# Existence of Multiple Solutions for ap-Kirchhoff Problem with Nonlinear Boundary Conditions

Authors: Zonghu Xiu; Caisheng Chen
Journal: The Scientific World Journal
 (2013)

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## Abstract

The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem-M(∫Ω‍|x|-ap|∇u|p)div(|x|-ap|∇u|p-2∇u)=λh(x)|u|r-2u, x∈Ω, M(∫Ω‍|x|-ap|∇u|p)|x|-ap|∇u|p-2 (∂u/∂ν) = g(x)|u|q-2u, on ∂Ω, where 1<(N+1)/2<p<N, a<(N-p)/p. By the variational method on the Nehari manifold, we prove that the problem  has at least two positive solutions when some conditions are satisfied.


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## Body

## 1. Introduction and Main Result

In this paper, we consider the existence of multiple solutions for the singular elliptic problem:(1)-M(∫Ω‍|x|-ap|∇u|pdx)div(|x|-ap|∇u|p-2∇u)=λh(x)|u|r-2u,x∈Ω,M(∫Ω‍|x|-ap|∇u|p)|x|-ap|∇u|p-2∂u∂ν=g(x)|u|q-2u,on∂Ω,
where 1<(N+1)/2<p<N, a<(N-p)/p, λ>0, Ω is an exterior domain of ℝN: that is, and Ω=ℝN∖D, where D is a bounded domain in ℝN with the smooth boundary ∂D(=∂Ω), and 0∈Ω. g(x) and h(x) are continuous functions, M(s)=αs+β with the parameters α,β>0.Problem like (1) is usually called nonlocal problem because of the presence of the integral over the entire domain, and this implies that (1) is no longer a pointwise identity. In fact, such kind of problem can be traced back to the work of Kirchhoff. In [1], Kirchhoff investigated the model of the form
(2)ρ∂2u∂t2-(P0h+E2L∫0L‍|∂u∂x|2dx)∂2u∂x2=0,
where ρ, P0, h, E, and L are all positive constants. This equation extends the classical d’Alembert’s wave equation by considering the effects of changes in the length of the strings during the vibrations. Problem (1) is related to the stationary analogue of problem (2). After Kirchhoff’s work, various models of Kirchhoff-type have been studied by many authors: we refer the readers to [2–9]. In [4], by the variational methods, Bensedik and Bouchekif considered the problem
(3)-M(∫Ω‍|∇u|2dx)Δu=f(x,u),x∈Ω,u=0,on∂Ω,
where Ω is a bounded domain in ℝN. One of the assumptions made on f(x,t) in (3) is that (f1)f(x,t) is continuous function on Ω-×ℝ such that
(4)f(x,t)≥0,∀t≥0,x∈Ω-,f(x,t)=0,∀t≤0,x∈Ω-.
The authors proved that problem (3) has a positive solution or has no solution when some other assumptions are fulfilled. In our paper, however, the weight functions h(x) and g(x) are permitted to change sign. Thus, the methods in [4] cannot be directly applied on (1).In recent years, some other authors considered the Kirchhoff-type equations withp-Laplacian [10–13]. In fact, motivated by [4, 5] and our previous work [14], we consider the existence of multiple solutions for problem (1) on the Nehari manifold by variational methods. We prove that problem (1) has at least two positive solutions. Since Ω⊂ℝN is an unbounded domain and the problem is singular, the loss of compactness of the Sobolev embedding renders variational technique more delicate.In order to state our result, we introduce a weighted Sobolev spaceE=Wa1,p(Ω), which is the completion of the space C0∞(Ω) with the norm of
(5)∥u∥E=(∫Ω‍|x|-ap|∇u|pdx)1/p.
For ρ≥1 and f=f(x)>0 in Ω, we define the space Lρ(Ω,f) as being the set of Lebesgue measurable functions u:Ω→ℝ1, which satisfies
(6)∥u∥Lρ(Ω,f)=(∫Ω‍f(x)|u|ρdx)1/ρ<∞.
