# Existence and Uniqueness of Positive Solution forp-Laplacian Kirchhoff-Schrödinger-Type Equation

Authors: Wei Han; Jiangyan Yao
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We study the existence and uniqueness of positive solution for the followingp-Laplacian-Kirchhoff-Schrödinger-type equation: -a+b∫Ω∇up∆pu+λvxup-2u=hfu-μgu,inΩ,u>0,inΩ,u=0,on∂Ω, where Ω⊂RN(N≥3), λ,μ≥0 are parameters, v(x),f(u),g(u) and h are under some suitable assumptions. For the purpose of overcoming the difficulty caused by the appearance of the Schrödinger term and general singularity, we use the variational method and some mathematical skills to obtain the existence and uniqueness of the solution to this problem.


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## Body

## 1. Introduction and Main Results

In this paper, we discuss the following problem:(1)-a+b∫Ω∇up∆pu+λvxup-2u=hfu-μguinΩ,u>0,inΩ,u=0,on∂Ω,where Ω⊂RN(N≥3) is a smooth bounded domain with boundary ∂Ω, λ,μ≥0 are parameters, Δpu=div(|∇u|p-2∇u) is the p-Laplacian operator, and p≥2, a,b≥0 with a+b>0.In recent years, a lot of scholars have studied the singular Kirchhoff problem (for more details, we refer the reader to [1–4]), the Schrödinger-Poisson system (we refer the reader to [5–8]), and the Kirchhoff-Schrödinger-Poisson system (we refer the reader to [9–12]). The authors use various methods to obtain the properties of the solution, which makes such problems very interesting. Inspired by the above papers, later scholars begin to make some expanding study about the above problems. For example, in [13], Guo and Nie studied the existence and multiplicity of nontrivial solutions for p-Laplacian Schrödinger-Kirchhoff-type equations by variational methods. For a more complex situation, we refer the reader to [14]. The related studies on the elliptic equations also can be found in [15–26].However, up to now, no paper has appeared in the literature which discusses the existence and uniqueness of the positive solution for thep-Laplacian-Kirchhoff-Schrödinger-type problem. This paper attempts to fill this gap in the literature. Inspired by the above works, in this paper, we try to study the existence and uniqueness of solution to the problem (1) by using the variational method.Next, we will make some assumptions aboutv(x),f(u),g(u) and h.( f 0 )   f ∈ C ( ( 0 , ∞ ) , R + ) satisfies that there exists σ>0, such that f is nonincreasing on (0,σ], ∫0σf(s)ds<∞, and there exists α,γ∈(0,1) such that(2)lims→0+⁡fssα⁡=+∞,lims→∞⁡fssγ=0.( h 0 )   h ∈ L p ∗ / ( p ∗ - 1 - γ ) ( Ω ) satisfies h(x)>0,  a.e.  x∈Ω.