# Existence Results for apx-Kirchhoff-Type Equation without Ambrosetti-Rabinowitz Condition

Authors: Libo Wang; Minghe Pei
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We consider the existence and multiplicity of solutions for thepx-Kirchhoff-type equations without Ambrosetti-Rabinowitz condition. Using the Mountain Pass Lemma, the Fountain Theorem, and its dual, the existence of solutions and infinitely many solutions were obtained, respectively.


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## Body

## 1. Introduction

The Kirchhoff equation(1)ρ∂2u∂t2-(ρ0h+E2L∫0L|∂u∂x|dx)∂2u∂x2=0
was introduced by Kirchhoff [1] in the study of oscillations of stretched strings and plates, where ρ,ρ0,h,E, andL are constants. The stationary analogue of the Kirchhoff equation, that is, (1), is as follows:
(2)-(a+b∫Ω‍|∇u|2dx)Δu=f(x,u).
After the excellent work of Lions [2], problem (2) has received more attention; see [3–10] and references therein.Thep(x)-Laplace operator arises from various phenomena, for instance, the image restoration [11], the electro-rheological fluids [12], and the thermoconvective flows of non-Newtonian fluids [13, 14]. The study of the p(x)-Laplace operator is based on the theory of the generalized Lebesgue space Lp(x)(Ω) and the Sobolev space Wm,p(x)(Ω), which we called variable exponent Lebesgue and Sobolev space. We refer the reader to [15–19] for an overview on the variable exponent Sobo-lev space, and to [20–29] for the study of the p(x)-Laplacian-type equations.Recently, there has been an increasing interest in studying the Kirchhoff equation involving thep(x)-Laplace operator. Autuori et al. [30, 31] have dealt with the nonstationary Kirchhoff-type equation involving the p(x)-Laplacian of the form
(3)utt-M(∫Ω‍1p(x)|∇u|p(x)dx)Δp(x)u+Q(t,x,u,ut)+f(x,u)=0,utt-M(∫Ω‍1p(x)|∇u|p(x)dx)Δp(x)u+μ|∇u|p(x)-2u+Q(t,x,u,ut)=f(t,x,u).In [32–35], applying variational method and Ambrosetti-Rabinowitz (AR) condition, Guowei Dai has studied the existence and multiplicity of solutions for the p(x)-Kirchhoff-type equations with Dirichlet or Neumann boundary condition. In [36], by using (S+) mapping theory and the Mountain Pass Lemma, Fan has discussed the nonlocal p(x)-Laplacian Dirichlet problem with the nonvariational form
(4)-A(u)Δp(x)u=B(u)f(x,u),inΩ,u=0,on∂Ω,
and the p(x)-Kirchhoff-type equation with the variational form
(5)-a(∫Ω‍1p(x)|∇u|p(x)dx)Δp(x)u=b(∫Ω‍F(x,u)dx)f(x,u),inΩ,u=0,on∂Ω,
under (AR) condition, where A,B are two functionals defined on W01,p(x)(Ω), and F(x,t)=∫0t‍f(x,s)ds.Motivated by the above works, the purpose of this paper is to study thep(x)-Kirchhoff-type equation
(6)-(a+b∫Ω‍1p(x)|∇u|p(x)dx)Δp(x)u=f(x,u),inΩ,u=0,on∂Ω,
without (AR) condition, where Ω is a smooth bounded domain in ℝN, a,b are two positive constants, Δp(x)u=div(|∇u(x)|p(x)-2∇u(x)), p∈C0,β(Ω-) for some β∈(0,1), and
(7)1<p-:=infΩp(x)≤p+:=supΩp(x)<+∞.
By taking the famous Mountain Pass Lemma, the Fountain Theorem, and its dual, we obtain the existence of solutions and infinitely many solutions for the p(x)-Kirchhoff-type equation  (6) under no (AR) condition.


