# The Existence of a Nontrivial Solution for ap-Kirchhoff Type Elliptic Equation in ℝN

Authors: Zonghu Xiu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Using Mountain Pass lemma, under some appropriate assumptions, we establish the existence of one nontrivial solution for a class ofp-Kirchhoff-type elliptic equations in ℝN.


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## Body

## 1. Introduction and Main Result

In this paper, we consider the existence of solution for the following elliptic problem:(1)-(a+b∫ℝN‍g(x)|∇u|pdx)div(g(x)|∇u|p-2∇u)mm+h(x)|u|p-2u=f(x,u),x∈ℝN,u(x)⟶0as|x|⟶∞,
where a>1,b>0,N≥3, g(x), and h(x) are positive functions, f(x,t):ℝN×R→R is continuous function, and further assumptions will be listed later. Problems like (1) originally came from the stationary problem of a model introduced by Kirchhoff [1]. Due to the existence of the integration over the whole space, problems like (1) are also called nonlocal problems.In recent years, the Kirchhoff-type equations withp-Laplacian operator has been considered by many authors; see [2–6]. When h(x)≡0, the authors in [6] considered the following similar Kirchhoff-type elliptic problem on the bounded domain Ω⊂ℝN(2)-[M(∫Ω‍|∇u|pdx)]p-1div(|∇u|p-2∇u)m=f(x,u),x∈Ω,u(x)=0on∂Ω.The functionM(t) in (2) is required to meet the condition of m0≤M(t)≤m1 for some constants m0,m1>0. The authors proved that problem (2) has at least one positive solution under some other additional conditions. Problem (2) was also considered in [3], where the function f(x,u) is odd about u. For p=2, we refer to [7–9]. The authors in [8] studied the following Kirchhoff type problem:
(3)-(∫Ω‍a+b|∇u|2dx)Δu+u=f(x,u),x∈ℝN,u(x)⟶0as|x|⟶+∞.By the Fountain theorem, the author proved the existence of infinitely many solutions. Note that one of the assumptions made on the functionf(x,u) in (3) is that f(x,u)=-f(x,-u) for any (x,u)∈(ℝN,ℝ). In the present paper, however, the function f(x,u) is not required to be odd about u as that of [3, 8]. When a=b=0 and g(x)≡1, problem (1) becomes the p-Laplacian elliptic equations without nonlocal term, and this kind of problem is also studied by many authors. For these works, we refer to [10–14] and the references therein. In [15], Liu discussed the following elliptic problem:
(4)-div(|∇u|p-2∇u)+V(x)|u|p-2u=f(x,u),u(x)∈W1,p(ℝN).The author proved that problem (4) has at least one ground state. The weight function V(x) is required to be bounded; more precisely, there exist constants α,β>0 such that
(5)V(x)∈C(ℝN),0<α≤V(x)≤β<+∞.We point out that the weight functionh(x) in problem (1) is permitted to tends to infinity.In this paper, inspired by [4, 8, 15], we consider the existence of solution of problem (1). By the variational method, we will prove that problem (1) has at least one nontrivial weak solution. Since problem (1) is considered in the whole space ℝN, the loss of compactness of the Sobolev embedding renders the variational technique more delicate.In this paper, we make the following hypothesis:(A1)
g(x)≥1, h(x)≥h0>0 and h(x)→+∞ as |x|→∞,(A2) there exists μ≥2p such that 0≤μF(x,t)≤tf(x,t), where F(x,t)=∫0t‍f(x,s)ds,(A3) there exist constants c>0 and p<q<p* such that |f(x,t)|≤c|t|q-1.Remark 1.
Throughout this paper, we denote byc the constant which may vary from line to line but remains independent of the relevant quantities. Note that there exist many functions g(x),h(x), and f(x,t) such that satisfy the assumptions of Theorem 3, for example, g(x)=1+x4, h(x)=ex2, f(x,t)=t5p-1/(1+x4).
LetW01,p(ℝN) be the usual Sobolev space with the norm of
(6)∥u∥=(∫ℝN‍|∇u|p+|u|pdx)1/p.
Denote(7)E={u∈W01,p(ℝN):∫ℝN‍(|∇u|p+h(x)|u|p)dx<∞}.
Then,E is a Sobolev space with the norm of
(8)∥u∥E=(∫ℝN‍(|∇u|p+h(x)|u|p)dx)1/p.
We give another space(9)X={u∈E:∫ℝN‍(g(x)|∇u|p+h(x)|u|p)dx<∞}
endowed with the norm
(10)∥u∥X=(∫ℝN‍(g(x)|∇u|p+h(x)|u|p)dx)1/p.
It is not difficult to check thatX is a Banach space. The Euler functional of problem (1) is
(11)J(u)=ap∫ℝN‍g(x)|∇u|pdx+b2p(∫ℝN‍g(x)|∇u|pdx)2+1p∫ℝN‍h(x)|u|pdx-∫ℝN‍F(x)dx,
where F(u)=∫0u‍f(x,t)dt. Then, the assumptions (A1)–(A3) imply that J(u)∈C1(X,ℝ1), and for any φ∈X, there holds
(12)〈J′(u),φ〉=a∫ℝN‍g(x)|∇u|p-2∇u∇φdx+b∫ℝN‍g(x)|∇u|pdx∫ℝN‍g(x)|∇u|p-2∇u∇φdx+∫ℝN‍h(x)|u|p-2uφdx-∫ℝN‍f(x,u)φdx.
Particularly,(13)〈J′(u),u〉=a∫ℝN‍g(x)|∇u|pdx+b(∫ℝN‍g(x)|∇u|pdx)2+∫ℝN‍h(x)|u|pdx-∫ℝN‍f(x,u)udx.Definition 2.
A functionu∈X is said to be a weak solution of (1) if and only if (12) holds for any φ∈X.Our main result is listed later.Theorem 3.
Assume(A1)–(A3). Then, problem (1) has at least one nontrivial weak solution in X.This paper is organized as follows. In Section2, we introduce some definitions and prove several lemmas which will be used later. In Section 3, we give the proof of Theorem 3 by making use of the Mountain Pass lemma.


