R E S E A R C H
Open Access
Multiplicity of solutions for fractional
Schrödinger equations with perturbation
Liu Yang
**Correspondence:
[email protected] Department of Mathematics and Computing Sciences, Hengyang Normal University, Hengyang, Hunan 421008, People’s Republic of China
Department of Mathematics, Hunan University, Changsha, Hunan 410075, People’s Republic of China
Abstract
In this paper, we investigate a class of fractional Schrödinger equations with
perturbation. By using the mountain pass theorem and Ekeland’s variational principle, we see that such equations possess two solutions. Recent results in the literature are generalized and significantly improved.
MSC: 34B15
Keywords: fractional Schrödinger equations; mountain pass theorem; critical point
1 Introduction
In this paper, we consider the following class of fractional Schrödinger equations:
(–)αu+V(x)u=f(x,u) +λh(x)|u|p–u, x∈RN, (.)
where <α< , α<N, ≤p< ,f ∈C(RN ×R,R),h∈L–p(RN),V ∈C(RN,R), and (–)αuis defined pointwise forxinRNby
(–)αu(x) = –
RN
u(x+y) +u(x–y) – u(x)
|y|N+α dy
along any rapidly decaying functionuof classC∞(RN); see Lemma . of [].
Recently, a lot of attention has been focused on the study of fractional and non-local problems; see [–]. This may be due to its concrete applications in different fields, such as the thin obstacle problem, optimization, finance, phase transitions, stratified materi-als, anomalous diffusion, deblurring and denoising of images, and so on; see [, –]. For standing wave solutions of fractional Schrödinger equations in the whole spaceRN,
there were also many works; see [–]. The fractional Schrödinger equation is a fun-damental equation of fractional quantum mechanics. The fractional quantum mechanics has been discovered as a result of expanding the Feynman path integral, from Brownian-like to Lévy-Brownian-like quantum mechanical paths. In [], Laskin formulated the fractional Schrödinger equations as follows:
i∂tψ= (–)αψ+V(x)ψ–|ψ|p–ψ, x∈RN,t∈R, (.)
where <α < ,ψ is the wavefunction andV(x) denotes the potential energy. We let ψ(x,t) =eiωtu(x) be standing waves solutions for (.). Thenuis a solution of an equation
of type of (.).
For the fractional Schrödinger equations, variational methods are available. In [], Felmer et al.studied the existence and regularity of solutions for a class of fractional Schrödinger equations under the Ambrosetti-Rabinowitz condition,i.e., there existsθ> such that
<θF(x,t)≤tf(x,t).
In [], Secchi obtained the existence of ground state solutions of a class of fractional Schrödinger equations under the Ambrosetti-Rabinowitz condition and the following condition:
(V) V∈C(RN),infx∈RNV(x) =V> andlim|x|→∞V(x) =∞.
In [], Torres studied the existence of solutions for the following equations:
(–)αu+V(x)u=f(u) +h(x), x∈RN, (.)
under the conditions of (V) and the Ambrosetti-Rabinowitz condition forf.
As far as we know, there are few works on problem (.), of which nonlinearity involves a combination of superlinear or asymptotically linear terms and a sublinear perturbation. Motivated by the above facts, we investigate this case in this paper.
Before stating our results we introduce some notations. Throughout this paper, we de-note byrtheLr-norm, ≤r≤ ∞, andh±=max{±h, }. If we take a subsequence of a
sequence{un}we shall denote it again by{un}.
Now we state our main result.
Theorem . Assume that h∈L–p\ {}with h+= , (V
),and the following conditions are satisfied:
(F)f(x,s)is a continuous function onRN ×Rsuch that f(x,s)≡ for all s< and x∈RN.Moreover,there exists b∈L∞(RN,R+)with|b|
∞<S
such that
lim
s→+
f(x,s)
sk =b(x) uniformly in x∈R N,
and
f(x,s)
sk ≥b(x) ∀s> and x∈R N,
where Sr is the best constant for the embedding of X in Lr(RN);see Lemma.and Re-mark.in Section;
(F)there exists q∈L∞(RN,R+)with|q|∞>csuch that
lim
s→∞ f(x,s)
sk =q(x) uniformly in x∈R N,
(F)there exist two constantsθ,dsatisfyingθ> and≤d<S
(θ–)
θ such that
F(x,s) –
θf(x,s)s≤ds
∀s> and x∈RN,
where F(x,s) =sf(x,τ)dτ.
