# Multiple Solutions for Kirchhoff Equations under the Partially Sublinear Case

Authors: Wenjun Feng; Xiaojing Feng
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We prove the infinitely many solutions to a class of sublinear Kirchhoff type equations by using an extension of Clark’s theorem established by Zhaoli Liu and Zhi-Qiang Wang.


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## Body

## 1. Introduction and Main Results

In this paper we study the existence and multiplicity of solutions for the following Kirchhoff type equations:(1)a+∫RN∇u2+b∫RNu2-Δu+bu=Kxfx,u,inR3,where a, b are positive constants.WhenΩ is a smooth bounded domain in R3, the problem(2)-a+b∫Ω∇u2dxΔu=fx,u,inΩu=0,on∂Ω,has been studied in several papers. Perera and Zhang [1] considered the case where f(x,·) is asymptotically linear at 0 and asymptotically 4-linear at infinity. They obtained a nontrivial solution of the problems by using the Yang index and critical group. Then, in [1] they considered the cases where f(x,·) is 4-sublinear, 4-superlinear, and asymptotically 4-linear at infinity. By various assumptions on f(x,·) near 0, they obtained multiple and sign changing solutions. Cheng and Wu [2] and Ma and Rivera [3] studied the existence of positive solutions of (2) and He and Zou [4] obtained the existence of infinitely many positive solutions of (2), respectively; Mao and Luan [5] obtained the existence of signed and sign-changing solutions for problem (2) with asymptotically 4-linear bounded nonlinearity via variational methods and invariant sets of descent flow; Sun and Tang [6] studied the existence and multiplicity results of nontrivial solutions for problem (2) with the weaker monotony and 4-superlinear nonlinearity. For (2), Sun and Liu [7] considered the cases where the nonlinearity is superlinear near zero but asymptotically 4-linear at infinity, and the nonlinearity is asymptotically linear near zero but 4-superlinear at infinity. By computing the relevant critical groups, they obtained nontrivial solutions via Morse theory.Comparing with (1) and (2), R3 is in place of the bounded domain Ω⊂R3. This makes the study of problem (1) more difficult and interesting. Wu [8] considered a class of Schrödinger Kirchhoff type problem in RN and a sequence of high energy solutions are obtained by using a symmetric Mountain Pass Theorem. In [9], Alves and Figueiredo study a periodic Kirchhoff equation in RN; they get the nontrivial solution when the nonlinearity is in subcritical case and critical case. Liu and He [10] obtained multiplicity of high energy solutions for superlinear Kirchhoff equations in R3. Li et al. in [11] proved the existence of a positive solution to a Kirchhoff type problem on RN by using variational methods and cutoff functional technique.In [12], Jin and Wu consider the following problem: (3)-a+b∫RN∇u2dxΔu+u=fx,u,inRN,u∈H1RN,where constants a>0, b>0, N=2 or 3, and f∈C(RN×R,R). By using the Fountain Theorem, they obtained the following theorem.Theorem A (see [12]).
Assume that the following conditions hold.
If the following assumptions are satisfied,(H1)
f(x,u)=o(|u|) as |u|→0 uniformly for any x∈RN,(H2)
there are constants1<p<2∗-1 and c>0 such that (4)fx,u≤c1+up,∀x,u∈RN×R,where(5)2∗-1=N+2N-2,N≥3;+∞,N=1,2,(H3)
there existsμ>4 such that(6)μFx,u=μ∫0ufx,sds≤ufx,u,∀x,u∈RN×R,(H4)
(7)infx∈RN,u=1Fx,u>0,(H5)
f(gx,u)=f(x,u) for each g∈O(N) and for each (x,u)∈RN×R, where O(N) is the group of orthogonal transformations on RN,(H6)
f(x,-u)=-f(x,u) for any (x,u)∈RN×R,then problem (3) has a sequence {uk} of radial solutions.Recently, Liu and Wang [13] obtained an extension of Clark’s theorem as follows.Theorem B (see [13]).
LetX be a Banach space, Φ∈C1(X,R). Assume Φ is even and satisfies the (PS) condition, bounded from below, and Φ(0)=0. If, for any k∈N, there exists a k-dimensional subspace Xk of X and ρk>0 such that supXk∩SρkΦ<0, where Sρ={u∈X∣u=ρ}, then at least one of the following conclusions holds.(i)
There exists a sequence of critical points{uk} satisfying Φ(uk)<0 for all k and uk→0 as k→∞.(ii)
There existsr>0 such that for any 0<a<r there exists a critical point u such that u=a and Φ(u)=0.Theorem A obtained the existence of infinitely many solutions under the case thatf(t,u) is sublinear at infinity in u. It is worth noticing that there are few papers concerning the sublinear case up to now. Motivated by the above fact, in this paper our aim is to study the existence of infinitely many solutions for (1) when f(t,u) satisfies sublinear condition in u at infinity. Our tool is extension of Clark’s theorem established in [13]. Now, we state our main result.Theorem 1.
Assume thatf satisfies (H6) and the following conditions:(f1)
There existδ>0, 1≤γ<2, C>0 such that f∈C(R3×[-δ,δ],R) and |f(x,z)|≤C|z|γ-1.(f2)
Considerlimz→0F(x,z)/|z|2=+∞ uniformly in some ball Br(x0)⊂R3, where F(x,z)=∫0zf(x,s)ds.(f3)
K:R3→R+ is a positive continuous function such that K∈L2/(2-γ)(R3)∩L∞(R3).
Then (1) possesses infinitely many solutions {uk} such that ukL∞→0 as k→∞.Remark 2.
Throughout the paper we denote byC>0 various positive constants which may vary from line to line and are not essential to the problem.The paper is organized as follows: in Section2, some preliminary results are presented. Section 3 is devoted to the proof of Theorem 1.