## 2. Preliminary Results

In this section, we give some important lemmas, which will be needed in the proof of our main result. Particularly, one result of compact embedding on unbounded domain will be proved.Lemma 4.
Assume(A1) and p≤q<p*. Then, the embedding X↪Lq(ℝN) is compact.Proof.
We split the proof into two cases.
C
a
s
e
1 (q=p). Let
(14)BR={u∈X:∥u∥X<R},BRc=ℝN∖BR
and η=infBRch(x); then (A1) shows that η→+∞ as R→∞. Furthermore, one gets that
(15)∫BRc‍|u|pdx≤η-1∫BRc‍h(x)|u|pdx,
which implies that
(16)(∫BRc‍|u|pdx)1/p≤η-1/p(∫BRc‍h(x)|u|pdx)1/p≤η-1/p(∫ℝN‍h(x)|u|pdx)1/p≤η-1/p∥u∥x.
It follows from (16) that
(17)limR→∞supu∈X∖{0}∥u∥Lp(BRc)∥u∥X=0.
Let{un} be a bounded sequence of X such that ∥un∥X≤c. Then, there exists u∈X such that un⇀u. For any ε>0, it follows from (17) that there exists Rε large enough such that
(18)∥un∥Lp(BRεc)≤εc-1∥un∥X≤ε(n=1,2,…).
Note that the embeddingX↪Lp(BRε) is compact. Therefore, for these ε and Rε, there exists Nε>0 large enough such that
(19)∥un-u∥Lp(BRε)≤ε
for all n>Nε. Thus, one can get from (18) and (19) that
(20)∥un-u∥Lp(ℝN)≤∥un-u∥Lp(BRε)+∥un∥Lp(BRεc)+∥u∥Lp(BRεc)<3ε,
which implies that un→u strongly in Lq(ℝN).
C
a
s
e
2  (p<q<p*). By the Hölder and Young inequalities, one can get from (A1) that
(21)∫BRc‍|u|qdx≤(∫BRc‍|u|pdx)(p*-q)/(p*-p)×(∫BRc‍|u|p*dx)(q-p)/(p*-p)≤η-(p*-q)/(p*-p)(∫BRc‍h(x)|u|pdx)(p*-q)/(p*-p)×(∫BRc‍|u|p*dx)(q-p)/(p*-p)≤cη-(p*-q)/(p*-p)[(∫BRc‍h(x)|u|pdx)q/pmmmmmmmm+(∫BRc‍|u|p*dx)q/p*]≤cη-(p*-q)/(p*-p)[(∫BRc‍h(x)|u|pdx)q/pmmmmmmmm+(∫BRc‍g(x)|∇u|pdx)q/p]≤cη-(p*-q)/(p*-p)∥u∥Xq.
Sincep<q<p* and η→+∞ as R→∞, then
(22)limR→∞supu∈X∖{0}∥u∥Lq(BRc)∥u∥X=0.
Therefore, similar to the proof ofCase1, the embedding E↪Lq(ℝN) is compact.Now, we give the definitions of Palais-Smale (simply(PS)c) sequence and (PS)c condition.Definition 5.
Letc∈R1, J(u)∈C1(X,R), and X be a Banach space. The sequence {un}⊂X is said to be a (PS)c sequence if there holds
(23)J(un)⟶c,J′(un)⟶0inX*asn⟶∞.
A functionalJ(u) is said to satisfy the (PS)c condition if any (PS)c sequence in X contains a convergent subsequence.Next, we will prove that the functionalJ(u) satisfies the (PS)c condition.Lemma 6.
Assume(A1)–(A3). Then, J(u) satisfies the (PS)c condition on X for any c∈ℝ1.Proof.
Let{un} be an any (PS)c sequence in X. We divide the proof into two steps.
S
t
e
p
1 ({un} is bounded in X). Note that a>1; then, it follows from (23) and (A2) that
(24)c+∥un∥X≥J(un)-12p〈J′(un),un〉=(1p-12p)a∫ℝN‍g(x)|∇un|pdxm+(1p-12p)∫ℝN‍h(x)|un|pdxm+∫ℝN‍(12pf(x,un)un-F)dx≥12p∫ℝN‍g(x)|∇un|pdxm+12p∫ℝN‍h(x)|un|pdx=12p∥un∥Xp,