The following weighted Sobolev-Hardy inequality is due to Caffarelli et al. [15], which is called the Caffarelli-Kohn-Nirenberg inequality. There is a constant S1>0 such that
(7)(∫ℝN‍|x|-bp*|u|p*dx)1/p*≤S1(∫ℝN‍|x|-ap|∇u|pdx)1/p,∀u∈C0∞(ℝN),
where -∞<a<(N-p)/p, a≤b<a+1, d=a+1-b, and p*=pN/(N-pd). Throughout this paper, we make the following assumptions: (A1)h(x)|x|br∈Lη(Ω)∩L∞(Ω) with η=p*/(p*-r), h±=max{±h(x),0}≢0,(A2)g(x)∈C(∂Ω)∩L∞(∂Ω),(A3)1<r<p<2p≤q<p*=p(N-1)/(N-p).Now, we give the definition of weak solution for problem (1).Definition 1.
A functionu∈E is said to be a weak solution of problem (1) if for any φ∈C0∞(Ω)(8)(α∥u∥Ep+β)∫Ω‍|x|-ap|∇u|p-2∇u·∇φdx-∫∂Ω‍g(x)|u|q-2uφdσ=λ∫Ω‍h(x)|u|r-2uφdx.
The assumptions (A1)–(A3) mean that all the integrals in (8) are well defined and convergent.
In view of(A3), it follows from the compact trace embedding Wa1,p(Ω)↪Lq(∂Ω) [16] that
(9)∫∂Ω‍g(x)|u|qdσ≤S2∥g∥∞∥u∥Eq,
for some constant S2>0, and ∥g∥∞≜∥g∥L∞(∂Ω).Our main result is in the following.Theorem 2.
Assume(A1)–(A3); there exists λ*>0 such that problem (1) has at least two positive solutions for all λ>0 satisfying 0<λ<λ*.This paper is organized as follows. In Section2, we give some properties of the Nehari manifold and set up the variational framework for problem (1). In Section 3, we consider the multiplicity results and prove Theorem 2.


## 2. Preliminary Results

It is clear that problem (1) has a variational structure. Let Jλ(u):E→ℝ1 be the corresponding Euler functional of problem (1), which is defined by
(10)Jλ(u)=1pM^(∥u∥Ep)-1q∫∂Ω‍g(x)|u|qdσ-1r∫Ω‍h(x)|u|rdx,
where M^(t)=∫0t‍M(s)ds. Then, we see that the functional Jλ(u)∈C1(E,ℝ1), and for ∀φ∈E, there holds
(11)〈Jλ′(u),φ〉=M(∥u∥Ep)∫Ω‍|x|-ap|∇u|p-2∇u·∇φdx-∫∂Ω‍g|u|q-2uφdσ-λ∫Ω‍h|u|r-2uφdx.
Particularly, we have
(12)〈Jλ′(u),u〉=M(∥u∥Ep)∫Ω‍|x|-ap|∇u|pdx-∫∂Ω‍g(x)|u|qdσ-λ∫Ω‍h(x)|u|rdx.
It is well known that the weak solution of problem (1) is the critical point of Jλ(u). Thus, to prove the existence of weak solutions for problem (1), it is sufficient to show that Jλ(u) admits a sequence of critical points. Since Jλ(u) is not bounded below on E, it is useful to consider the functional Jλ(u) on the Nehari manifold [17, 18]:
(13)Nλ={u∈E∖{0}∣〈Jλ′(u),u〉=0},
where 〈,〉 denotes the usual duality. Then, it follows from (12) that u∈Nλ if and only if
(14)M(∥u∥Ep)∥u∥Ep-∫∂Ω‍g(x)|u|qdσ-λ∫Ω‍h(x)|u|rdx=0.
Then, we get from (10)–(14) that
(15)Jλ(u)=(12p-1r)α∥u∥E2p-(1r-1p)β∥u∥Ep+(1r-1q)∫∂Ω‍g(x)|u|qdσ(16)=(12p-1q)α∥u∥E2p+(1p-1q)β∥u∥Ep-(1r-1q)λ∫Ω‍h(x)|u|rdx.
