Vol. 22 (2021), No. 1, pp. 351–362 DOI: 10.18514/MMN.2021.3429
EXISTENCE RESULTS FOR A KIRCHHOFF-TYPE PROBLEM WITH SINGULARITY
M. KHODABAKHSHI, S.M. VAEZPOUR, AND M. R. HEIDARI TAVANI Received 11 September, 2020
Abstract. In this work, using an infinitely many critical points theorem we establish the existence of a sequence of weak solutions for a Kirchhoff-type problem with singular term. This approach is based on variational methods and critical point theory.
2010Mathematics Subject Classification: 35J35; 35J60
Keywords: singularity, Kirchhoff type problems, variational methods, critical point
1. INTRODUCTION
In 1883, the stationary problem ρ∂2u
∂t2 −
ρ0
h + E 2L
L
Z
0
∂u
∂x
2
dx
∂2u
∂x2 =0,
was proposed by Kirchhoff [14] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. In recent years, the study of elliptic problems involving Kirchhoff type operators have been studied in many works, we refer to [1,3,5–7,16–18,20,22]. For instance, in [17], Molica Bisci and Pizzimenti considered the following problem
−
a+bR
Ω
|∇u|pdx
∆pu+α(x)|u|p−2u=λh(x)f(u) inΩ,
u=0 on∂Ω.
They obtained the existence of infinitely many weak solutions by using variational methods. Also, in [5], the authors studied the non-local problem
−M
R
Ω
|∇u|pdx
∆pu= f(x,u) inΩ,
u=0 on∂Ω.
We gratefully thank the Iran National Science Foundation (INFS) for financial support.
© 2021 Miskolc University Press
By using Browder Theorem, the writers proved the existence and uniqueness of solu- tions. On the other hand, singular elliptic problems have been intensively studied in the last decads. Among others, we mention the works [8–12,15,19,21]. Recently, motivated by this large interest, Ferrara and Molica Bisci in [8] studied the existence of at least one non-trivial weak solution for the following elliptic Dirichlet problem
−∆pu=µ|u|p−2u
|x|p +λf(x,u) inΩ, u|∂Ω=0,
whereλ>0 andµ≥0 are two real parameters,Ωis a bounded domain inRN(N≥2) containing the origin and with smooth boundary∂Ω, 1<p<N and f :Ω×R→R is a Carath´eodory function satisfying a suitable subcritical growth condition.
The aim of this paper is to investigate the existence of infinitely many weak solu- tions for the following problem
−M
R
Ω
|∇u|pdx
∆pu+|u|
q−2u
|x|q =λf(x,u) inΩ,
u=0 on∂Ω,
(1.1) where∆pu:=div(|∇u|p−2∇u) denotes the p-Laplace operator, Ωis a bounded do- main in RN(N ≥ 2) containing the origin and with smooth boundary ∂Ω, 1<q<N<p,M:[0,+∞)→Ris a continuous function satisfying
(f1) there are two positive constantsm0,m1, such that m0≤M(t)≤m1, ∀t≥0, and f :Ω×R→Ris anL1-Carath´eodory function.
Recall that a function f:Ω×R→Ris said to be anL1-Carath´eodory function, if (C1) the functionx7→ f(x,t)is measurable for everyt∈R;
(C2) the functiont7→ f(x,t)is continuous for a.e.x∈Ω;
(C3) for everyρ>0 there exists a functionlρ∈L1(Ω)such that sup
|t|≤ρ
|f(x,t)| ≤lρ(x), for a.e.x∈Ω.
A special case of our main result is the following theorem.
Theorem 1. Assume that f :R→Ris a non-negative continuous function such that
lim inf
ξ→+∞
f(ξ)
ξp−1 =0 and lim sup
ξ→+∞
Rξ 0 f(t)dt
ξp = +∞.
Then, the problem
−
1+2
R
Ω
|∇u|pdx 2
1+
R
Ω
|∇u|pdx 2
∆pu+|u||x|q−2qu= f(u) inΩ,
u=0 on∂Ω
admits a sequence of weak solutions which is unbounded in X. 1.1. Preliminary considerations
LetΩbe a bounded domain inRN(N≥2)containing the origin and with smooth boundary∂Ω. Further, denote byXthe spaceW01,p(Ω)endowed with the norm
kuk:=
Z
Ω
|∇u(x)|pdx
!1/p
. Also, letk · k1denotes the usual norm ofL1(Ω); i.e.,
kuk1:=
Z
Ω
|u(x)|dx.