which implies that {un} is bounded in X.
S
t
e
p
2 ({un} converges strongly in X). Since {un} is bounded in the separable X, there exist u0∈X and a subsequence of {un}, still denoted by {un}, such that un⇀u0 in X. Now, we want to prove that
(25)∫ℝN‍f(x,un)undxm-∫ℝN‍f(x,un)u0dx⟶0,asn⟶∞.
In fact, it follows from(A3) that
(26)|∫ℝN‍f(x,un)undx-∫ℝN‍f(x,un)u0dx|m≤∫ℝN‍|un|q-1|un-u0|dx.
Thus, in order to prove (25), we need only to prove that
(27)∫ℝN‍|un|q-1|un-u0|dx⟶0,asn⟶∞.
Here, we divide the proof (27) into two cases:
(i)(28)p≤q<(N-1)pN-p.
By the Hölder inequality, we obtain that(29)∫ℝN|un|q-1‍|un-u0|dxm≤(∫ℝN‍|un|p(q-1)/(p-1)dx)(p-1)/pmmmm×(∫ℝN|un-u0|p‍dx)1/p.
It is easy to check thatp≤p(q-1)/(p-1)<p*. Then, the compact embedding in Lemma 4 shows that
(30)(∫ℝN‍|un|p(q-1)/(p-1)dx)(p-1)/p<∞,(∫ℝN‍|un-u0|pdx)1/p⟶0asn⟶∞,
which gives (27).
(ii)(31)(N-2)pN-p≤q<p*.
Letλ1=p*/(q-2). Since 1<p<N and p(N-2)/(N-p)≤q<p*, there exist p≤λ2=λ3<p* such that
(32)1λ1+1λ2+1λ3=1.
The Hölder inequality shows that(33)∫ℝN‍|un|q-1|un-u0|dx≤(∫ℝN‍|un|(q-2)λ1dx)1/λ1×(∫ℝN‍|un|λ2)1/λ2×(∫ℝN‍|un-u0|λ3dx)1/λ3.
Therefore, the embedding in Lemma4 implies that
(34)(∫ℝN‍|un|(q-2)λ1dx)1/λ1<∞,(∫ℝN|un|λ2‍)1/λ2<∞,(∫ℝN‍|un-u0|λ3dx)1/λ3⟶0,
which also gives (27).
Note that(N-1)p/(N-p)<p* and
(35)(N-2)pN-p≥p,p≥2,(N-2)pN-p<p,1<p<2.
Therefore, the previous cases (i) and (ii) imply that (25) holds for all p≤q<p*.
Denote(36)Dn=〈J′(un),un〉-〈J′(un),u0〉+∫ℝN‍f(x,un)undx-∫ℝN‍f(x,un)u0dx.
Then, it follows from (23) and (25) that Dn=o(1) and
(37)o(1)=Dn=a∫ℝN‍g(x)|∇un|p-2∇un∇(un-u0)dx+b(∫ℝN‍g(x)|∇un|pdx)m×∫ℝN‍g(x)|∇un|p-2∇u∇(un-u0)dx+∫ℝN‍h(x)|un|p-2un(un-u0)dx+∫ℝN‍f(x,un)undx-∫ℝN‍f(x,un)u0dx.
Sinceun⇀u0 in X, one obtains from the Hölder inequality that
(38)o(1)=-a∫ℝN‍g(x)|∇u0|p-2∇u0∇(un-u0)dx-b(∫ℝN‍g(x)|∇un|pdx)m×∫ℝN‍g(x)|∇u0|p-2∇u0∇(un-u0)dx-∫ℝN‍h(x)|u0|p-2u0(un-u0)dx.
Thus, it follows from (37) and (38) that
(39)o(1)=a∫ℝN‍g(x)(|∇un|p-2∇unmmmmm-|∇u0|p-2∇u0)∇(un-u0)dx+b(∫ℝN‍g(x)|∇un|pdx)m×∫ℝN‍g(x)(|∇un|p-2∇unmmmmmmm-|∇u0|p-2∇u0)∇(un-u0)dx+∫ℝN‍h(x)(|un|p-2un-|u0|p-2u0)(un-u0)dx+∫ℝN‍f(x,un)undx-∫ℝN‍f(x,un)u0dx.
Sincea≥1, b>0, and g(x), h(x) are positive functions, we get from (25) and (39) that
(40)∫ℝN‍g(x)(|∇un|p-2∇unmmmmm-|∇u0|p-2∇u0)∇(un-u0)dx⟶0,∫ℝN‍h(x)(|un|p-2un-|u0|p-2u0)(un-u0)dx⟶0.
Therefore, we can deduce from (40) and the following standard inequalities in [16]:
(41)|ξ-ζ|p≤{c〈|ξ|p-2ξ-|ζ|p-2ζ,ξ-ζ〉forp≥2,c〈|ξ|p-2ξ-|ζ|p-2ζ,ξ-ζ〉p/2×(|ξ|p+|ζ|p)(2-p)/2for1<p<2
that
(42)∥un-u0∥X⟶0asn⟶∞.
Then, we complete the proof.