## 2. Preliminary

We recall in this section some definitions and properties of variable exponent Lebesgue-Sobolev space. The variable exponent Lebesgue spaceLp(x)(Ω) is defined by
(8)Lp(x)(Ω)={∫Ω‍u:u:Ω→ℝismeasurable,∫Ω‍|u|p(x)dx<∞}
with the norm
(9)|u|Lp(x)=|u|p(x)=inf{σ>0:∫Ω‍|uσ|p(x)dx≤1}.
The variable exponent Sobolev space W1,p(x)(Ω) is defined by
(10)W1,p(x)(Ω)={u∈Lp(x)(Ω):|∇u|∈Lp(x)(Ω)}
with the norm
(11)∥u∥W1,p(x)=∥u∥1,p(x)=|u|p(x)+|∇u|p(x).
Denote by W01,p(x)(Ω) the closure of C0∞(Ω) in W1,p(x)(Ω). |∇u|p(x) is an equivalent norm on W01,p(x)(Ω). In this paper we use the notation ∥u∥=|∇u|p(x) for u∈W01,p(x)(Ω). Define
(12)p*(x)={Np(x)N-p(x)ifp(x)<N,+∞ifp(x)≥N.
We refer the reader to [36–38] for the elementary properties of the space W1,p(x)(Ω).Proposition 1 (see [38]).
Setρ(u)=∫Ω‍|u(x)|p(x)dx. For any u∈Lp(x)(Ω), then the following are given:(1)
|u|p(x)=λ⇔ρ(u/λ)=1  if  u≠0;(2)
|u|p(x)<1(=1;>1)⇔ρ(u)<1(=1;>1);(3)
|u|p(x)p-≤ρ(u)≤|u|p(x)p+  if  |u|p(x)>1;(4)
|u|p(x)p+≤ρ(u)≤|u|p(x)p-  if    |u|p(x)<1;(5)
limk→+∞|uk|p(x)=0⇔limk→+∞ρ(uk)=0;(6)
limk→+∞|uk|p(x)=+∞⇔limk→+∞ρ(uk)=+∞.