Then we have the following results: (i) ifk= andμ< with
μ=inf
RN
RN
|u(x) –u(z)|
|x–z|N+α dx dz+
RNV(x)u(x)
dxu∈HαRN,
RNq(x)u(x)
dx=
,
then there exists> such that for everyλ∈(,),problem(.)has at least two nontrivial solutions;
(ii) if <k< ∗α– ,then there exists> such that for everyλ∈(,),problem(.)
has at least two nontrivial solutions,where∗α=N–Nα.
Remark . Theorem . extends the perturbationhin [] to the caseλh(x)|u|p–uand
(F) is weaker than the Ambrosetti-Rabinowitz condition. Moreover, ourf is allowed to
be asymptotically linear at infinity whenk= , which is not the same as that in [], where they needlim sups→+ f(x,s)s <lim infs→+∞f(x,s)s .
The paper is organized as follows. In Section , we present some preliminaries. In Sec-tion , we give the proof of our main results.
2 Preliminaries
In order to prove our main results, we first give some properties of spaceXon which the variational setting for problem (.) is defined. Let
H=HαRN:=
u∈LRN:
RN
RN
|u(x) –u(z)|
|x–z|N+α dx dz<∞
with the inner product and the norm
u,vH=
RN
RN
[u(x) –u(z)][v(x) –v(z)]
|x–z|N+α dx dz+
RNu(x)v(x)dx, uH=u,u
H.
Letting
X=
u∈HαRN:
RN
RN
|u(x) –u(z)|
|x–z|N+α dx dz+
RN
V(x)u(x)dx< +∞
,
thenXis a Hilbert space with the inner product
u,v=
RN
RN
[u(x) –u(z)][v(x) –v(z)]
|x–z|N+α dx dz+
and the corresponding normu=u,u. Note that
X⊂HαRN
and
X⊂LrRN
for all r∈[, ∗α] with the embedding being continuous. It is easy to get the following lemma.
Lemma . Assume that the condition(V)holds.Then there exists a constant c> such
that
RN
RN
|u(x) –u(z)|
|x–z|N+α dx dz+
RNV(x)u
dx≥c
uH, ∀u∈H
αRN. (.)
Lemma .(see [, ]) Assume that the condition(V)holds.Then X is compactly em-bedded in Lr(RN)for all r∈[, ∗
α).
Remark . By Lemma ., we have
Srur≤ u,
whereSris the best constants for the embedding ofXinLr(RN).
Now we begin describing the variational formulation of problem (.). Consider the functionalJ:X→Rdefined by
J(u) = u
–
RNF(x,u)dx– λ
p
RNh(x)|u|
pdx. (.)
By the continuity off,gand Lemma .,J∈C(X,R) and its derivative is given by
J(u)v=
RN
RN
[u(x) –u(z)][v(x) –v(z)]
|x–z|N+α dx dz+
RNV
(x)u(x)v(x)dx+
–
RNf
x,u(x)v(x)dx–λ
RNh(x)|u|
p–uv dx (.)
for allu,v∈X. In addition, any critical point ofJonXis a solution of problem (.). Next, we give the variant version of the mountain pass theorem which is important for the proof of our main results.
Theorem .(see []) Let E be a real Banach space with its dual space E∗,and suppose that I∈C(E,R)satisfies
max I(),I(e)≤μ<η≤ inf
for someμ<η,ρ> and e∈E withe>ρ.Letˆc≥ηbe characterized by
ˆ
c=inf γ∈max≤τ≤I
γ(τ),
where={γ ∈C([, ],E) :γ() = ,γ() =e}is the set of continuous paths joiningand e,
then there exists a sequence{un} ⊂E such that
I(un)→ ˆc≥η and
+unI(un)E∗→, as n→ ∞.