We define
(17)Φλ(u)=〈Jλ′(u),u〉.
Then, (14) implies that
(18)〈Φλ′(u),u〉=pM′(∥u∥Ep)∥u∥E2p+pM(∥u∥Ep)∥u∥Ep-q∫∂Ω‍g(x)|u|qdσ-rλ∫Ω‍h(x)|u|rdx(19)=(2p-q)α∥u∥E2p+(p-q)β∥u∥Ep+(q-r)λ∫Ω‍h(x)|u|rdx(20)=(2p-r)α∥u∥E2p+(p-r)β∥u∥Ep-(q-r)∫∂Ω‍g(x)|u|qdσ.
It is natural to split Nλ into three parts:
(21)Nλ+={u∈Nλ∣〈Φλ′(u),u〉>0},Nλ-={u∈Nλ∣〈Φλ′(u),u〉<0},Nλ0={u∈Nλ∣〈Φλ′(u),u〉=0}.
Now, we give some important properties of Nλ+, Nλ-, and Nλ0.Lemma 3.
Let1<r<p<q and q≥2p. Then, Jλ(u) is coercive and bounded below on Nλ.Proof.
For1<r<p<q, we obtain from (A1), the Hölder and Caffarelli-Kohn-Nirenberg inequalities that
(22)∫Ω‍h(x)|u|rdx≤(∫Ω‍(|h(x)||x|br)ηdx)1/η×(∫Ω‍|x|-bp*|u|p*dx)r/p*≤S1rhη(∫Ω‍|x|-ap|∇u|pdx)r/p≤S1rhη∥u∥Er,
where hη=(∫Ω‍(|g(x)||x|br)ηdx)1/η, η=p*/(p*-r). Thus, we get from (16) that
(23)Jλ(u)=(12p-1q)α∥u∥E2p+(1p-1q)β∥u∥Ep-(1r-1q)∫Ω‍h(x)|u|rdx≥∥u∥Ep[(12p-1q)α∥u∥Ep+(1p-1q)β]-(1r-1q)λS1rhη∥u∥Er.
Then, one can obtain by the Young inequality and q≥2p that Jλ(u) is coercive and bounded below on Nλ.
Let(24)λ0={2(q-2p)(q-p)(q-r)S2rhη×[2αβ(2p-r)(p-r)(q-r)∥g∥∞](3p-2r)/(2q-3p),q>2p,pβ(2p-r)S1rhη×[2αβ(2p-r)(p-r)(2p-r)∥g∥∞]2(p-r)/p,q=2p.
Then, we have the following result.Lemma 4.
Letr<p<q,q≥2p. Then, Nλ0=∅ for 0<λ<λ0.Proof.
Suppose that there existsu∈Nλ0. If q>2p, then it follows from (19) and (21) that
(25)2β(q-2p)(q-p)∥u∥E3p/2≤(q-2p)α∥u∥E2p+β(q-p)∥u∥Ep=(q-r)λ∫Ω‍h(x)|u|rdx≤(q-r)λS1rhη∥u∥Er,
which implies that
(26)∥u∥E≤[(q-r)λS1rhη2(q-2p)(q-p)]2/(3p-2r).
On the other hand, we can similarly get from (20) and (21) that
(27)2αβ(2p-r)(p-r)∥u∥E3p/2≤(2p-r)α∥u∥E2p+(p-r)∥u∥Ep=(q-r)∫∂Ω‍g(x)|u|qdσ≤(q-r)∥g∥∞∥u∥Eq,
which yields that
(28)∥u∥E≥[2αβ(2p-r)(p-r)(q-r)∥g∥∞]2/(2q-3p).
Thus, inequalities (26) and (28) show that λ≥λ0, which contradicts the hypothesis of λ.
Forq=2p, we can similarly obtain from (19) and (21) that
(29)∥u∥E≤[(q-r)λS1rhηpβ]1/(p-r).