We recall classical Hardy’s inequality, which says that Z
Ω
|u(x)|q
|x|q dx≤ 1 H
Z
Ω
|∇u(x)|qdx, (∀u∈X), (1.2) whereH:= (N−qq )q; see, for instance, the paper [2].
Let us defineF(x,ξ):=R0ξf(x,t)dt, for every(x,ξ)inΩ×R.Moreover we intro- duce the functionalIλ:X→Rassociated with (1.1),
Iλ(u):=Φ(u)−λΨ(u), for everyu∈X,where
Φ(u):= 1
pMˆ(kukp) +1 q Z
Ω
|u(x)|q
|x|q dx, and
Ψ(u):=
Z
Ω
F(x,u(x))dx, for everyu∈X,where ˆM(t):=
t
R
0
M(s)ds, t≥0. By standard arguments, one has thatΦis well defined (by Hardy’s inequality), Gˆateaux differentiable and sequentially weakly lower semicontinuous, and its Gˆateaux derivative is the functionalΦ0(u)∈X∗ given by
Φ0(u)(v) =M
Z
Ω
|∇u|pdx
Z
Ω
|∇u(x)|p−2∇u(x)∇v(x)dx+
Z
Ω
|u(x)|q−2
|x|q u(x)v(x)dx,
for every v∈X and clearly Φ is coercive. It is easy to prove that Φ is strongly continuous. On the other hand, standard arguments show thatΨis well defined and continuously Gˆateaux differentiable functional whose Gˆateaux derivative
Ψ0(u)(v) = Z
Ω
f(x,u(x))v(x)dx, for everyv∈X, is a compact operator fromXto the dualX∗.
Fixing the real parameterλ, a functionu:Ω→Ris said to be a weak solution of (1.1) ifu∈X and
M
Z
Ω
|∇u|pdx
Z
Ω
|∇u(x)|p−2∇u(x)∇v(x)dx
+ Z
Ω
|u(x)|q−2
|x|q u(x)v(x)dx−λ Z
Ω
f(x,u(x))v(x)dx=0, for every v∈X. Hence, the critical points ofIλ are exactly the weak solutions of (1.1).
Our main tool to investigate the existence of infinitely many solutions for the prob- lem (1.1) is a smooth version of [4, Theorem 2.1] which is a more precise version of Ricceri’s variational principle [19, Theorem 2.5], which we now recall.
Theorem 2. Let X be a reflexive real Banach space, let Φ,Ψ:X →R be two Gˆaateaux differentiable functionals such that Φis sequentially weakly lower semi- continuous, strongly continuous and coercive, andΨ is sequentially weakly upper semicontinuous. For every r>infXΦ,put
ϕ(r):= inf
Φ(u)<r
supΦ(v)<rΨ(v)
−Ψ(u)
r−Φ(u) ,
γ:=lim inf
r→+∞ϕ(r), and δ:= lim inf
r→(infXΦ)+ϕ(r).
Then the following properties hold:
(a) For every r>infXΦand everyλ∈]0,1/ϕ(r)[,the restriction of the functional Iλ:=Φ−λΨ
toΦ−1(]−∞,r[)admits a global minimum, which is a critical point(local minimum)of Iλin X .
(b) Ifγ<+∞,then for eachλ∈]0,1/γ[, the following alternative holds: either (b1) Iλpossesses a global minimum, or
(b2) there is a sequence{un}of critical points(local minima)of Iλsuch that
n→+∞lim Φ(un) = +∞.
(c) Ifδ<+∞,then for eachλ∈]0,1/δ[, the following alternative holds: either
(c1) there is a global minimum ofΦwhich is a local minimum of Iλ, or (c2) there is a sequence{un}of pairwise distinct critical points(local minima)
of Iλwhich weakly converges to a global minimum ofΦ,with
n→+∞lim Φ(un) =inf
u∈XΦ(u).
