsloc(x)(RN), and hence,
n→+lim∞ Z
|x|≤Rε
|un−u|s(x)dx=0. (3.5)
LetBε ={x∈RN :|x| ≤ Rε}andBcε =RN\Bε. Using Proposition2.1and (3.4), we have
|b|
L
s(x) s(x)−q(x)(Bcε)
<ε. (3.6)
Also from Proposition2.1and (3.5), there existsn0∈ Nsuch that
|un−u0|Ls(x)(Bε) <ε, forn≥ n0. (3.7)
Now it is easily seen that
hLun− Lu0,un−u0i= hϕ0λn(un)−ϕ0λ1(u0),un−u0i +
Z
RN(λnf(x,un)− f(x,un))(un−u0)dx. (3.8)
We will estimate the right-hand side of (3.8). One clearly has
hϕ0λn(un)−ϕ0λ1(u0),un−u0i=hϕ0λn(un),un−u0i − hϕ0λ1(u0),un−u0i →0. (3.9)
Moreover, we have
Z
RN(λnf(x,un)− f(x,un))(un−u0)dx
≤ q+ Z
RNb(x)(λn|un|q(x)−1+|u0|q(x)−1)|un−u0|dx
≤ q+ Z
RNb(x)|un|q(x)−1|un−u0|+q+ Z
RNb(x)|u0|q(x)−1|un−u0|dx
= q+ Z
RN
b(x) V
q(x)−1 s(x)
V
q(x)−1
s(x) |un|q(x)−1|un−u0|dx +q+
Z
RN
b(x) V
q(x)−1 s(x)
V
q(x)−1
s(x) |u0|q(x)−1|un−u0|dx
≤ q
+
K0 Z
RNb(x)V
q(x)−1
s(x) |un|q(x)−1|un−u0|dx +q
+
K0 Z
RNb(x)V
q(x)−1
s(x) |u0|q(x)−1|un−u0|dx
≤ q
+
K0
Z
Bε
b(x)Vq
(x)−1
s(x) |un|q(x)−1|un−u0|dx +
Z
Bεcb(x)Vq
(x)−1
s(x) |un|q(x)−1|un−u0|dx
+q
+
K0 Z
Bε
b(x)Vq
(x)−1
s(x) |u0|q(x)−1|un−u0|dx +
Z
Bcε b(x)V
q(x)−1
s(x) |u0|q(x)−1|un−u0|dx
=: q+
K0[I1+I2] +q
+
K0[I3+I4], (3.10)
where
K0:=min
V
q− −1 s+
0 ,V
q+−1 s− 0
.
Using the Proposition2.5, Hölder’s inequality and (3.7), we have I1=
Z
Bε
b(x)V
q(x)−1
s(x) |un|q(x)−1|un−u0|dx
≤3|b|
L
s(x) s(x)−q(x)(Bε)
Vq
(x)−1
s(x) |un|q(x)−1
L
s(x) q(x)−1(Bε)
|un−u0|Ls(x)(Bε)
≤3ε|b|
L
s(x) s(x)−q(x)(Bε)
V
s(x)−1
q(x) |un|q(x)−1
L
s(x) q(x)−1(Bε)
≤3ε|b|
L
s(x) s(x)−q(x)(RN)
V
s(x)−1
q(x) |un|q(x)−1
L
s(x) q(x)−1(RN)
≤3ε|b|
L
s(x) s(x)−q(x)(RN)
V(x)|un|s(x)
q+−1 s−
1 , if
V(x)|un|s(x)
1 ≤1,
V(x)|un|s(x)
q− −1 s+
1 , if
V(x)|un|s(x)
1 ≥1,
≤3ε|b|
L
s(x) s(x)−q(x)(RN)
kunk
(q+−1)s+ s−
E , if
V(x)|un|s(x)
1 ≤1,kunkE ≥1, kunkqE+−1, ifV(x)|un|s(x)
1
≤1,kunkE ≤1, kunkqE−−1, if
V(x)|un|s(x)
1 ≥1,kunkE ≥1, kunk
(q− −1)s− s+
E , if
V(x)|un|s(x)
1 ≥1,kunkE ≤1.