( g )   g ∈ C ( R + , R + ) and there exists a constant c>0, such that(3)gs≤csp-1+sp∗-1,s∈R+.( v 1 )   v ( x ) ∈ C ( Ω , R ) , v ( x ) > 0 and the minimum of v(x) can be achieved in Ω. In other words, there exists a constant c′, such that c′=infx∈Ω⁡v(x).( h 1 )  h is bounded in Ω satisfies hx>0,a.e.x∈Ω.( f 1 ) There exists a constant k∈(0,λc′/h∞) such that(4)fs-ft≤ks-tp-1,σ≤t≤s,where ·∞ denotes the maximal value in Ω.In this paper, we will make full use of the following definitions.First, we define the spaceEλ=u∈W01,pΩ:∫Ωλvxup<∞ and the norm(5)uEλp=∫Ωa∇up+λvxup.We denote the norm inLp(Ω) by up=∫Ωup1/p.By(v1) and the Poincaré inequality, we can deduce that the embedding Eλ↪W01,p is continuous. Thus, according to the continuity of the embedding Eλ↪Ls(Ω)(p≤s≤p∗), there are constants cs>0 such that(6)us≤csuEλ.We make further assumptions for convenience. We assumef(s)=g(s)=0 for all s∈(-∞,0). Since lims→∞f(s)/sγ=0 in (f0), we know there exists c0>0, such that(7)fs≤c0sγ,s∈σ/4,∞,which implies(8)0≤Fs=∫0sftdt≤∫0sc0tγdt+∫0σftdt=c0s1+γ1+γ+c1,s∈R.Also, from the fact that∫0σf(s)ds<∞, we can get that F is continuous on R. Thus for any u∈Eλ, by the conditions (h0), (8),(g), (6) and Hölder inequality, we have(9)∫ΩhFu≤c01+γ∫Ωhu1+γdx+c1∫Ωhdx≤c01+γup∗1+γ·hp∗/p∗-1-γ+c1h1≤c21+γuEλ1+γ·hp∗/p∗-1-γ+c1h1,(10)0≤∫ΩGu≤c3upp+up∗p∗≤c4uEλp+uEλp∗,where G(s)=∫0sgtdt for all s∈R and c0,c1,c2,c3,c4 are some positive constants. Next, we can define the energy functional corresponding to problem (1):(11)Iu=ap∫Ω∇updx+b2p∫Ω∇updx2+1p∫Ωλvxup+∫ΩμGu-∫ΩhFu.By a simple computation, we can get(12)I′u,ν=a∫Ω∇up-2∇u·∇ν+b∫Ω∇up∫Ω∇up-2∇u·∇ν+∫Ωλvxup-2u·ν-∫Ωhfuν+∫Ωμguν.It is clear thatu with u(x)>0,a.e.x∈Ω,u(x)∈Eλ is called a weak solution of the problem (1) if for any ν∈Eλ it holds(13)I′u,ν=0.Finally, we will give the main results of the paper.Theorem 1.
Ifa,b≥0 with a+b>0 and the assumptions (f0),(h0),(g) and (v1) hold, then the problem (1) possesses a positive solution for any λ,μ∈R+. Moreover, this solution is a global minimizer of I.Theorem 2.
Ifa>0 and the assumptions (f0),(f1),(h1),(g) and (v1) hold. Moreover, assume that g is nondecreasing on R+, then the solution for problem (1) is unique for any λ,μ∈R+.Remark 3.
The result obtained in the paper is an expanding study of the Kirchhoff-Schrödinger-type equation(p=2); the difficulty is posed by the degenerate quasilinear elliptic operator. We mainly use the variational method to solve the problem.This paper is organized as follows. In Section2, we will give a preliminary. In Section 3, we will prove the main results.In this paper,c,ci denote various positive constants, which may vary from line to line.