## 2. Preliminary

In this section, we will give some notations that will be used throughout this paper.LetH1=H1(R3) be the completion of C0∞(R3) with respect to the inner product and norm (8)u,v=∫R3∇u∇v+buvdx,u=u,u1/2.Moreover, we denote the completion of C0∞(R3) with respect to the norm (9)uD12=∫R3∇u2dxby D1=D1(R3). To avoid lack of compactness, we need to consider the set of radial functions as follows: (10)H=Hr1R3=u∈H1R3∣ux=ux.Here we note that the continuous embedding H↪Lq(R3) is compact for any q∈(2,6).Define a functional by(11)J1u=a2u2+14u4-∫R3KxFx,u,u∈H.Then we have from (f1) that J1 is well defined on H and is of C1, and (12)J1u,v=au,v+u2u,v-∫R3Kxfx,uv,u,v∈H.It is standard to verify that the weak solutions of (1) correspond to the critical points of functional J1.


## 3. Proofs of the Main Result

Proof of Theorem1.
Choosef^∈C(RN×R,R) such that f^ is odd in u∈R, f^(x,u)=f(x,u) for x∈RN and |u|<δ/2, and f^(x,u)=0 for x∈RN and |u|>δ. In order to obtain solutions of (1) we consider (13)a+∫RN∇u2+b∫RNu2-Δu+bu=Kxf^x,u,inRN.
Moreover, (13) is variational and its solutions are the critical points of the functional defined in H by (14)Ju=12au2+14u4-∫R3KxF^x,udx.From (f1), it is easy to check that J is well defined on H and J∈C1(H,R), and (15)J′uv=au,v+u2u,v-∫R3Kxf^x,uvdx,v∈H.
Note thatJ is even, and J(0)=0. For u∈H, (16)∫R3KxF^x,udx≤C∫R3Kxuγdx≤CKL2/2-γR3uL2R3γ≤Cuγ.Hence, it follows from (14) that (17)Ju≥12u2-Cuγ,u∈H.We now use the same ideas to prove the (PS) condition. Let {un} be a sequence in H so that J(un) is bounded and J′(un)→0. We will prove that {un} contains a convergent subsequence. By (17), we claim that {un} is bounded. Assume without loss of generality that {un} converges to u weakly in H. Observe that (18)J′un-J′u,un-u=aun-u2+un2un-u2+un2-u2u,un-u-∫R3Kxf^x,un-f^x,uun-udx.Hence, we have (19)aun-u2≤J′un-J′u,un-u-un2-u2u,un-u+∫R3Kxf^x,un-f^x,uun-udx≡I1+I2+I3.
It is clear thatI1→0 and I2→0 as n→∞. In the following, we will estimate I3, by using (f3), for any R>0, (20)∫R3Kxf^x,un-f^x,uun-udx≤C∫R3∖BR0Kxunγ+uγdx+C∫BR0Kxunγ-1+uγ-1un-udx≤CunL2R3∖BR0γ+uL2R3∖BR0γKL2/2-γR3∖BR0+CKL2/2-γBR0unL2BR0γ-1+uL2BR0γ-1un-uL2BR0≤CKL2/2-γR3∖BR0+Cun-uL2BR0,which implies (21)limn→+∞∫R3Kxf^x,un-f^x,uun-udx=0.Therefore, {un} converges strongly in H and the (PS) condition holds for J. By (f2) and (f3), for any L>0, there exists δ=δ(L)>0 such that if u∈C0∞(Br(x0)) and |u|∞<δ then K(x)F^(x,u(x))≥L|u(x)|2, and it follows from (14) that (22)Ju≤a2u2+14u4-LuL2R32.This implies, for any k∈N, if Xk is a k-dimensional subspace of C0∞(Br(x0)) and ρk is sufficiently small then supXk∩SρkJ(u)<0, where Sρ={u∈R3∣u=ρ}. Now we apply Theorem B to obtain infinitely many solutions {uk} for (13) such that (23)uk⟶0,k⟶∞.
Finally we show thatukL∞→0 as k→∞. Let u be a solution of (13) and α>0. Let M>0 and set uM(x)=max{-M,min{u(x),M}}. Multiplying both sides of (13) with |uM|αuM implies (24)4aα+22∫R3∇uMα/2+12dx≤C∫R3uMα+1dx.By using the iterating method in [13], we can get the following estimate: (25)uL∞R3≤C1uL6R3ν,where ν is a number in (0,1) and C1>0 is independent of u and α. By (23) and Sobolev Imbedding Theorem [14], we derive that ukL∞(R3)→0 as k→∞. Therefore, uk are the solutions of (1) as k is sufficiently large. The proof is completed.



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*Source: 610858-2015-09-15.xml*