Thus, (28) and (29) imply that λ>λ0, which is also a contradiction. Therefore, we complete the proof.Lemma 5.
Ifu0 is a local minimizer of Jλ(u) on Nλ and u0∉Nλ0, then u0 is a critical point of Jλ(u).Proof.
Let(30)F(u)=M(∥u∥Ep)∥u∥Ep-∫∂Ω‍g(x)|u|qdσ-λ∫Ω‍h(x)|u|rdx,∀u∈E.
Consider the following minimizing problem:
(31)minu∈NλJλ(u),subjecttoF(u)=0.
By the Lagrange multiplier principle, there exists γ∈ℝ1 such that
(32)Jλ′(u0)=γF′(u0),(33)〈Jλ′(u0),u0〉=γ〈F′(u0),u0〉=γ〈Φλ′(u0),u0〉.
Since u0∈Nλ and u0∉Nλ0, it follows from (33) that γ=0; furthermore, Jλ′(u0)=0.Now, we writeNλ=Nλ+∪Nλ- for 0<λ<λ0 and define
(34)δλ+=infu∈Nλ+Jλ(u),δλ-=infu∈Nλ-Jλ(u).
Denote
(35)λ1=r(q-p)βp(q-r)S1rhη[β(p-r)S2(q-r)∥g∥∞](p-r)/(q-p).
Then, the following results on δλ+ and δλ- are established.Lemma 6.
Letr<p<q,q≥2p, and 0<λ<λ1; then(i)
δλ+<0,(ii)
there exists constantk0>0 such that δλ->k0.Proof.
(i) For∀u∈Nλ+, we have from (19) and (21) that
(36)(q-2p)α2pq∥u∥E2p+(q-p)βpq∥u∥Ep≤(q-r)λpq∫∂Ω‍g(x)|u|qdσ,λ∫∂Ω‍g(x)|u|qdσ≥(q-p)βq-r∥u∥Ep.
Thus, (36) and (16) give that
(37)Jλ(u)<(q-rpq-q-rqr)λ∫Ω‍h(x)|u|rdx≤-β(p-r)(q-p)qpr∥u∥Ep<0,
which implies that
(38)δλ+=infu∈Nλ+Jλ(u)<0.
(ii) Letu∈Nλ+; one can deduce from (20) and (21) that
(39)∥u∥E≥[(p-r)βS2(q-r)∥g∥∞]1/(q-p).
On the other hand, we obtain from (16), (26), and (39) that
(40)Jλ(u)≥(12p-1q)α∥u∥E2p+(1p-1q)β∥u∥Ep-(q-r)λrqS1rhη∥u∥Er≥∥u∥Er[(q-p)βpq∥u∥Ep-r-(q-r)λS1rhηqr]≥∥u∥Er[(q-p)βpq((p-r)βS2(q-r)∥g∥∞)(p-r)/(q-p)-(q-r)λS1rhηrq{((p-r)βS2(q-r)∥g∥∞)p-rq-p}].
Therefore, if 0<λ<λ1, there exists k0>0 such that δλ->k0.For eachu∈E with ∫∂Ω‍g(x)|u|qdσ>0, we define
(41)mα(t)=αt2p-r∥u∥E2p+βtp-r∥u∥Ep-tq-r∫∂Ω‍g|u|qdσ.
Then, mα(0)=0, mα(t)→-∞ as t→+∞, and mα(t) gets its unique maximum at the critical point tα,max. Particularly, m0(t) gets its unique maximum at
(42)t0,max=[β(p-r)∥u∥Ep(q-r)∫∂Ω‍g|u|qdσ]1/(q-p),
and we have the following results.Lemma 7.
Letr<p<q<p*, q≥2p, and 0<λ<(p/r)λ1. For each u∈E with ∫∂Ω‍g(x)|u|qdσ>0, one has the following:(i)
if∫Ω‍h(x)|u|rdx≤0, then there exists t->tα,max such that t-u∈Nλ- and
(43)Jλ(t-u)=supt≥0Jλ(tu);(ii)
if∫Ω‍h(x)|u|rdx>0, then there exists 0<t+<tα,max<t- such that t+u∈Nλ+,t-u∈Nλ-, and
(44)Jλ(t+u)=inf0<t≤tα,maxJλ(tu),Jλ(t-u)=supt≥tα,maxJλ(tu).Proof.