## 3. Positive Energy Solution

In this section we discuss the existence of weak solutions of (6). For simplicity we write X=W01,p(x)(Ω).First, we state the assumptions onf as follows. (f0) Let f:Ω×ℝ→ℝ be a continuous function, and there exist positive constants c1,c2 such that
(13)|f(x,t)|≤c1+c2|t|α(x)-1,
where α∈C(Ω-) and 1<α(x)<p*(x) for all x∈Ω-.(f0′) Let f:Ω×ℝ→ℝ be a continuous function, and there exist positive constants c1,c2 such that
(14)|f(x,t)|≤c1+c2|t|α(x)-1,
where α∈C(Ω-) and p+<α(x)<p*(x) for all x∈Ω-; tf(x,t)≥0 for all t>0.(f1) Let limt→+∞(F(x,t)/|t|2p+)=+∞, uniformly for x∈Ω-, where F(x,t)=∫0t‍f(x,s)ds.(f2) There exists θ≥1 such that θG(x,t)≥G(x,st) for (x,t)∈Ω×ℝ and s∈[0,1], where
(15)G(x,t)=tf(x,t)-2p+F(x,t).(f3) Let limt→0(F(x,t)/|t|p+)=0, uniformly on x∈Ω.(f3′) There exists δ>0, such that F(x,t)≤0 for x∈Ω-, |t|<δ.(f4) Let f(x,t)=-f(x,-t) for x∈Ω and t∈ℝ.(f5) Let limt→0(F(x,t)/|t|q+)=0, uniformly on x∈Ω-, where q∈C(Ω-) satisfies 1<q(x)<p(x) for x∈Ω-.Remark 2.
Condition(f2) was first introduced by Jeanjean [39] for the case p(x)=2. Let f(x,t)=2p+|t|2p+-2tln|t|, then
(16)F(x,t)=|t|2p+ln|t|-12p+|t|2p+,G(x,t)=|t|2p+.
It is easy to see that the function f does not satisfy (AR) condition, but it satisfies (f1)–(f5) and (f3′).
DefineI(u)=J(u)-Φ(u), where
(17)J(u)=(a+b2∫Ω‍1p(x)|∇u|p(x)dx)∫Ω‍1p(x)|∇u|p(x)dx,Φ(u)=∫Ω‍F(x,u)du.
Then I∈C1(X,ℝ).Proposition 3 (see [38]).
Assume that(f0) hold, then the functional J:X→ℝ is sequentially weakly lower semicontinuous, Φ:X→ℝ is sequentially weakly continuous, and I is sequentially weakly lower semicontinuous.Proposition 4 (see [37]).
Assume that(f0) hold, and let u0∈W01,p(x)(Ω) be a local minimizer (resp., a strictly local minimizer) of I in the C1(Ω-) topology. Then u0 is a local minimizer (resp., a strictly local minimizer) of I in the W01,p(x)(Ω) topology.Definition 5.
We say thatu∈X is a weak solution of (6), if
(18)(a+b∫Ω‍1p(x)|∇u|p(x)dx)∫Ω‍|∇u|p(x)-2∇u∇vdx=∫Ω‍f(x,u)vdx
for any v∈X.Definition 6.
LetX be a Banach space and I∈C1(X,ℝ). Given c∈ℝ. we say that I satisfies the Cerami c condition (we denote by(C)c the condition), if (i)
any bounded sequence{un}⊂X such that I(un)→c and I′(un)→0 has a convergent subsequence;(ii)
there exist constantsδ,R,β>0 such that
(19)∥u∥∥I′(u)∥≥β,∀u∈I-1[c-δ,c+δ],∥u∥≥R.IfI∈C1(X,ℝ) satisfies (C)c condition for every c∈ℝ, then we say that I satisfies (C) condition.Remark 7.
Although (PS) condition is stronger than(C) condition, the Deformation Theorem is still valid under (C) condition; we see that the Mountain Pass Lemma, the Fountain Theorem, and its dual are true under (C) condition.Lemma 8.
Assume that conditions(f0)–(f2) hold. Then I satisfies (C) condition.Proof.
From [36, Proposition 3.1], I satisfies (i) of (C) condition.
Now we check thatI satisfies (ii) of (C) condition. Arguing by contradiction, we may assume that, for some c∈ℝ,
(20)I(un)⟶c,∥un∥⟶∞,∥un∥∥I′(un)∥⟶0.
Then we have
(21)limn→∞{a∫Ω‍(1p(x)-12p+)|∇u|p(x)dx+b2∫Ω‍1p(x)|∇u|p(x)dx×∫Ω‍(1p(x)-1p+)|∇u|p(x)dx+12p+∫Ω‍G(x,u)dx}=limn→∞{I(un)-12p+〈I′(un),un〉}=c.
Let vn=un/∥un∥, then up to a subsequence we may assume that
(22)vn⇀vinX,vn⟶vinLα(x)(Ω),vn(x)⟶v(x)a.e.x∈Ω.
