# Existence of Solutions for Sublinear Kirchhoff Problems with Sublinear Growth

Authors: Wei Yang; Zhan Wang; Zhuo Yao
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, we consider the following sublinear Kirchhoff problems-a+b∫RN∇u2dxΔu+V(x)u=f(x,u), in RN, where a>0 and b≥0 with N≥3. A new sublinear growth condition is given. When f(x,u) is not odd in u and not integrable in x, we obtain the existence of solutions for the above problem.


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## Body

## 1. Introduction and Main Results

In this paper, we consider the following nonlinear Kirchhoff type problems:(1)-a+b∫RN∇u2dxΔu+Vxu=fx,u,inRN,where a,b≥0,f(x,u)∈C(RN×R,R). The Kirchhoff type problems with general potentials on a bounded domain are introduced by(2)-a+b∫Ω∇u2dxΔu=fx,u,inΩu=0,on∂Ωwhich is related to the stationary analogue of the Kirchhoff equation (3)utt-a+b∫RN∇u2dxΔu=gx,u.With the development of the variational methods in the last decades, many mathematicians tried to use these methods to search for the existence and multiplicity of solutions for differential equations (see [1–30]). After the work of Lions [14], the Kirchhoff problems have been studied by many mathematicians using the functional analysis methods.Whenf(x,t) is superlinear but subcritical with respect to t, many mathematicians obtained the existence and multiplicity of solutions for problem (1) (see [4, 9, 15–18, 20, 22–24, 30]). But there are only a few papers concerning the sublinear Kirchhoff type problems. In 2013, Ye and Tang [30] obtained the existence of infinitely many solutions for (1) by symmetrical mountain pass theorem with f(x,t)=νξ(x)tν-2t, where ν∈(1,2) and ξ∈L2/2-ν(RN,R+). In [2], Bahrouni obtained the infinitely many solutions for sublinear Kirchhoff equations with sign-changing potentials. In 2014, Duan and Huang [6] obtained the existence and multiplicity of solutions for problem (1) with more general sublinear nonlinearities. In a recent paper [19], Li and Zhong showed the existence of infinitely many solutions for problem (1) with local sublinear nonlinearities. However, in the above papers, the nonlinear term f(x,t) is assumed to belong to some Lp(RN) with respect to x for some p>1. Under a coercive condition, Wang and Han considered a class of sublinear nonlinearities for problem (1) and showed the existence of infinitely many solutions when f(x,t) is odd in t which is the following theorem.Theorem 1 (see [25]).
Suppose that the following conditions are satisfied:
( V )   inf x ∈ R N ⁡ V ( x ) > 0 and there are constants ϱ>0 and α>N such that, for any b>0, limy→+∞⁡measx∈Bϱ(y):V(x)/xα≤b=0, where meas(·) denotes the Lebesgue measure;
( s 1 ) there exists a constant δ>0 such that f∈C(RN×[-δ,δ],R) and f(x,-t)=-f(x,t) for all t≤δ and x∈RN;
( s 2 ) There is a ball Br0(x0) such that (4)lim⁡inft→0⁡infx∈Br0x0⁡Fx,tt2>-∞and (5)lim⁡supt→0⁡infx∈Br0x0⁡Fx,tt2=+∞,where F(x,t)=∫0tf(x,s)ds;
( s 3 ) There is a constant τ>0 and a function g∈C(-τ,τ,R+) such that fx,t≤g(t) for all t≤τ and x∈RN.
Then Equation (1) possesses a sequence of weak solutions un in X∩L∞(RN) such that un→0 in L∞(RN) as n→∞.With coercive condition(V), the authors obtained a new compact embedding theorem, stated as follows.Lemma 2 (see [25]).
Suppose thatV satisfies conditions (V). Then E is compact embedded into Lp(RN) for any p∈[1,2∗).Remark 3.
From Lemma2, for any p∈[1,2∗), there exists a constant ηp>0 such that(6)up≤ηpuforallu∈E.A natural question is whether there exists solution for problem (1) if there is no symmetrical condition on f(x,t) and f(x,t) is not integrable in x. In this paper, we try to give an existence theorem on this problem. Motivated by the above papers, we consider problem (1) with some new sublinear nonlinearities and, before we state our result, we assume that Γ is a continuous function space such that, for any δ(s)∈Γ, there exists a constant l0>0 such that(i)δ(s)>0 for all s>0; (ii) ∫l0l1/sδ(s)ds→+∞ as l→+∞.In order to obtain the critical points of the corresponding functional, we consider the following function space in this paper:(7)E≔u∈H1RN:∫RN∇u2+Vxu2dx<∞with the inner product (8)u,vE=∫RN∇u·∇v+Vxuvdx.For any u∈E, it follows from (6) that (9)c~=infu∈E,u2=1⁡u>0.In order to show that the infimum can be achieved, consider a minimizing sequence {um}⊂E such that (10)um2=1,um→c~asm→∞.It is easy to see that there exists u~∈E such that um⇀u~. From Brézis-Lieb Lemma and Lemma 2, we obtain u~2=1, u~=c~ and(11)u2≤1c~ufor  allu∈E.Then we state our main result.Theorem 4.
Suppose thatV and f satisfy (V) and the following conditions:
( f 1 ) letting F(x,t)=∫0tf(x,v)dv, there exist δ(s)∈Γ with 0<1/δs<2 and l∞>0 such that (12)0≤tfx,t≤2-1δtFx,tforallx∈RNandt≥l∞,
( f 2 ) there exists d~>c~2/2(a+1), d1>0, and μ∈[1,2∗) such that (13)Fx,t≥d~t2-d1tμ∀x,t∈RN×R;
( f 3 )   lim t → 0 ⁡ f ( x , t ) = 0 uniformly in x;
( f 4 ) there is S>0 such that (14)supx∈RN,t=l∞⁡Fx,t≤S.
Then there exists at least one nontrivial solution for problem (1).Remark 5.
Let(15)Fx,t=c~2a+1t2fort≤1Ptfor1≤t≤2t2+2ln⁡t2+2fort≥2,where P(t)∈C1(R) such that F(x,t) is a C1 class function. It is easy to see that F(x,t) satisfies (f1)–(f4). However, since limt→0⁡F(x,t)/t2=c~2(a+1), we see that F(x,t) does not satisfy (s2)Remark 6.
Condition(f1) is introduced by Wang and Xiao in [26] to prove the existence of periodic solutions for a class of subquadratic Hamiltonian systems. As we know, this is the first time to use such condition on the existence of solutions for sublinear Kirchhoff equations.