(i) For∀u∈E, we define
(45)ϕ0(t)=〈Jλ′(tu),tu〉=t2pα∥u∥E2p+tpβ∥u∥Ep-tq×∫∂Ω‍g(x)|u|qdσ-trλ∫Ω‍h(x)|u|rdx,(46)ϕ1(t)=〈Φλ′(tu),tu〉=2pαt2p∥u∥E2p+tpβ∥u∥Ep-tq∫∂Ω‍g|u|qdσ-trλ∫Ω‍h(x)|u|rdx,(47)ϕ2(t)=Jλ(tu)=α2pt2p∥u∥E2p+βptp∥u∥Ep-1qtq∫∂Ω‍g|u|qdσ-λrtr∫Ω‍h(x)|u|rdx.
Since ∫Ω‍h(x)|u|rdx≤0, there exists unique t->tα,max such that
(48)mα′(t-)<0,mα(t-)=∫Ω‍h(x)|u|rdx.
Thus, ϕ0(t-)=tr(mα(t-)-∫Ω‍h(x)|u|rdx)=0, which implies that t-u∈Nλ. It follows from (20) that
(49)ϕ1(t-)=(2p-r)α(t-)2p∥u∥E2p+(p-r)β(t-)p∥u∥Ep-(q-r)(t-)q∫∂Ω‍g|u|qdσ=(t-)r+1mα′(t-)<0;
that is, t-u∈Nλ-. By (47), we obtain that
(50)ϕ2′(t)=αt2p-1∥u∥E2p+βtp-1∥u∥Ep-tq-1∫∂Ω‍g|u|qdσ-λtr-1∫Ω‍h(x)|u|rdx=tr-1[mα(t)-λ∫Ω‍h(x)|u|rdx],
which shows that ϕ2(t) increases for t∈[0,t-] and decreases for t∈[t-,+∞). Therefore,
(51)Jλ(t-u)=ϕ2(t-)=supt≥0Jλ(tu).
(ii) We firstly want to prove that0<∫Ω‍h(x)|u|rdx≤mα(tα,max). In fact,
(52)m0(t0,max)=β[β(p-r)∥u∥Ep(q-r)∫∂Ω‍g|u|qdσ](p-r)/(q-p)∥u∥Ep-[β(p-r)∥u∥Ep(q-r)∫∂Ω‍g|u|qdσ](q-r)/(q-p)×∫∂Ω‍g|u|qdσ=β(q-r)/(q-p)∥u∥Ep(q-r)/(q-p)(p-rq-r)(p-r)/(q-p)×(q-pq-r)[1∥g∥∞S2∥u∥Eq](p-r)/(q-p)≥β(q-r)/(q-p)∥u∥Er(q-pq-r)×[p-r(q-r)∥g∥∞](p-r)/(q-p).
Then, if 0<λ<(p/r)λ1, we have that
(53)mα(0)=0<λ∫Ω‍h(x)|u|rdx≤β(q-r)/(q-p)∥u∥Er(q-pq-r)[p-r(q-r)∥g∥∞](p-r)/(q-p)≤m0(t0,max)<mα(tα,max).
Since ∫Ω‍h(x)|u|rdx>0, there exists 0<t+<tα,max<t- such that
(54)mα(t+)=mα(t-)=λ∫Ω‍h(x)|u|rdx,
and mα′(t+)>0, mα′(t-)<0. Similar to the proof of (i), we get that t+u∈Nλ+, t-u∈Nλ-. We can deduce from (50) that ϕ2(t) decreases for t∈[0,t+] and increases for t∈[t+,tα,max). Therefore,
(55)Jλ(t+u)=ϕ2(t+)=inf0≤t≤tα,maxJλ(tu).