If v=0, inspired by [13, 14], then we define
(23)I(tnun)=maxt∈[0,1]I(tun).
For any m>1/2p+, let wn=(2mp+)1/p-vn. Since wn→0 in Lα(x)(Ω) and
(24)|F(x,t)|≤c5+c6|t|α(x),
by the continuity of F(x,·), F(·,wn)→0 in L1(Ω), thus,
(25)limn→0∫Ω‍F(·,wn)dx=0.
Then for n large enough, (2mp+)1/p-/∥un∥∈(0,1) and
(26)I(tnun)≥I(wn)=a∫Ω‍1p(x)|∇wn|p(x)dx+b2(∫Ω‍1p(x)|∇wn|p(x)dx)2-∫Ω‍F(x,wn)dx=a∫Ω‍1p(x)((2mp+)1/p-|∇vn|)p(x)dx+b2(∫Ω‍1p(x)((2mp+)1/p-|∇vn|)p(x)dx)2-∫Ω‍F(x,wn)dx≥2map+∫Ω‍|∇vn|p(x)dx+2m2b(p+)2(∫Ω‍|∇vn|p(x)dx)2-∫Ω‍F(x,wn)dx≥2map++2m2b(p+)2-∫Ω‍F(x,wn)dx.
That is, I(tnun)→∞. From I(0)=0 and I(un)→c, we know that tn∈(0,1) and
(27)a∫Ω‍|∇tnun|p(x)dx+b(∫Ω‍1p(x)|∇tnun|p(x)dx)∫Ω‍|∇tnun|p(x)dx-∫Ω‍f(x,tnun)undx=〈I′(tnun),tnun〉=tnddt|t=tnI(tun)=0.
Therefore, from (f2), we have
(28)a∫Ω‍(1p(x)-12p+)|∇un|p(x)dx+b2∫Ω‍1p(x)|∇un|p(x)dx×∫Ω‍(1p(x)-1p+)|∇un|p(x)dx+12p+∫Ω‍G(x,un)dx≥a∫Ω‍(1p(x)-12p+)|∇un|p(x)dx+b2∫Ω‍1p(x)|∇un|p(x)dx×∫Ω‍(1p(x)-1p+)|∇un|p(x)dx+12p+∫Ω‍G(x,tnun)θdx≥aθ∫Ω‍(1p(x)-12p+)tnp(x)|∇un|p(x)dx+b2θ∫Ω‍1p(x)tnp(x)|∇un|p(x)dx×∫Ω‍(1p(x)-1p+)tnp(x)|∇un|p(x)dx+12p+∫Ω‍G(x,tnun)θdx=1θ(I(tnun)-12p+〈I′(tnun),tnun〉)=1θI(tnun)→∞.
This contradicts (21).
Ifv≠0, from (20), when ∥un∥≥1,
(29)ap-∥un∥p++b2(p-)2∥un∥2p+-(c+o(1))≥∫Ω‍F(x,un)dx.
Then from (f1) we have
(30)ap-1∥un∥p++b2(p-)2-c+o(1)∥un∥2p+≥∫Ω‍F(x,un)∥un∥2p+dx=(∫vn≠0‍+∫vn=0‍)F(x,un)|un|2p+|vn|2p+dx=∫vn≠0‍F(x,un)|un|2p+|vn|2p+dx.
For x∈Θ:={x∈Ω:v(x)≠0}, |un(x)|→+∞. By (f1) we have
(31)F(x,un)|un|p+|vn|p+⟶+∞.
Note that the Lebesgue measure of Θ is positive; using the Fatou Lemma, we have
(32)∫vn≠0‍F(x,un)|un|2p+|vn|2p+dx⟶+∞.
This contradicts (30).
The technique used in this lemma was first introduced by [39, 40].Theorem 9.
Assume that conditions(f0)–(f2) and (f3) (or (f3′)) hold. Then (6) has a nontrivial solution with positive energy.Proof.
From Lemma8, I satisfies (C) condition. Let us show that the functional I has a Mountain-Pass-type geometry.
Note thatI(0)=0. By (f3), there exists δ>0,  and  for  any u∈X with |u|L∞(Ω)<δ,
(33)I(u)=a∫Ω‍1p(x)|∇u|p(x)dx+b2(∫Ω‍1p(x)|∇u|p(x)dx)2-∫Ω‍F(x,u)dx≥ap+∥u∥p++b(p+)2∥u∥2p+-∫Ω‍F(x,u)dx>0.
This shows that 0 is a strictly local minimizer of I in the C(Ω-) topology, and hence 0 is a strictly local minimizer of I in the C1(Ω-) topology. By [37, Theorem 1.1], 0 is a strictly local minimizer of I in the W01,p(x)(Ω) topology. Thus there exists r>0 such that I(u)>0 for every u∈X∖{0} with ∥u∥≤r.
We claim thatinf∥u∥=rI(u)>0. To prove this claim, arguing by contradiction, assume that there exists a sequence {un}⊂X with ∥un∥=r such that I(un)→0 as n→∞. We may assume that un⇀u0 in X. Since I is sequentially weakly lower semicontinuous, we have that I(u0)=0, and hence u0=0. Since Φ is sequentially weakly continuous, then we have that Φ(un)→Φ(0)=0, and hence J(un)=I(un)+Φ(un)→0. It follows from this that un→0 in X which contradicts with ∥un∥=r.
Lety∈X with y>0 in Ω and ∥y∥=1. By (f0) and (f1), for s≥1 we have
(34)I(sy)=a∫Ω‍1p(x)|∇sy|p(x)dx+b(∫Ω‍1p(x)|∇sy|p(x)dx)2-∫Ω‍F(x,sy)dx≤ap-sp++b(p-)2s2p+-c1s2p+∫Ω‍|y|2p+dx+c2⟶-∞ass⟶+∞.
We set e=sy. Then for s large, we obtain
(35)∥e∥>r,I(e)<0.
Hence by the famous Mountain Pass Lemma, problem (6) has a nontrivial weak solution with positive energy.