2. MAIN RESULTS
Put
k:= sup
u∈X,u6=0
maxx∈Ω¯|u(x)|
kuk
. (2.1)
Since the embeddingX,→C(Ω)¯ is compact, one hask<+∞. Fixx0∈ΩandD>0 such thatB(x0,D)⊂ΩandB(x0,D)not containing the origin, whereB(x0,D)denotes the ball with center atx0and radiusD.
Put
ω:=m1 p
2 D
p
m
DN−D 2
N
, (2.2)
α:=
Z
B(x0,D2)
1
|x|qdx , β:=
2 D
qZ
B(x0,D)\B(x0,D2)
(D− |x−x0|)q
|x|q dx, (2.3) wherem:=Γ(1+πN/2N
2).HereΓis the Gamma function defined by Γ(t):=
Z +∞
0
zt−1e−zdz (∀t>0). Put
A:=lim inf
ξ→+∞
klξk1 ξp−1, and
B:=lim sup
ξ→+∞
R
B(x0,D2)F(x,ξ)dx ξp ,
wherelξ∈L1(Ω)satisfies condition(C3)on f(x,t)for everyξ>0.
Our main result is the following.
Theorem 3. Assume that M:[0,+∞[→R is a continuous function satisfying(f1).
Also let f :Ω×R→Rbe an L1-Carath´eodory function such that (i) F(x,t)≥0for every(x,t)∈Ω×R+,
(ii) A< 1
pωkpB,where k andωare given by(2.1)and(2.2),respectively.
Then, for every λ∈Λ:=
i
ω B,pk1pA
h
,the problem (1.1) admits a sequence of weak solutions which is unbounded in X .
Proof. Fix λ∈i
ω B,pk1pA
h
. Our aim is to apply Theorem 2 part (b) with X :=
W01,p(Ω)and whereΦandΨare the functionals introduced in Section 2. As seen be- fore, the functionalsΦandΨsatisfy the regularity assumptions requested in Theorem 2. Now, we look on the existence of critical points of the functionalIλ:=Φ−λΨin X. To this end, we take{ξn} ⊂R+such that limn→+∞ξn= +∞,and
n→+∞lim klξnk1
ξp−1n
=A.
Setrn:= m0ξ
p n
pkp for alln∈N.From (2.1) we get max
x∈Ω¯
|u(x)| ≤kkuk, (2.4)
for everyu∈X.Then, for eachu∈X withΦ(u)<rn,we have max
x∈Ω¯
|u(x)| ≤k( p
m0Φ(u))1/p<k( p
m0rn)1/p=ξn. Then, sinceΦ(0) =Ψ(0) =0,we have
ϕ(rn) = inf
Φ(v)<rn
supΦ(u)<rnΨ(u)
−Ψ(v) rn−Φ(u)
≤supΦ(u)<rnRΩF(x,u(x))dx rn
≤ξnklξnk1 m0 ξ
np
pkp
.
Hence, it follows that γ≤lim inf
n→+∞ϕ(rn)≤ p
m0kplim inf
n→+∞
klξnk1 ξnp−1
= p
m0kpA<+∞,
since condition(ii)yieldsA<+∞.Now, we claim that the functionalIλis unbounded from below. Let{dn}be a real sequence such that limn→+∞dn= +∞and
n→+∞lim R
B(x0,D2)F(x,dn)dx dnp
=B. (2.5)
Further, for eachn≥1,definevn∈X given by
vn(x):=
0, x∈Ω\B(x0,D),
dn, x∈B(x0,D2), 2dn
D (D− |x−x0|), x∈B(x0,D)\B(x0,D2).
(2.6)
By using condition(i), we infer Ψ(vn) =
Z
Ω
F(x,vn(x))dx≥ Z
B(x0,D2)
F(x,dn)dx, for everyn≥1.Then, we have
Iλ(vn)≤ωdnp+α+β q dnq−λ
Z
B(x0,D2)
F(x,dn)dx.
IfB<+∞, let
δ∈i ω λB,1h
. By (2.5), there existsNδsuch that
Z
B(x0,D2)
F(x,dn)dx>δBdnp, (∀n>Nδ).
Consequently, one has
Iλ(vn)<ωdnp+α+β
q dnq−λδBdnp
= (ω−λδB)dnp+α+β q dnq, for everyn>Nδ.Then, it follows that
n→+∞lim Iλ(vn) =−∞, sinceq<p.