## 2. Preliminary

To prove the main results in this paper, we will employ the following important lemma.Lemma 4.
If the assumptions(f0), (h0), (g), and (v1) hold, then I attains the global minimum in Eλ; that is, there exists u0∈Eλ such that I(u0)=m:=infEλI, and m<0.Proof.
For anyu∈Eλ, by (5), (9)–(11), we can get(14)Iu=ap∫Ω∇updx+b2p∫Ω∇updx2+1p∫Ωλvxup+∫ΩμGu-∫ΩhFu≥ap∫Ω∇updx+1p∫Ωλvxup-c21+γuEλ1+γ·hp∗/p∗-1-γ-c1h1=1puEλp-c2hp∗/p∗-1-γ1+γ·uEλ1+γ-c1h1.Since p≥2 and γ∈(0,1), we can obtain that I is coercive and bounded from below on Eλ. The definition that m=infEλI makes some sense.
Sincelims→0+fs/s-α→∞ in the condition (f0), there exists σ1>0 such that(15)fs≥s-α,Fs≥s1-α1-α,s∈0,σ1.
Choosing a nonnegative functionφ∈C0∞(Ω)\{0} with maxΩφ≤σ1, then for any t∈(0,1],tφ∈(0,σ1], by (5), (10), (11), (15), we have(16)Itφ=atpp∫Ω∇φpdx+bt2p2p∫Ω∇φpdx2+tpp∫Ωλvxφp+∫ΩμGtφ-∫ΩhFtφ≤tppφEλp+bt2p2p∫Ω∇φpdx2+μc4φEλptp+φEλp∗tp∗-t1-α1-α∫Ωhφ1-α.Since 1-α∈(0,1) and h(x)>0,a.e.x∈Ωin(h0),p∗>p≥2, we can get that I(tφ)<0 for t>0 small enough. That is m<0.
On the basis of the definition ofm, we can deduce that there exists a sequence {un}⊂Eλ such that limn→∞⁡I(un)=m. Since I is coercive and m<0, {un} is bounded in Eλ. Going if necessary to a subsequence, still denoted by {un}, there exists u0∈Eλ such that(17)un⇀u0,weakly  inEλ,un→u0,strongly  inLsΩ,1≤s<p∗unx→u0xa.e.inΩ,as n→∞. It follows from (8) and Sobolev embedding theorem that {F(un)} is bounded in Lp∗/1+γ. Moreover, from the continuity of F, we can get that F(un(x))→F(uo(x)),a.e.x∈Ω. Thus, we obtain F(un)⇀F(u0) in Lp∗/(1+γ)(Ω).
Sinceh∈Lp∗/(p∗-1-γ)(Ω), we have(18)limn→∞⁡∫ΩhFun⁡=∫ΩhFu0.
Moreover, by Fatou’s lemma, we have(19)lim⁡infn→∞⁡∫ΩGun≥∫ΩGu0.
According to the weakly lower semicontinuity of the norm, (18) and (19), we have(20)m=lim⁡infn→∞⁡Iun=lim⁡infn→∞⁡ap∫Ω∇unp+b2p∫Ω∇unp2+1p∫Ωλvxunp+∫ΩμGun-∫ΩhFun≥1pu0Eλp+b2p∫Ω∇u0p2+∫ΩμGu0-∫ΩhFuo=Iu0≥m,which yields I(u0)=m. The proof is completed.