## 3. Existence of Solution

In this section, the proof of Theorem3 is mainly based on the following Mountain Pass lemma [17] (also see [18]).Lemma 7 (Mountain Pass lemma).
LetX be a Banach space and J∈C1(X,ℝ) satisfies (PS)c condition. Suppose that J(0)=0 and (I1)
there are constantρ,α>0 such that J∂Bρ≥α,(I2)
there is ane∈X∖Bρ such that J(e)≤0.Then, J possesses a critical value c≥α. Moreover, c can be characterized as
(43)c=infg∈Γmaxu∈g[0,1]J(u)≥α>0,
where
(44)Γ={g∈C([0,1],X):g(0)=0,g(1)=e}.In view of Lemma6, the functional J(u) satisfies the (PS)c condition. It is obvious that J(0)=0; then, in order to apply Lemma 7, we need only to prove that J(u) satisfies the geometric conditions of the Mountain Pass lemma.Lemma 8.
Assume(A1)–(A2). Then, (I1)
there existρ,α>0 such that J(u)≥α>0 with ∥u∥X=ρ,(I2)
there is ane∈X such that ∥e∥X>ρ and J(e)≤0.Proof.
It follows from(A2), (A3), and the embedding in Lemma 4 that
(45)J(u)≥ap∫ℝN‍g(x)|∇u|pdx+b2p(∫ℝN‍g(x)|∇u|pdx)2+1p∫ℝN‍h(x)|u|pdx-1μ∫ℝN‍|u|qdx≥1p∫ℝN‍(g(x)|∇u|p+h(x)|u|p)dx-c∥u∥Xq=1p∥u∥Xp-c∥u∥Xq.
Since p<q, there exists a sufficient small ρ>0 and α>0 such that J(u)≥α>0 with ∥u∥X=ρ. On the other hand, for fixed u0∈X and t>0, it follows from (A2) that
(46)J(tu0)≤atpp∫ℝN‍g(x)|∇u|pdx+bt2p2p(∫ℝN‍g(x)|∇u|pdx)2+tpp∫ℝN‍h(x)|u|pdx-tμ∫ℝN‍|u0|μdx⟶-∞ast⟶+∞.
Then, there is a large t0>0 such that ∥t0u0∥X>ρ and J(t0u0)<0. One may choose e=t0u0; then, ∥e∥X>ρ and J(e)<0. Thus, the proof of Lemma 8 is complete.Proof of Theorem3.
It follows from Lemmas7 and 8 that the functional J(u) has a critical point u such that J(u)=c>0; that is, problem (1) has at least one weak solution. On the other hand, J(0)=0; then, problem (1) has at least one nontrivial weak solution.



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