JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2015/610858 610858 Research Article Multiple Solutions for Kirchhoff Equations under the Partially Sublinear Case Feng Wenjun 1 Feng Xiaojing 2 Infante Gennaro 1 College of Applied Mathematics Shanxi University of Finance and Economics Taiyuan 030006 China sxufe.edu.cn 2 School of Mathematical Sciences Shanxi University Taiyuan 030006 China sxu.edu.cn 2015 1592015 2015 16 06 2015 12 08 2015 26 08 2015 1592015 2015 Copyright © 2015 Wenjun Feng and Xiaojing Feng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction and Main Results

In this paper we study the existence and multiplicity of solutions for the following Kirchhoff type equations:(1)a+RNu2+bRNu2-Δu+bu=Kxfx,u,in  R3,where a, b are positive constants.

When Ω is a smooth bounded domain in R3, the problem(2)-a+bΩu2dxΔu=fx,u,in  Ωu=0,on  Ω,has been studied in several papers. Perera and Zhang  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  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  and Ma and Rivera  studied the existence of positive solutions of (2) and He and Zou  obtained the existence of infinitely many positive solutions of (2), respectively; Mao and Luan  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  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  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 ΩR3. This makes the study of problem (1) more difficult and interesting. Wu  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 , 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  obtained multiplicity of high energy solutions for superlinear Kirchhoff equations in R3. Li et al. in  proved the existence of a positive solution to a Kirchhoff type problem on RN by using variational methods and cutoff functional technique.

In , Jin and Wu consider the following problem: (3)-a+bRNu2dxΔu+u=fx,u,in  RN,uH1RN,where constants a>0, b>0, N=2 or 3, and fC(RN×R,R). By using the Fountain Theorem, they obtained the following theorem.

Theorem A (see [<xref ref-type="bibr" rid="B9">12</xref>]).

Assume that the following conditions hold.

If the following assumptions are satisfied,

f(x,u)=o(|u|) as |u|0 uniformly for any xRN,

there are constants 1<p<2-1 and c>0 such that (4)fx,uc1+up,x,uRN×R,

where (5)2-1=N+2N-2,N3;+,N=1,2,

there exists μ>4 such that(6)μFx,u=μ0ufx,sdsufx,u,x,uRN×R,

(7)infxRN,u=1Fx,u>0,

f(gx,u)=f(x,u) for each gO(N) and for each (x,u)RN×R, where O(N) is the group of orthogonal transformations on RN,

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  obtained an extension of Clark’s theorem as follows.

Theorem B (see [<xref ref-type="bibr" rid="B10">13</xref>]).

Let X be a Banach space, ΦC1(X,R). Assume Φ is even and satisfies the (PS) condition, bounded from below, and Φ(0)=0. If, for any kN, there exists a k-dimensional subspace Xk of X and ρk>0 such that supXkSρkΦ<0, where Sρ={uXu=ρ}, then at least one of the following conclusions holds.

There exists a sequence of critical points {uk} satisfying Φ(uk)<0 for all k and uk0 as k.

There exists r>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 that f(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 . Now, we state our main result.

Theorem 1.

Assume that f satisfies (H6) and the following conditions:

There exist δ>0, 1γ<2, C>0 such that fC(R3×[-δ,δ],R) and |f(x,z)|C|z|γ-1.

Consider limz0F(x,z)/|z|2=+ uniformly in some ball Br(x0)R3, where F(x,z)=0zf(x,s)ds.

K:R3R+ is a positive continuous function such that KL2/(2-γ)(R3)L(R3).

Then (1) possesses infinitely many solutions {uk} such that ukL0 as k.

Remark 2.

Throughout the paper we denote by C>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 Section 2, 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.

Let H1=H1(R3) be the completion of C0(R3) with respect to the inner product and norm (8)u,v=R3uv+buvdx,u=u,u1/2.Moreover, we denote the completion of C0(R3) with respect to the norm (9)uD12=R3u2dxby D1=D1(R3). To avoid lack of compactness, we need to consider the set of radial functions as follows: (10)H=Hr1R3=uH1R3ux=ux.Here we note that the continuous embedding HLq(R3) is compact for any q(2,6).

Define a functional by (11)J1u=a2u2+14u4-R3KxFx,u,uH.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,vH.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 Theorem <xref ref-type="statement" rid="thm1.1">1</xref>.

Choose f^C(RN×R,R) such that f^ is odd in uR, f^(x,u)=f(x,u) for xRN and |u|<δ/2, and f^(x,u)=0 for xRN and |u|>δ. In order to obtain solutions of (1) we consider (13)a+RNu2+bRNu2-Δu+bu=Kxf^x,u,in  RN.

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 JC1(H,R), and (15)Juv=au,v+u2u,v-R3Kxf^x,uvdx,vH.

Note that J is even, and J(0)=0. For uH, (16)R3KxF^x,udxCR3KxuγdxCKL2/2-γR3uL2R3γCuγ.Hence, it follows from (14) that (17)Ju12u2-Cuγ,uH.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)Jun-Ju,un-u=aun-u2+un2un-u2+un2-u2u,un-u-R3Kxf^x,un-f^x,uun-udx.Hence, we have (19)aun-u2Jun-Ju,un-u-un2-u2u,un-u+R3Kxf^x,un-f^x,uun-udxI1+I2+I3.

It is clear that I10 and I20 as n. In the following, we will estimate I3, by using (f3), for any R>0, (20)R3Kxf^x,un-f^x,uun-udxCR3BR0Kxunγ+uγdx+CBR0Kxunγ-1+uγ-1un-udxCunL2R3BR0γ+uL2R3BR0γKL2/2-γR3BR0+CKL2/2-γBR0unL2BR0γ-1+uL2BR0γ-1un-uL2BR0CKL2/2-γR3BR0+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 uC0(Br(x0)) and |u|<δ then K(x)F^(x,u(x))L|u(x)|2, and it follows from (14) that (22)Jua2u2+14u4-LuL2R32.This implies, for any kN, if Xk is a k-dimensional subspace of C0(Br(x0)) and ρk is sufficiently small then supXkSρkJ(u)<0, where Sρ={uR3u=ρ}. Now we apply Theorem B to obtain infinitely many solutions {uk} for (13) such that (23)uk0,k.

Finally we show that ukL0 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+22R3uMα/2+12dxCR3uMα+1dx.By using the iterating method in , we can get the following estimate: (25)uLR3C1uL6R3ν,where ν is a number in (0,1) and C1>0 is independent of u and α. By (23) and Sobolev Imbedding Theorem , we derive that ukL(R3)0 as k. Therefore, uk are the solutions of (1) as k is sufficiently large. The proof is completed.

Conflict of Interests

The authors declare no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere gratitude to one anonymous referee for his/her constructive comments for improving the quality of this paper. This work was supported by the National Natural Science Foundation of China (Grant nos. 11071149, 11271299, and 11301313), Natural Science Foundation of Shanxi Province (2012011004-2, 2013021001-4, and 2014021009-1), and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (no. 2015101).

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