(3.11)
On the other hand, using Proposition 2.4, Proposition 2.5, Hölder’s inequality and (3.6), we have
I2 =
Z
Bcεb(x)V
q(x)−1
s(x) |un|q(x)−1|un−u0|dx
≤3|b|
L
s(x) s(x)−q(x)(Bεc)
V
q(x)−1
s(x) |un|q(x)−1
L
s(x) q(x)−1(Bcε)
|un−u0|Ls(x)(Bcε)
≤3|b|
L
s(x) s(x)−q(x)(Bεc)
Vq
(x)−1
s(x) |un|q(x)−1
L
s(x) q(x)−1(RN)
|un−u0|Ls(x)(RN)
≤3Cεkun−u0kEV
q(x)−1
s(x) |un|q(x)−1
L
s(x) q(x)−1(RN)
≤3Cεkun−u0kE
V(x)|un|s(x)
q+−1 s−
1 , if
V(x)|un|s(x)
1
≤1,
V(x)|un|s(x)
q− −1 s+
1 , if
V(x)|un|s(x)
1≥1,
≤3Cε(kunkE+ku0kE)
kunk
(q+−1)s+ s−
E , if
V(x)|un|s(x)
1
≤1, kunkE ≥1, kunkqE+−1, if
V(x)|un|s(x)
1 ≤1, kunkE ≤1, kunkqE−−1, if
V(x)|un|s(x)
1 ≥1, kunkE ≥1, kunk
(q− −1)s− s+
E , if
V(x)|un|s(x)
1
≥1, kunkE ≤1.
(3.12)
Similarly, we also have that I3=
Z
Bεb(x)Vq
(x)−1
s(x) |u0|q(x)−1|un−u0|dx
≤3ε|b|
L
s(x) s(x)−q(x)(RN)
ku0k
(q+−1)s+ s−
E , if
V(x)|u0|s(x)
1 ≤1, ku0kE ≥1, ku0kqE+−1, if
V(x)|u0|s(x)
1 ≤1, ku0kE ≤1, ku0kqE−−1, if
V(x)|u0|s(x)
1 ≥1, ku0kE ≥1, ku0k
(q− −1)s− s+
E , if
V(x)|u0|s(x)
1 ≥1, ku0kE ≤1,
(3.13)
and I4=
Z
Bcε b(x)V
q(x)−1
s(x) |un|q(x)−1|un−u0|dx
≤3Cε(kunkE+ku0kE)
ku0k
(q+−1)s+ s−
E , if
V(x)|u0|s(x)
1 ≤1, ku0kE ≥1, ku0kqE+−1, if
V(x)|u0|s(x)
1 ≤1, ku0kE ≤1, ku0kqE−−1, if
V(x)|u0|s(x)
1 ≥1, ku0kE ≥1, ku0k
(q− −1)s− s+
E , if
V(x)|u0|s(x)
1 ≥1, ku0kE ≤1.
(3.14)
Sinceεis arbitrary, it follows from (3.10)–(3.14) that Z
RN(λnf(x,un)− f(x,un))(un−u0)dx→0 asn→+∞. (3.15) According to (3.8), (3.9) and (3.15) we obtain
hLun− Lu0,un−u0i →0 asn →+∞, (3.16) which implies un → u0 in E from Proposition 2.6(ii). Thus, from the last assertion of Lemma 2.7, we know that ϕ = ϕ1 has infinitely many nontrivial critical points. Therefore, problem (P) possesses infinitely many nontrivial solutions. The proof of Theorem1.1 is com- pleted.
Acknowledgements
Q. M. Zhou was supported by the Fundamental Research Funds for the Central Universities (nos. DL12BC10, 2015), the New Century Higher Education Teaching Reform Project of Hei- longjiang Province in 2012 (no. JG2012010012), the Humanities and Social Sciences Foundation of the Educational Commission of Heilongjiang Province of China (no. 12544026), the Post- doctoral Research Startup Foundation of Heilongjiang (no. LBH-Q14044), the Science Research Funds for Overseas Returned Chinese Scholars of Heilongjiang Province (no. LC201502).
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