Similarly,
(56)Jλ(t-u)=ϕ2(t-)=supt≥tα,maxJλ(tu).
Then, we complete the proof.For eachu∈E with ∫Ω‍h(x)|u|rdx>0, we define
(57)m-α(t)=αt2p-q∥u∥E2p+βtp-q∥u∥Ep-λtr-q∫Ω‍h(x)|u|rdx.
Then, m-α(t)→-∞ as t→0+ and m-α(t)→0 as t→∞. Furthermore, m-α(t) gets its unique maximum at some certain point t-α,max. Particularly, m-0(t) gets its unique maximum at the critical point
(58)t-0=t-0,max(u)=[λ(q-r)∫Ω‍h(x)|u|rdxβ(q-p)∥u∥Ep]1/(p-r),
then we have the following results.Lemma 8.
Let1<r<p<q, q≥2p, and 0<λ<pλ1/r. For each u∈E with ∫Ω‍h(x)|u|rdx>0 one has the following:(i)
if∫∂Ω‍g|u|qdσ<0, then there exists unique 0<t+<t-α,max such that t+u∈Nλ+ and
(59)Jλ(t+u)=inft≥0Jλ(tu);(ii)
if∫∂Ω‍g|u|qdσ≥0, then there exists 0<t+<t-α,max<t- such that t+u∈Nλ+, t-u∈Nλ-, and
(60)Jλ(t+u)=inf0<t≤t-α,maxJλ(tu),Jλ(t-u)=supt≥t-α,maxJλ(tu).Proof.
(i) Since∫∂Ω‍g|u|qdσ<0, there exists unique 0<t+<t-α,max such that
(61)m-α′(t+)>0,m-α(t+)=∫∂Ω‍g|u|qdσ.
Note that (45) can be rewritten as
(62)ϕ0(t)=tq[m-α(t)-∫∂Ω‍g|u|qdσ].
Thus, (61) shows that t+u∈Nλ. By virtue of (14) and (46), we get that
(63)ϕ1(t+)=〈Φλ′(t+u),t+u〉=(t+)q+1[{∫Ω‍h(x)}(2p-q)α(t+)2p-q-1+(p-q)β(t+)p-q-1∥u∥Ep-λ(r-q)(t+)r-q-1∫Ω‍h(x)|u|rdx]=(t+)q+1m-α′(t+)>0,
which implies that t+u∈Nλ+. We have from (47) that
(64)ϕ2′(t)=tq-1[m-α(t)-∫∂Ω‍g|u|qdσ].
Then, ϕ2(t) decreases for t∈[0,t+] and increases for t∈[t+,+∞); that is,
(65)Jλ(t+u)=inft≥0Jλ(tu).
(ii) We need also to prove that0<∫∂Ω‍g|u|qdσ<m-α(t). In fact, we have that
(66)m-α(tα,max)≥m-0(t-0)=β[λ(q-r)∫Ω‍h|u|rdxβ(q-p)∥u∥Ep](p-q)/(p-r)∥u∥E2p-λ[λ(q-r)∫Ω‍h|u|rdxβ(q-p)∥u∥Ep](r-q)/(p-r)=λ(p-q)/(p-r)β(q-r)/(p-r)∥u∥Ep(q-r)/(p-r)×(q-pq-r)(q-p)/(p-r)×(p-rq-r)(∫Ω‍h|u|rdx)(p-q)/(p-r)≥λ(p-q)/(p-r)β(q-r)/(p-r)(q-rp-r)×(q-p(q-r)S1rhη)(q-p)/(p-r)∥u∥Eq≥∫∂Ω‍g|u|qdσ≥0
for 0<λ≤(p/r)λ1.
The rest of the proof is similar to that of (ii) in Lemma7, and here we omit the proof.Lemma 9.
Assume(A1)–(A3). If un⇀u in E, then there exists a subsequence of {un}, still denoted by {un}, such that
(67)∫Ω‍h(x)|un|rdx⟶∫Ω‍h(x)|u|rdx,∫∂Ω‍g(x)|un|qdσ⟶∫∂Ω‍g(x)|u|qdσ.Proof.