## 3. Proof of Main Results

Proof of Theorem1.
Sincem≤I(u0+)≤I(u0)=m, then I(u0+)=I(u0). Thus we may assume u0≥0. Owing to m<0, we know u0≠0. Next we will give the two-step proof.
(i) Firstly, we shall proveu0(x)>0,a.e.x∈Ω.
For anyν∈Eλ with ν(x)≥0,a.e.  x∈Ω and t>0, we have(21)0≤Iu0+tν-Iu0t=apt∫Ω∇u0+tνp-∫Ω∇u0p+b2pt∫Ω∇u0+tνp2-∫Ω∇u0p2+λpt∫Ωvxu0+tνp-vxu0p+μt∫ΩGu0+tν-Gu0-1t∫ΩhFu0+tν-hFu0.Letting t→0+, we can get(22)lim⁡inft→0+⁡∫ΩhtFu0+tν-Fu0≤a∫Ω∇u0p-2∇u0∇ν+b∫Ω∇u0p∫Ω∇u0p-2∇u0∇ν+λ∫Ωvxu0p-2u0ν+μ∫Ωgu0ν.Thus, by Fatou’s lemma and Lemma  2.3 in [27], we can get(23)∫Ωhfu0ν≤a∫Ω∇u0p-2∇u0∇ν+b∫Ω∇u0p∫Ω∇u0p-2∇u0∇ν+λ∫Ωvxu0p-2u0ν+μ∫Ωgu0ν.
Lete1∈Eλ be the first eigenfunction of the operator -Δp with the Dirichlet boundary and e1(x)>0 for all x∈Ω. Taking ν=e1 in (23), by (v1) and the condition (g), we have(24)∫Ωhfu0e1≤a∫Ω∇u0p-2∇u0∇e1+b∫Ω∇u0p∫Ω∇u0p-2∇u0∇e1+λ∫Ωvxu0p-2u0e1+μ∫Ωgu0e1≤a∫Ω∇u0p-2∇u0∇e1+b∫Ω∇u0p∫Ω∇u0p-2∇u0∇e1+λ∫Ωvxu0p-2u0e1+μc∫Ωu0p-1+u0p∗-1e1<∞,which implies u0(x)>0,a.e.x∈Ω by the condition (h0). If not, there exists E⊂Ω such that m(E)>0 and u0(x)=0 for all x∈E. Then by Lemma  2.3 in [27], we can get(25)∫Ωhfu0e1≥∫Ehfu0e1=∞.It is a contradiction. So the claim u0(x)>0 is true.
(ii)u0 is exactly a solution of the problem (1); that is, u0 satisfies (13):(26)a∫Ω∇u0p-2∇u0·∇ν+b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇ν+∫Ωλvxu0p-2u0·ν-∫Ωhfu0ν+∫Ωμgu0ν=0.
To obtain the conclusion, we define a functionΨ:R→R,Ψ(t)=I(u0+tu0); that is,(27)Ψt=a1+tpp∫Ω∇u0p+b1+t2p2p∫Ω∇u0p2+1+tpp∫Ωλvxu0p+∫ΩμGu0+tu0-∫ΩhFu0+tu0.
From the above discussion, we knowΨ(t) attains its minimum at t=0. By Lemma  2.4 in [27], we can get that Ψ(t) is differentiable at t=0 and Ψ′(0)=0; that is,(28)a∫Ω∇u0p+b∫Ω∇u0p2+∫Ωλvxu0p+∫Ωμgu0u0-∫Ωhfu0u0=0.
For eachυ∈Eλ and ε>0, we define υε=u0+ευ and(29)Ω+=x∈Ω:u0+ευ≥0,Ω-=x∈Ω:u0+ευ<0.Then υε-|Ω+=0 and υε-|Ω-=u0+ευ.
Insertingυε+ into (23) and using (28), we can get that(30)0≤a∫Ω∇u0p-2∇u0·∇υε++b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇υε++λ∫Ωvxu0p-2u0υε++μ∫Ωgu0υε+-∫Ωhfu0υε+=a∫Ω∇u0p-2∇u0·∇υε+b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇υε+λ∫Ωvxu0p-2u0υε+μ∫Ωgu0υε-∫Ωhfu0υε-a∫Ω∇u0p-2∇u0·∇υε-+b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇υε-+λ∫Ωvxu0p-2u0υε-+μ∫Ωgu0υε--∫Ωhfu0υε-=εa∫Ω∇u0p-2∇u0·∇υ+b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇υ+λ∫Ωvxu0p-2u0υ+μ∫Ωgu0υ-∫Ωhfu0υ-a∫Ω-∇u0p-2∇u0·∇u0+ευ+b∫Ω∇u0p∫Ω-∇u0p-2∇u0·∇u0+ευ+λ∫Ω-vxu0p-2u0u0+ευ+μ∫Ω-gu0u0+ευ-∫Ω-hfu0u0+ευ≤εa+b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇υ+λ∫Ωvxu0p-2u0υ+μ∫Ωgu0υ-∫Ωhfu0υ-εa+b∫Ω∇u0p∫Ω-∇u0p-2∇u0·∇υ+λ∫Ω-vxu0p-2u0υ+μ∫Ω-gu0υ,
