# Multiple Nontrivial Solutions for a Nonlocal Problem with Sublinear Nonlinearity

Authors: Zhigao Shi; Xiaotao Qian
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this paper, we study the following nonlocal problem−a−b∫Ω∇u2dxΔu=λu+fxup−2u,x∈Ω,u=0,x∈∂Ω, where a,b>0 are constants, 1<p<2, λ>0, f∈L∞Ω is a positive function, and Ω is a smooth bounded domain in ℝN with N≥3. By variational methods, we obtain a pair of nontrivial solutions for the considered problem provided f∞ is small enough.


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## Body

## 1. Introduction and Main Results

This paper is concerned with the existence and multiplicity of nontrivial solutions for the following nonlocal problem with Dirichlet boundary value conditions:(1)−a−b∫Ω∇u2dxΔu=λu+fxup−2u,x∈Ω,u=0,x∈∂Ω,where a,b>0, 1<p<2, λ>0, f∈L∞Ω is a positive function, and Ω is a smooth bounded domain in ℝN with N≥3.In the past two decades, the following Kirchhoff type problems with Dirichlet boundary value conditions(2)−a+b∫Ω∇u2dxΔu=fx,u,x∈Ω,u=0,x∈∂Ω,have attracted great attention of many researchers. Such problems are often viewed as nonlocal because of the appearance of the term a+b∫Ω∇u2dx, which implies that (2) is no longer a pointwise identity. It is worthwhile pointing out that the equation in (2) arises in various models of physical and biological systems. Indeed, problem (2) is related to the stationary analogue of the following equation:
(3)ρ∂2u∂t2−P0h+E2L∫0L∂u∂xdx∂2u∂x2=0,which was first presented by Kirchhoff [1] as an extension of the classical d’Alembert wave equation for free vibrations of elastic strings, where u=ux,t denotes the lateral displacement, ρ the mass density, P0 the initial tension, h the cross-section area, E the Young modulus of the material, and L the length of the string. Under different assumptions on fx,u, many interesting results on the existence of solutions to (2) were obtained. We refer the interested readers to [2–14] and the references therein.However, we now face a new nonlocal terma−b∫Ω∇u2dx, which is different from the well-known Kirchhoff type nonlocal term a+b∫Ω∇u2dx. Now, there has been some results on the existence and multiplicity of nontrivial solutions to this new nonlocal problem (see [15–22]).In particular, Yin and Liu [15] firstly studied this kind of problem:
(4)−a−b∫Ω∇u2dxΔu=up−2u,x∈Ω,u=0,x∈∂Ω,where 2<p<2∗, and obtained the existence and multiplicity of solutions for the problem.In [16], Lei et al. considered
(5)−a−b∫Ω∇u2dxΔu=fλxup−2u,x∈Ω,u=0,x∈∂Ω,where 1<p<2, and proved under certain condition on fλx, that there are at least two positive solutions. After this, the authors also studied the problem with singularity [17].Wang et al. [20] investigated the nonlocal problem with critical exponent
(6)−a−b∫ℝ4∇u2dxΔu=u2u+μgx,x∈ℝ4,u∈D1,2ℝ4.Whenμ is a nonnegative parameter and gx∈L4/3ℝ4, they showed the existence of multiple positive solutions.Recently, Zhang and Zhang [22] studied the nonlocal problem
(7)−a−b∫Ω∇u2dxΔu=λu+gx,u,x∈Ω,u=0,x∈∂Ω,where the parameter 0<λ<aλ1, λ1 is the first eigenvalue of operator −Δ, and g is a superlinear function with subcritical growth. By using the Mountain Pass Theorem, the authors obtained the existence of a nontrivial solution.As far as we know, there is no work on the existence of solution to (1), which is just our purpose here. Moreover, we extend λ≥aλ1 and without assuming nonlinearity is superlinear.Our main result can be stated as follows.Theorem 1.
Assume thata,b>0, 1<p<2, λ>0, f∈L∞Ω is a positive function, then problem (1) has at least a pair of nontrivial solutions if f∞ is small enough.The paper is organized as follows. In Section2, we give some notations and preliminaries. In Section 3, we prove Theorem 1 for the case of 0<λ<aλ1. Section 4 is devoted to the proof of Theorem 1 for the case of λ≥aλ1.