3 Proof of the main results
To prove our main results, we first give the following lemma.
Lemma . For any real number∗α– >k≥,assume that the conditions(V), (F)-(F) hold.Then there exists> such that for everyλ∈(,)there are two positive constants
ρ,ηsuch that J(u)|u=ρ≥η> .
Proof For any> , it follows from the conditions (F)-(F) that there existC > and ∗α>r>max{,k}such that
F(x,s)≤|b|∞+ s
+C
r |s|
r, ∀s∈R. (.)
By (.) and (.), Sobolev’s inequality, and Hölder’s inequality, one has
J(u) = u
–
RNF(x,u)dx– λ
p
RNh(x)|u|
pdx
≥
u
–
RN
|b|∞+
u(x)
dx–
RN
C
r u(x) rdx–λ
p
RN
h(x)u(x)pdx
≥
u
–|b|∞+
S u – C
rSr r
ur–λS
–p
p h–pu
p
=up
–|b|∞+ S
u–p–C
rSr r
ur–p–λS
–p p h–p
(.)
for allu∈X. Take=S
–|b|∞and define l(t) =
t
–p–C
r–S–rr tr–p, ∀t≥.
It is easy to prove that there existsρ> such that
max
t≥l(t) =l(ρ) = r– (r–p)
( –p)rSr r
(r–p)C
–p r– .
Consider the minimum problem
μ=inf
RN
RN
|u(x) –u(z)|
|x–z|N+α dx dz+
RNV(x)u(x)
dx:u∈HαRN,
RNq(x)u(x)
dx=
. (.)
Then we have the following results.
Lemma . There exist a constant c> andφ∈Hα(RN)with
RNq(x)φ(x)dx= such thatμ≥cand
μ=
RN
RN
|φ(x) –φ(z)|
|x–z|N+α dx dz+
RNV(x)φ(x)
dx, (.)
i.e. the minimum(.)is achieved.
Proof For anyu∈Hα(RN) with
RNq(x)u(x)dx= , by Lemma . and Sobolev’s embed-ded theorem, we have
RN
RN
|u(x) –u(z)|
|x–z|N+α dx dz+
RN
V(x)u(x)dx≥cuH≥cu≥ c
|q|∞ > .
Therefore, there exists a constantc> such thatμ≥c. Let{un} ⊂Hα(RN) be a
minimiz-ing sequence of (.). Clearly,RNq(x)un(x)dx= and{un}is bounded. Then there exist
a subsequence{un}andφ∈Hα(RN) such thatun φweakly inHα(RN) andun→φ
strongly inL(RN). So it is easy to verify that
RNq(x)un(x)dx→
RNq(x)φ(x)dxas n→ ∞andRNq(x)φ(x)dx= . Therefore,
μ≤
RN
RN
|φ(x) –φ(z)|
|x–z|N+α dx dz+
RN
V(x)φ(x)dx
≤lim inf
n→∞
RN
RN
|u(x) –u(z)|
|x–z|N+α dx dz+
RNV(x)u(x)
dx
≤μ. (.)
This implies that
μ=
RN
RN
|φ(x) –φ(z)|
|x–z|N+α dx dz+
RNV(x)φ(x)
dx.
Lemma . For any real number∗α– >k≥,assume that the conditions(V), (F)-(F) hold.Letρ,> be as in Lemma..Then we have the following results:
(i) Ifk= andμ< ,then there existse∈Xwithe>ρsuch thatJ(e) < for all
λ∈(,).
Proof (i) In casek= . Sinceμ< , we can choose a nonnegative function ϕ∈Hα(RN)
withRNq(x)ϕ(x)dx= such that
ϕ=
RN
RN
|ϕ(x) –ϕ(z)|
|x–z|N+α dx dz+
RNV(x)ϕ(x)
dx< .