By the assumption(A2), we have h(x)|x|br∈Lη(Ω)∩L∞(Ω), and then for any ε>0, there exists Rε>0 large enough such that
(68)∫BRεc‍(|h(x)||x|br)ηdx≤εη,
where BR={x∈ℝN∣|x|≤R}, BRc={x∈ℝN∣|x|>R} for R>0.
The compact embedding theoremWa1,p(Ω)↪Lr(Ω,|x|-α) (Theorem 2.1 in [19]) implies that
(69)un⟶ua.e.inΩ.
Inequality (7) shows that {un} is bounded in Lp*(ℝN,|x|br), which implies that un⇀u in Lp*(ℝN,|x|br). Thus, we can obtain that
(70)∫BRε∖D‍|x|-bp*|un-u|p*dx<εp*,∫Ω‍|x|-bp*|un|p*dx≤Mp*,∫Ω‍|x|-bp*|u|p*dx≤Mp*
for some M>0. On the other hand, we get from the Höder inequality and (68)–(70) that
(71)∫BRεc‍h|un-u|rdx≤(∫BRεc‍(|h||x|br)ηdx)1/η×(∫BRεc‍|x|-bp*|un-u|p*dx)r/p^*≤2rMrε,∫BRε∖D‍h|un-u|rdx≤(∫BRε∖D‍(|h||x|br)ηdx)1/η×(∫BRε∖D‍|x|-bp*|un-u|p*dx)r/p^*≤c0εr
for some constant c0>0 and large n.
Thus, (71) implies that
(72)∫Ω‍h(x)|un-u|rdx⟶0asn⟶∞;
that is,
(73)∫Ω‍h(x)|un|rdx⟶∫Ω‍h(x)|u|rdxasn⟶∞.
Since ∂Ω is compact and g∈C(∂Ω)∩L∞(∂Ω), we obtain by the trace embedding theorem in [20] that
(74)λ∫∂Ω‍g(x)|un|qdσ⟶λ∫∂Ω‍g(x)|u|qdσasn⟶∞.
This concludes the proof.


## 3. Existence of Solutions

In this part, we will give the proof of the existence of nonnegative and nontrivial solutions. Before this, we need to prove the following two important lemmas.Lemma 10.
Assume(A3) and 0<λ<(p/r)λ1. Then, the functional Jλ(u) has a minimizer u0+∈Nλ+ and(i)
Jλ(u0+)=δλ+,(ii)
u0+ is a positive weak solution of problem (1).Proof.
SinceJλ(u) is bounded in Nλ+, there exists a minimizing sequence of {uk}∈Nλ+ such that
(75)limk→∞Jλ(uk)=infu∈Nλ+Jλ(u).
We can get that {uk} is bounded in E and uk⇀u0+ weakly in E since Jλ(u) is coercive. It follows from Lemma 6 that
(76)Jλ(uk)⟶δλ+<0ask⟶+∞.
Furthermore, equality (16) and Lemma 9 imply that ∫Ω‍h(x)|u0+|rdx>0. In the following, we prove that uk→u0+ in E. Suppose otherwise that
(77)∥u0+∥E<limk→+∞inf∥uk∥E.
Let
(78)mα,u0+(t)=αt2p-q∥u0+∥E2p+βtp-q∥u0+∥Ep-λtr-q∫Ω‍h(x)|u0+|rdx,φ0(t)=αt2p-q∥u0+∥E2p+βtp-q∥u0+∥Ep-λtr-q∫Ω‍h(x)|u0+|rdx-∫∂Ω‍g(x)|u0+|qdσ.
It is obvious that mα,u0+(t) has the unique critical point tα,max(u0+) and mα,u0+(t) increases for t∈[0,tα,max(u0+)] and decreases for t∈[tα,max(u0+),∞). Particularly, m0,u0+(t) has the unique critical point at t0(u0+)=t-0,max(u0+). Since ∫Ω‍h(x)|u0+|rdx>0, it follows from Lemma 8 that there exists t0+<tα,max(u0+) such that t0+u0+∈Nλ+ and
(79)Jλ(t0+u0+)=inf0≤t≤tα,max(u0+)Jλ(tu0+).