If B = +∞, let us considerL>ωλ.By (2.5), there existsNLsuch that Z
B(x0,D2)
F(x,dn)dx>Ldnp, (∀n>NL).
So, we have
Iλ(vn)<ωdnp+α+β
q dnq−λLdnp
= (ω−λL)dnp+α+β
q dnq (∀n>NL).
Taking into account the choice ofL,also in this case, one has
n→+∞lim Iλ(vn) =−∞,
sinceq<p. Therefore owing to Theorem2(b), the functionalIλadmits an unboun- ded sequence{un} ⊂Xof critical points. Then the problem (1.1) admits a sequence
of weak solutions which is unbounded inX.
Among the consequences of Theorem3, we point out the following result.
Corollary 1. Let f :Ω×R→Rbe an L1-Carath´eodory function. Assume that condition(i)of Theorem3holds. Further, require that
(iii) A< pk1p and B>ω,where k andωare given by(2.1)and(2.2), respectively.
Then the following problem
−M
Z
Ω
|∇u|pdx
∆pu+|u|q−2u
|x|q = f(x,u), inΩ,
u=0, on∂Ω,
admits a sequences of weak solutions which is unbounded in X.
Remark1. We note that assumption(ii) in Theorem3 could be replaced by the following more general hypothesis:
(ii0) There exists two positive sequences{an}and{bn}such that 1
p
ω m1anp
Z
0
M(s)ds+(α+β)
q aqn<m0bnp
pkp , (∀n≥1) and limn→+∞bn= +∞such that
Ae< B ω,
wherek,ωandα,βare given by (2.1), (2.2) and (2.3), respectively, and Ae:= lim
n→+∞
bnklbnk1−RB(x
0,D2)F(x,an)dx
m0bpn pkp −1p
ω m1anp
R
0
M(s)ds−(α+β)q aqn
.
Then, for every
λ∈ ω
B,1 Ae
,
the problem (1.1) admits a sequence of weak solutions which is unbounded inX.
Indeed, from(ii0) we obtain(ii), by choosingan=0 for alln∈N.Moreover, if we assume(ii0)instead of(ii)and setrn:=mpk0bppn for everyn≥1, one has
ϕ(rn):= inf
Φ(v)<rn
supΦ(u)<rnRΩF(x,u(x))dx
−RΩF(x,v(x))dx rn−Φ(v)
≤bnklbnk1−RΩF(x,vn(x))dx
m0bnp
pkp −Φ(vn)
≤ bnklbnk1−RB(x
0,D2)F(x,an)dx
m0bpn pkp −1p
ω m1anp
R
0
M(s)ds−(α+β)q aqn
,
by choosing
vn(x):=
0, x∈Ω\B(x0,D),
an, x∈B(x0,D2), 2an
D (D− |x−x0|), x∈B(x0,D)\B(x0,D2),
for everyn≥1. Therefore, since by assumption(ii0)one hasAe<+∞,we obtain γ≤lim inf
n→+∞ϕ(rn)≤Ae<+∞.
From now on, arguing as in the proof of Theorem3, the conclusion follows.
Now, we present the other main result. First, put A0:=lim inf
ξ→0+
klξk1
ξp−1, B0:=lim sup
ξ→0+
R
B(x0,D2)F(x,ξ)dx ξp .
Arguing as in the proof of Theorem3but using conclusion(c)of Theorem2instead of(b), the following result holds.
Theorem 4. Assume that f :Ω×R→Rbe an L1-Carath´eodory function such that hypothesis(i)in Theorem3holds, and
(iv) A0< pωkm0pB0.
Then, for every λ∈Λ0 :=i
ω B0,pkmp0A0
h
, the problem (1.1) has a sequence of weak solutions, which strongly converges to zero in X.
Proof. Fixλ∈Λ0.We takeΦ,ΨandIλas in Section 2.Now, as it has been pointed out before, the functionals Φ and Ψ satisfy the regularity assumptions reqired in Theorem2. As first step, we will prove thatλ<1/δ.Then, let{ξn}be a sequence of positive numbers such that limn→+∞ξn=0 and
n→+∞lim klξnk1
ξnp
=A0.