## 2. Notations and Preliminaries

Throughout this paper, we make use of the following notations.H01Ω and LqΩ are standard Sobolev spaces with the usual norm ∥u∥2=∫Ω∇u2dx, uqq=∫Ωuqdx. Brx (∂Brx) denotes an open ball (the sphere) centered at x with radius r>0. ⟶ and ⇀ denote strong and weak convergence, respectively. For each 1≤r≤2∗, we denote by Sq the best Sobolev constant for the embedding of H01Ω into LqΩ. Let 0<λ1<λ2≤⋯≤λk<⋯ be the sequence of eigenvalues of −Δ on H01Ω satisfying limk⟶∞λk=+∞, and let e1,e2,⋯, be the corresponding orthonormal eigenfunctions in L2Ω.By the Sobolev Theorem andf∈L∞Ω, the functional
(8)Iu=a2∥u∥2−b4∥u∥4−λ2∫Ωu2dx−1p∫Ωfxupdxis well defined on H01Ω. Furthermore, it belongs to C1H01Ω,ℝ, and its critical points are precisely the weak solutions of (1). Here, we say u∈H01Ω is a weak solution to (1), if for any v∈H01Ω, it holds
(9)a−b∫Ω∇u2dx∫Ω∇u∇vdx−λ∫Ωuvdx−∫Ωfxup−2uvdx=0.Following [15], we first prove that the functional I satisfies the PSc condition for any c<a2/4b.Lemma 2.
Under the assumptions of Theorem1, I satisfies the PSc condition with c<a2/4b.Proof.
Letun⊂H01Ω be a PSc sequence for I with c<a2/4b; that is,
(10)Iun⟶c,I′un⟶0,asn⟶∞.
By the Sobolev Theorem and (10),
(11)c+1+o1∥un∥≥Iun−12I′un,un=b4∥un∥4−1p−12∫Ωfxunpdx≥b4∥un∥4−1p−12f∞Sp−p/2∥un∥p.
Sincep<2, we conclude that un is bounded in H01Ω. Up to a subsequence (still denoted by un), we may assume that
(12)un⇀u,inH01Ω,un⟶u,inLrΩ,1≤r<2∗,un⟶u,a.e.inΩ.
By using Hölder’s inequality, it follows from (12) that
(13)∫Ωfxunp−2unun−udx≤f∞unpp−1un−up⟶0,as n⟶∞. Similarly, we also have
(14)∫Ωunun−udx≤un2un−u2⟶0.
From the two above convergences, we get(15)o1=I′unun−u=a−b∥un∥2∫Ω∇un∇un−u+o1,as n⟶∞. We claim that a−b∥un∥2⟶0 is false. If, to the contrary, namely, ∥un∥2⟶ab, define a functional by
(16)ϕu=λ2∫Ωu2dx+1p∫Ωfxupdx,u∈H01Ω.
Then,(17)ϕ′u,v=λ∫Ωuvdx+∫Ωfxup−2uvdx,u,v∈H01Ω.
By using Hölder’s inequality again, we obtain(18)∣∫Ωfxunp−2un−up−2uvdx∣≤f∞∫Ωunp−2un−up−2u∣v∣dx≤∣f∞unp−2un−∣uup-2p/p−1Sp−p/2∥v∥p,∣∫Ωunv−uvdx∣≤∫Ωun−uvdx≤un−u2λ1−1∥v∥2.
Hence, by using (12), we obtain
(19)∥ϕ′un−ϕ′u∥≤un−u2λ1−1+∣f∞unp−2un−∣uup-2p/p−1Sp−p/2⟶0,as n⟶∞. This shows ϕ′un⟶ϕ′u.
On the other hand, from(20)o1=I′un,v=a−b∥un∥2∫Ω∇u∇vdx−ϕ′un,vand ∥un∥2⟶a/b, we have ϕ′un⟶0.
Thus, we can deduce that(21)ϕ′u,v=λ∫Ωuvdx+∫Ωfxup−2uvdx=0,∀v∈H01Ω.
By the variational method fundamental lemma [23], we further obtain
(22)λux+fxuxp−2ux=0,a.e.x∈Ω.
Sincefx>0, it then follows that u=0.
By (12) and f∈L∞Ω, we can use the Vitali Convergence Theorem to obtain
(23)limn⟶∞∫Ωfxun∣pdx=∫Ωfxupdx,and consequently,
(24)ϕun=λ2∫Ωun2dx+1p∫Ωfxunpdx⟶λ2∫Ωu2dx+1p∫Ωfxupdx=0.
This and∥un∥2⟶a/b provide
(25)Iun=a2∥un∥2−b4∥un∥4−λ2∫Ωun2dx−1p∫Ωfxunpdx⟶a24b,which contradicts Iun⟶c<a2/4b. Thus, the claim follows. In turn, we have from (15) that ∫Ω∇un∇un−udx⟶0, and hence, ∥un∥⟶∥u∥. Combining this with the weak convergence of un in H01Ω, we deduce that un⟶u in H01Ω.☐