Therefore, by the condition (F) and Fatou’s lemma, we have
lim
t→+∞ J(tϕ)
t ≤
ϕ
– lim t→+∞
RN
F(x,tϕ)
tϕ ϕ
dx– lim t→+∞
λ
pt–p
RNh(x)| ϕ|pdx
≤
ϕ
–
RNq(x)ϕ
dx
=
ϕ– < . (.)
So,J(tϕ)→–∞ast→+∞, then there existse∈Xwithe>ρsuch thatJ(e) < for all λ∈(,).
(ii) In casek> .q∈L∞(RN,R+) withq+= , we can choose a functionω∈Hα(RN) such
that
RN
q(x)|ω|k+dx> .
Therefore, by the condition (F) and Fatou’s lemma, we have
lim
t→+∞ J(tω)
tk+ ≤
ω tk– –t→lim+∞
F(x,tω)
tk+ωk+ϕ
k+dx– lim t→+∞
λ
ptk+–p
ξ(x)|ω|pdx
≤–
k+
q(x)ωk+dx
< . (.)
So,J(tω)→–∞ast→+∞; then there existse∈Xwithe>ρsuch thatJ(e) < for
allλ∈(,).
Next, we define
β=inf γ∈max≤τ≤J
γ(τ),
where ={γ ∈C([, ],E) :γ() = ,γ() =e}. Then by Theorem ., Lemma ., and Lemma ., there exists a sequence{un} ⊂Xsuch that
J(un)→β and
+unJ(un)E∗→, asn→ ∞. (.)
Then we have the following results.
Lemma . For any real number∗α– >k≥,assume that the conditions(V), (F)-(F) hold. Let> be as in Lemma..Then {un}defined by(.)is bounded in X for all
Proof Fornlarge enough, by Hölder’s inequality and Lemma ., one has
β+ ≥J(un) –
θ
J(un),un = – θ
un–
RN
F(x,un) –
θf(x,un)un
dx –λ p–
θ RNh(x)|un|
pdx
≥θ–
θ un
–d
RN
u ndx–λ
p–
θ RN
h(x)|un|pdx
≥θ–
θ un
–d Sun
–λ p– θ
S–p h –pun
p
≥
θ– θ –
d S
un– p– θ
S–p h –pun
p, (.)
which implies that{un}is bounded inX, since ≤p< .
DenoteBρ={u∈X:u<ρ}, whereρis given by Lemma .. Then by Ekeland’s vari-ational principle and Lemma ., we have the following lemma, which shows thatJhas a local minimum ifλis small.
Lemma . For any real number∗α– >k≥,assume that the conditions(V), (F)-(F). Let> be as in Lemma..Then for everyλ∈(,),there exists u∈X such that
J(u) =inf J(u) :u∈ ¯Bρ
< ,
and uis a solution of problem(.).
Proof Sinceh∈L–p\ {}withh+= , we can choose a functionψ∈Hα(RN) such that
RNh(x)|
ψ|pdx> .
Hence, we have
J(tψ) =t
ψ
–
RNF(x,t
ψ)dx–λt
p
p
RNh(x)| ψ|pdx
≤t
ψ
–λtp p
RN
h(x)|ψ|pdx< (.)
fort> small enough, which impliesθ:=inf{J(u) :u∈ ¯Bρ}< . By Ekeland’s variational principle, there exists a minimizing sequence{un} ⊂ ¯Bρsuch thatJ(un)→θandJ(un)→
asn→ ∞. Hence Lemma . implies that there existsu∈Xsuch thatJ(u) = and
J(u) =c< .
¯
uis another solution of problem (.). Thus, combining with Lemma ., we prove that problem (.) has at least two solutionsu,u¯∈XsatisfyingJ(u) < andJ(u¯) > .
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This work was supported by the Natural Science Foundation of Hunan Province (12JJ9001), Hunan Provincial Science and Technology Department of Science and Technology Project (2012SK3117) and Construct program of the key discipline in Hunan Province.
Received: 17 January 2015 Accepted: 17 March 2015
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