which implies that(31)a+b∫Ω∇u0p∫Ω-∇u0p-2∇u0·∇υ+λ∫Ω-vxu0p-2u0υ+μ∫Ω-gu0υ≤a+b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇υ+λ∫Ωvxu0p-2u0υ+μ∫Ωgu0υ-∫Ωhfu0υ.
Next we defineΛn={x∈Ω:u0(x)>0,υ(x)>-∞,u0(x)+υ(x)/n<0} for all n. By simple computation, we can deduce that {Λn} is a nonincreasing sequence of measurable sets and limn→∞Λn=⋂n=1∞Λn=∅. Thus we have(32)limn→∞⁡mΛn⁡=mlimn→∞⁡Λn=0.
Letε=1/n; then Ω-⊂x∈Ω:u0x≤0∪x∈Ω:υx=-∞∪Λn and m(Ω-)=m(Λn)→0 as n→∞. Selecting ε=1/n→0 in (31), we have(33)0≤a+b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇υ+λ∫Ωvxu0p-2u0υ+μ∫Ωgu0υ-∫Ωhfu0υ.
According to the arbitrariness ofυ∈Eλ, this inequality also holds for -υ. Combining (33), we can get that, for any υ∈Eλ,(34)a+b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇υ+λ∫Ωvxu0p-2u0υ+μ∫Ωgu0υ-∫Ωhfu0υ=0.Thus, u0 is exactly a weak solution of the problem (1). By Lemma 4, we know I(u0)=infEλI. Therefore, u0 is exactly a global minimizer solution.Proof of Theorem2.
Assume thatu1 is also a solution of problem (1). Letting ν=u0-u1, according to the definition of the weak solution and (26), we can get(35)0=a+b∫Ω∇u0p∫Ω∇u0p-2∇u0·∇u0-u1+∫Ωλvxu0p-2u0u0-u1+∫Ωμgu0u0-u1-∫Ωhfu0u0-u1,(36)0=a+b∫Ω∇u1p∫Ω∇u1p-2∇u1·∇u0-u1+∫Ωλvxu1p-2u1u0-u1+∫Ωμgu1u0-u1-∫Ωhfu1u0-u1,
which implies(37)∫Ωhu0-u1fu0-fu1=a∫Ω∇u0-∇u1∇u0p-2∇u0-∇u1p-2∇u1dx+b∫Ω∇u0-u1∇u0∇u0p-2∫Ω∇u0p-∇u1∇u1p-2∫Ω∇u1p+∫Ωλvxu0-u1u0u0p-2-u1u1p-2+∫Ωμu0-u1gu0-gu1.Next, we will make some estimates for the equation.
(i)(38)∫Ω∇u0-u1∇u0∇u0p-2∫Ω∇u0p-∇u1∇u1p-2∫Ω∇u1p≥0.In fact, we estimate as follows.(39)∫Ω∇u0-u1∇u0∇u0p-2∫Ω∇u0p-∇u1∇u1p-2∫Ω∇u1p=∫Ω∇u0p2-∫Ω∇u0p-2∇u0·∇u1∫Ω∇u0p+∫Ω∇u1p2-∫Ω∇u1p∫Ω∇u1p-2∇u0·∇u1≥∫Ω∇u0p2-∫Ω∇u0p-1∇u1∫Ω∇u0p+∫Ω∇u1p2-∫Ω∇u1p∫Ω∇u1p-1∇u0≥max⁡∫Ω∇u0p,∫Ω∇u1p∫Ω∇u0-∇u1∇u0p-1-∇u1p-1≥0.
(ii)(40)∫Ωvxu0-u1u0u0p-2-u1u1p-2≥∫Ωvxu0-u1p.
(iii)(41)∫Ω∇u0-∇u1∇u0p-2∇u0-∇u1p-2∇u1=∫Ω∇u0p-2∇u02-∇u0·∇u1-∫Ω∇u1p-2∇u0·∇u1-∇u12≥∫Ω∇u0p-2∇u02-∇u0·∇u1-∫Ω∇u1p-2∇u0·∇u1-∇u12=∫Ω∇u0-∇u1∇u0p-1-∇u1p-1≥0.
( i v ) Since g is nondecreasing on R+, we have(42)∫Ωgu0-gu1u0-u1≥0.
( v ) Next, we estimate the left side of (37), according to the conditions (f0) and (f1), we can prove that(43)fs-ft≤ks-tp-1,0<t≤s<∞.Thus by a simple deduction and (h1),(f1), one has(44)∫Ωhfu0-fu1u0-u1≤k∫Ωhu0-u1p≤khx∞∫Ωu0-u1p.
It follows from (37), (38), (40)–(42), and (44) and we have(45)λc′-kh∞∫Ωu0-u1p≤0,where c′=infx∈Ωv(x). Since k∈(0,λc′/h∞) in the condition (f1), we can get that ∫Ωu0-u1p=0, that is, u0=u1. Therefore, the solution of the problem (1) is unique.



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