## 3. Proof of Theorem1 for 0<λ<aλ1

In this section, we will use the Mountain Pass Theorem to prove the existence of a pair of nontrivial solutions of the considered problem for0<λ<aλ1.Lemma 3.
Assume that0<λ<aλ1. If f∞ is sufficiently small, then there is a sequence un⊂H01Ω such that un≥0, Iun⟶c∗ and I′un⟶0, where 0<c∗<a2/4b.Proof.
By the Sobolev Theorem, we have that(26)Iu=a2∥u∥2−b4∥u∥4−λ2∫Ωu2dx−1p∫Ωfxupdx≥12a−λλ1∥u∥2−b4∥u∥4−1pf∞Sp−p/2∥u∥p=∥u∥p12a−λλ1∥u∥2−p−b4∥u∥4−p−1pf∞Sp−p/2.
ForA=1/2a−λ/λ1 and B=b/4, the function g:0,+∞⟶ℝ defined by
(27)gt≔At2−p−Bt4−pattains its maximum value at
(28)ρ=2−pA4−pB1/2.
TakeD0=gρ and note that, for any u∈H01Ω, ∥u∥=ρ, there holds
(29)Iu≥ρpD0−1pf∞Sp−p/2≥ρp2D02≕θ>0,whenever
(30)f∞≤D0p2Spp/2.
On the other hand, letu∈H01Ω\0, then we have that
(31)limt⟶+∞Itu=limt⟶+∞a2t2∥u∥2−b4t4∥u∥4−λ2t2∫Ωu2dx−1ptp∫Ωfxupdx=−∞.
Thus, there existsv1∈H01Ω such that ∥v1∥>ρ and Iv1<0.
ByIu=I∣u∣, we may assume that un≥0. By applying the Mountain Pass Theorem without PS condition [24], we construct a sequence un⊂H01Ω satisfying Iun⟶c∗ and I′un⟶0 for
(32)c∗≔infγ∈Γmaxt∈0,1Iγt≥θ>0,where
(33)Γ≔γ∈C0,1,H01Ω:γ0=0,γ1=v1.
From easy calculations, we get(34)maxt≥0Itv1=maxt≥0a2t2∥v1∥2−b4t4∥v1∥4−λ2t2∫Ωv12dx−1ptp∫Ωfxv1pdx<maxt≥0a2t2∥v1∥2−b4t4∥v1∥4≤a24b.
This and the definition ofc∗ yield that 0<c∗<a2/4b. Thus, we complete the proof of Lemma 3.☐Proposition 4.
Assumea,b>0, 0<λ<aλ1 and fx∈L∞Ω is a positive function. Then, problem (1) admits a pair of nontrivial solutions if f∞ is small enough.Proof.
By Lemma3, we obtain a sequence un⊂H01Ω such that un≥0, Iun⟶c∗ and I′un⟶0, provided f∞ is small enough. It then follows from Lemma 2 that there exists u∈H01Ω such that un⟶u in H01Ω with Iu=c∗ and I′u=0, which implies that u is a nontrivial nonnegative solution of (1). By the symmetry of functional I, we further deduce that −u is a nontrivial nonpositive solution of (1). This completes the proof.☐