By the fact that infXΦ=0 and the definition ofδ,we haveδ:=lim infr→0+ϕ(r).Put rn:=mpk0ξpnp for alln∈N.Then, for allu∈XwithΦ(u)<rn,taking (2.4) into account, one haskuk∞<ξn.Thus, for alln∈N,
ϕ(rn)≤supΦ(u)<rnΨ(u) rn
≤ pkp m0
klξnk1 ξnp−1
.
Hence,
δ≤lim inf
n→+∞ϕ(rn)≤ pkp m0 lim inf
n→+∞
klξ
nk1 ξnp−1
= pkpA0
m0 <+∞, and thereforeΛ0⊂]0,1/δ[.
Letλbe fixed. We claim that the functionalIλdoes not have a local minimum at zero. Let{dn}be a sequence of positive numbers such that limn→+∞dn=0 and
n→+∞lim R
B(x0,D2)F(x,dn)dx dnp
=B0.
For alln∈N,let vn ∈X defined by (2.6) with the above dn.Now, with the same argument as in the proof of Theorem3, we achieveIλ(vn)<0 for everyn∈Nlarge enough. Then, since limn→+∞Iλ(vn) =Iλ(0) =0, we see that zero is not a local minimum of Iλ. This, together with the fact that zero is the only global minimum ofΦ,we deduce that the energy functionalIλdoes not have a local minimum at the unique global minimum ofΦ.Therefore, by Theorem2(c), there exists a sequence {un} of critical points of Iλ which converges weakly to zero. In view of the fact that the embeddingX ,→C(Ω)¯ is compact, we know that the critical points converge
strongly to zero, and the proof is complete.
We end this paper with the following example to illustrate our results.
Example1. Letr>0 be a real number and{tn}, {sn}be two strictly increasing sequences of real numbers that defined by induction
t1=r, s1=2r and forn≥1,
t2n= 22n+1−1
t2n−1, t2n+1=
2− 1 22n+1
t2n,
s2n=t2n 2n =
2− 1
22n
s2n−1, s2n+1=2n+1t2n+1= 22n+2−1 s2n. Let f :Ω×R→Rbe the function defined by
f(x,t):=
( 2ϕ(x)t, (x,t)∈Ω×[0,t1],
ϕ(x)
sn−1+stn−sn−1
n−tn−1 (t−tn−1)
, (x,t)∈Ω×[tn−1,tn]for somen>1, whereϕ:Ω→Ris a positive continuous function with 0<m≤ϕ(x)≤M. Then f is anL1-Carath´eodory function and since f(x,t)is strictly increasing with respect to targument at everyx∈Ω, the functionlξ(x):= f(x,ξ)satisfies in condition(C3)on
f; i.e.,
sup
|t|≤ξ
|f(x,t)| ≤lξ(x), for a.e.x∈Ω.
Arguing as in [13], we have lim sup
ξ→+∞
R
B(x0,D2)F(x,ξ)dx ξ
7 3
= +∞, lim inf
ξ→+∞
klξk1 ξ
4 3
=0,
for everyx0∈ΩandD>0 such thatB(x0,D)⊂ΩandB(x0,D)not containing the origin, whereΩis a bounded domain inR2 containing the origin and with smooth boundary∂Ω. Hence, by Theorem3, for everyλ∈]0,+∞[, the following problem
−M
Z
Ω
|∇u|73dx
∆7
3u+|u|−12 u
|x|32 =λf(x,u), inΩ,
u=0, on∂Ω,
possesses an unbounded sequence of weak solutions inW1,
7 3
0 (Ω).
ACKNOWLEDGEMENT
We would like to show our gratitude to the anonymous reviewer for their valuable comments and suggestions to improve the paper.
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Authors’ addresses
M. Khodabakhshi
Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran
E-mail address:m.khodabakhshi11@gmail.com
S.M. Vaezpour
Department of Mathematics and Computer Science, Amirkabir University of Thechnology, Tehran, Iran
E-mail address:vaez@aut.ac.ir
M. R. Heidari Tavani
Department of Mathematics, Ramhormoz Branch, Islamic Azad University, Ramhormoz, Iran E-mail address:m.reza.h56@gmail.com