By virtue of t0+u0+∈Nλ+, we obtain that
(80)φ0(t0+)=(t0+)-q〈Jλ(t0+u0+),t0+u0+〉=0.
Similarly, we set
(81)φk(t)=αt2p-q∥uk∥E2p+βtp-q∥uk∥Ep-λtr-q∫Ω‍h(x)|uk|rdx-∫∂Ω‍g(x)|uk|qdσ.
In view of (77), we get that φk(t0+)>φ0(t0+)=0 for large k. Since φk(1)=0 and φk′(t)≥mα,u0+′(t)>0 for t∈(0,tα,max(u0+)], we obtain that 1<t0+<tα,max(u0+) and φk(t)<0 for t∈[0,1]. Then, by (77), (79), and Lemma 9, we get that
(82)Jλ(t0+u0+)≤Jλ(u0+)<limk→+∞Jλ(uk)=δλ+,
which is a contradiction. Hence, uk→u0+ strongly in E, and
(83)Jλ(uk)⟶Jλ(u0+)=δλ+ask⟶+∞.
Thus, u0+ is a minimum of Jλ(u) on Nλ+. Since Jλ(u0+)=Jλ(|u0+|) and |u0+|∈Nλ+, we may assume by Lemma 5 that u0+ is a positive solution of problem (1).Lemma 11.
Assume(A3) and 0<λ<(p/r)λ1. Then, the functional Jλ(u) has a minimizer u0-∈Nλ- and (i)
Jλ(u0-)=δλ-,(ii)
u0- is a positive weak solution of problem (1).Proof.
SinceJλ(u) is bounded on Nλ-, there exists a minimizing sequence {uk}∈Nλ- such that
(84)limk→∞Jλ(uk)=infu∈Nλ-Jλ(u).
Similar to the proof of Lemma 10, we may assume that uk⇀u0- weakly in E. For uk∈Nλ-, we deduce by Lemma 6 and (15) that ∫∂Ω‍g(x)|uk|qdσ>0; furthermore, ∫∂Ω‍g(x)|u0-|qdσ>0. We also want to prove that uk→u0- strongly in E. In fact, if not, we have
(85)∥u0+∥E<limk→+∞inf∥uk∥E.
By virtue of Lemma 7, there exists t0->0 such that
(86)Jλ(t0-u)=supt≥0Jλ(tuk),
and a simple transformation shows that
(87)Jλ(uk)≥Jλ(tuk),∀uk∈Nλ-.
Therefore, Lemma 9 together with (84)–(87) gives that
(88)Jλ(t0-u0-)<limk→+∞Jλ(t0-uk)≤limk→+∞Jλ(uk)=δλ-,
which is a contradiction, and we complete the proof.Proof of Theorem2.
We setλ*=min{λ1,λ0}. When 0<λ<λ*, by Lemmas 10 and 11, we obtain that problem (1) has two nontrivial nonnegative solutions u0+∈Nλ+, u0-∈Nλ-. Lemma 4 and the assumptions of Theorem 2 imply that Nλ+∩Nλ-=∅; then u0+ and u0- are distinct. Furthermore, since Jλ(u0±)=Jλ(|u0±|) and |u0±|∈Nλ±, we can assume that the solutions u0+ and u0- are positive. This completes the proof.


## 4. Conclusions

The object of this paper is to prove the existence of multiple solutions for the nonlinear Kirchhoff-type problem (1). By the variational methods, we discuss the problem on the Nehari manifold and give the sufficient conditions for the existence of solutions. We overcome the difficulty due to the loss of compactness on the unbounded domain. In the future work, we are interested to consider similar problems, but the term on the right will be replaced by abstract functions.



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*Source: 516093-2013-07-25.xml*