## 4. Proof of Theorem1 for λ≥aλ1

Sinceλ≥aλ1, the method used in the previous section does not work here. Indeed, we shall apply the following Linking Theorem [25] to establish the existence of a pair of nontrivial solutions for problem (1) when λ≥aλ1.Theorem 5.
LetX be a real Banach space with X=Y⊕Z and dimY<+∞. Suppose that I∈C1X,ℝ satisfies the following:
(I1) There are ρ~,θ~>0 such that I∂Bρ~0∩Z≥θ~
(I2) There are e∈∂Bρ~0∩Z and R>ρ~ such that I∂M≤0 with M=BR0∩Y⊕te:0<t<R
Then, there exists a sequenceun⊂X satisfying Iun⟶c and I′un⟶0 for
(35)c~∗≔infγ~∈Γ~maxu∈MIγ~u,where
(36)Γ~≔γ~∈CM,X:γ~∂M=id.
As the sequence of eigenvaluesλk goes to infinity, there is n∈ℕ such that aλ1≤λ<aλn+1. Set
(37)Y=spane1,⋯,en,Z=Y⊥.
Clearly,X=Y⊕Z.Lemma 6.
There existsz∈Z\0 such that
(38)maxu∈Y⊕ℝzIu<a24b.Proof.
Since, for anyu∈H01Ω\0,
(39)maxt≥0Itu=maxt≥0a2t2∥u∥2−b4t4∥u∥4−λ2t2∫Ωu2dx−1ptp∫Ωfxupdx<maxt≥0a2t2∥u∥2−b4t4∥u∥4≤a24b,then there exists u∗∈H01Ω such that
(40)z∗∈Z\0,maxu∈Y⊕ℝz∗Iu<a24b,where
(41)z∗=u∗−∑k=1n∫Ωu∗ekdxek.
This completes the proof of Lemma6.☐Proposition 7.
Assumea,b>0, λ≥aλ1 and fx∈L∞Ω is a positive function. Then, problem (1) admits a pair of nontrivial solutions if f∞ is small enough.Proof.
Firstly, we have for anyu∈Z,
(42)Iu=a2∥u∥2−b4∥u∥4−λ2∫Ωu2dx−1p∫Ωfxupdx≥12a−λλn+1∥u∥2−b4∥u∥4−f∞1pSp−p/2∥u∥p,and therefore, as in the proof of Lemma 3, we can prove that I satisfies the condition I1 of Theorem 5 when f∞ is small.
Secondly, sinceλ≥aλn, we also have for any u∈Y,
(43)Iu=a2∥u∥2−b4∥u∥4−λ2∫Ωu2dx−1p∫Ωfxupdx≤12aλn−λu22≤0.
Moreover, ifz∈Z\0 is given by Lemma 6, we can apply the equivalence of norms in the finite dimensional space, to obtain for u∈Y⊕ℝz,
(44)Iu⟶−∞,as∥u∥⟶+∞.
Thus, the conditionI2 is satisfied for R large enough.
Finally, by Theorem5, Lemma 6, and I∣u∣=Iu, we conclude that there is a sequence un⊂H01Ω satisfying un≥0, Iun⟶c~∗ and I′un⟶0, provided f∞ is small enough. Then, we can argue as in the proof of Proposition 4 to obtain a pair of nontrivial solutions of (1). Thus, the proof of Proposition 7 is complete.☐


## 5. Conclusions

In this paper, we consider a new nonlocal problem different from the well-known Kirchhoff type problem. Compared with the existing literature, we extendλ≥aλ1 and without assuming nonlinearity is superlinear, which causes the methods employed in the previous papers cannot be directly applied here. To overcome this difficulty, we distinguish two cases: (1) 0<λ<aλ1 and (2) λ≥aλ1. In the former case, the existence of a nontrivial nonnegative solution is established via the Mountain Pass Theorem, while the Linking Theorem for the latter case. Furthermore, the symmetry of the corresponding functional brings us another nontrivial nonpositive solution.



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