## 4. Infinitely Many Solutions

SinceX is a reflexive and separable Banach space, then there exists {ej}⊂X and {ej*}⊂X* such that
(36)X=span{ej:j=1,2,…}¯,X*=span{ej*:j=1,2,…}¯,〈ei,ej*,〉={1,i=j,0,i≠j.
For convenience, we write Xj=span{ej}, Yk=⊕j=1kXj, Zk=⊕j=k∞Xj¯.Lemma 10 (see [21]).
Ifα∈C(Ω-),  1<α(x)<p* for any x∈Ω-,  denote
(37)βk=sup{|u|α(x):∥u∥=1,u∈Zk}.
Then limk→+∞βk=0.Proposition 11 (Fountain Theorem).
Assume thatI∈C1(X,ℝ) is an even functional. If, for any k∈ℕ, there exists ρk>rk>0 such that(A1)ak=maxu∈Yk,∥u∥=ρkI(u)≤0,(A2)bk=infu∈Zk,∥u∥=rkI(u)→+∞   as  k→∞,(A3)I  satisfies  (C)c  condition for everyc>0,then  I  has an unbounded sequence of critical values.Proposition 12 (Dual Fountain Theorem).
Assume thatI∈C1(X,ℝ) is an even functional. If, for any k≥k0, there exists ρk>rk>0 such that(B1)ak=infu∈Zk,∥u∥=ρkI(u)≥0,(B2)bk=maxu∈Yk,∥u∥=rkI(u)<0,(B3)dk=infu∈Zk,∥u∥≤ρkI(u)→0  as  k→∞,(B4)I  satisfies  (c)c*  condition for every  c∈[dk0,0],then  I  has a sequence of negative critical values converging to0.Theorem 13.
Assume that the conditions(f0′), (f1)–(f4) hold. Then (6) has infinitely many solutions {uk} such that I(uk)→∞ as k→∞.Proof.
By conditions(f0′), (f1), and (f3), for any ε>0, there exists Cε such that
(38)F(x,u)≥Cε|u|2p+-ε|u|p+,∀(x,u)∈Ω×ℝ.
For u∈Yk, when ∥u∥>1,
(39)I(u)=a∫Ω‍1p(x)|∇u|p(x)dx+b2(∫Ω‍1p(x)|∇u|p(x)dx)2-∫Ω‍F(x,u)dx≤ap-∥u∥p++b2(p-)2∥u∥2p+-Cε|u|2p+2p++ε|u|p+p+⟶-∞as∥u∥⟶+∞.
Then for some ρk>0 large enough,
(40)ak:=maxu∈Yk,∥u∥=ρkI(u)≤0.
On the other hand, by (f0′) and (f3), there exists Cε>0 such that
(41)|F(x,u)|≤ε|u|p++Cε|u|α(x),∀(x,u)∈Ω×ℝ.
Let βk:=supu∈Zk,∥u∥=ρk|u|α-. From Lemma 10,  βk→0 as k→∞. For u∈Zk, when ∥u∥≤1 and ε small enough,
(42)I(u)=a∫Ω‍1p(x)|∇u|p(x)dx+b2(∫Ω‍1p(x)|∇u|p(x)dx)2-∫Ω‍F(x,u)dx≥ap+∥u∥p-+b2(p+)2∥u∥2p--Cε|u|α-α--ε|u|p+p+≥a2p+∥u∥p--c|u|α-α-≥a2p+∥u∥p--cβk∥u∥α-.
If we choose rk:=(a/4cp+βkα-)1/(α--p-)→∞ as k→∞, then, for u∈Zk with ∥u∥=rk,
(43)I(u)≥a4p+(a4cp+βkα-)p-/(α--p-):=b-k,
which implies that bk:=infu∈Zk,∥u∥=rkI(u)≥b-k→+∞ as k→+∞.Theorem 14.
Assume that conditions(f0′), (f1), (f2), (f4), and (f5) hold. Then (6) has infinitely many solutions {uk} such that I(uk)→0 as k→∞.Proof.
By conditions(f0′), (f1), and (f5), for any ε>0, there exists Cε such that
(44)F(x,u)≥Cε|u|2p+-ε|u|q+,∀(x,u)∈Ω×ℝ.
For u∈Yk, when ∥u∥ is large enough,
(45)I(u)=a∫Ω‍1p(x)|∇u|p(x)dx+b2(∫Ω‍1p(x)|∇u|p(x)dx)2-∫Ω‍F(x,u)dx≤ap-∥u∥p++b(p-)2∥u∥2p+-Cε|u|2p+2p++ε|u|q-q-⟶-∞as∥u∥⟶+∞.
Then for some rk>0 large enough,
(46)bk:=maxu∈Yk,∥u∥=rkI(u)<0.
On the other hand, by (f5), there exists Cε>0 such that
(47)|F(x,u)|≤ε|u|q-+Cε|u|α(x),∀(x,u)∈Ω×ℝ.
Let βk:=supu∈Zk,∥u∥=rk|u|q-, then βk→0 as k→∞. For u∈Zk, when ∥u∥ and ε small enough,
(48)I(u)=a∫Ω‍1p(x)|∇u|p(x)dx+b2(∫Ω‍1p(x)|∇u|p(x)dx)2-∫Ω‍F(x,u)dx≥ap+∥u∥p+-cCε|u|α-α--cε|u|q+q+≥a2p+∥u∥p+-c|u|q+q+≥a2p+∥u∥p+-cβkq+∥u∥q+.
If we choose ρk:=(4cp+βkq+/a)1/(p+-q+)→0 as k→∞, then, for u∈Zk with ∥u∥=ρk,
(49)I(u)≥cβkq+(4cp+βkq+a)q+/(p+-q+):=a-k,
which implies that ak:=infu∈Zk,∥u∥=ρkI(u)≥a-k→0 as k→+∞.
Furthermore, ifu∈Zk with ∥u∥≤ρk, then
(50)I(u)≥-cβkq-ρkq-⟶0ask⟶∞,
which implies that dk=infu∈Zk,∥u∥≤ρkI(u)→0 as k→∞.



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