## 2. Preliminaries

Lemma 7.
Suppose that(f1) holds; then there exists d2>0 such that(16)Fx,t≤d2Qtt2forallx∈RNandt≥l∞,whereQt=exp⁡-∫l∞t1sδsds.Proof.
For anyx∈RN and s≥l∞/t, let w(s)=F(x,st). It follows from condition (f1) that(17)w′s≤1s2-1δstwsfor all s≥l∞/t. Set (18)Ts=w′s-1s2-1δstws.Then T(s)=0 yields that (19)ws=∫l∞/tswrr2δrtdr+wl∞/tt2l∞2s2Qst.For any s≥l∞/t, (17) shows that T(s)≤0, which implies that (20)ws≤wl∞/tt2l∞2s2Qst.Then we can obtain(21)Fx,t=w1≤wl∞/tt2l∞2ηt=Fx,t/tl∞l∞2Qtt2fort≥l∞.From (21) and (f4), we can obtain our conclusion.Remark 8.
By the definition ofQ, it follows from the properties of η that Qt→0 as t→+∞.Lemma 9.
Suppose thatV satisfies conditions (f1), (f3), and (f4); then, for any ε>0, there exists dε>0 such that(22)fx,t≤ε+dεtforallx∈RNandt∈Rand(23)Fx,t≤12εt+dεt2forallx∈RNandt∈R.Proof.
From Lemma7, there exists d3>0 such that (24)Fx,t≤d3t2fort≥l∞.For t≥l∞, it follows from (f1) that(25)fx,t≤2Fx,tt≤2d3t.We can easily obtain (22) from (f3) and (25). (23) can be deduced from (22) directly.


## 3. Proof of Theorem1

By standard arguments, we know that the functionalI:E→R, defined by(26)Iu=a2∫RN∇u2dx+b4∫RN∇u2dx2+12∫RNVxu2dx-∫RNFx,udx,is well defined and the critical points of I are the solutions for problem (1).Lemma 10.
Suppose that(V), (f1)–(f4) hold; then I satisfies the (PS) condition.Proof.
From Remark8 and (V1), there exists r∞≥l∞ such that(27)Qt≤infx∈RN⁡Vx8d2foranyt≥r∞.First, we show that un is bounded in E. It follows from the definition of I, (16) and (27), that(28)Iun=a2∫RN∇un2dx+b4∫RN∇un2dx2+12∫RNVxun2dx-∫RNFx,undx≥a2∫RN∇un2dx+12∫RNVxun2dx-∫un≥r∞Fx,undx-∫un≤r∞Fx,undx≥a2∫RN∇un2dx+12∫RNVxun2dx-d2∫un≥r∞Qunun2dx-12∫un≤r∞εun+dεun2dx≥a2∫RN∇un2dx+38∫RNVxun2dx-12ε+dεr∞∫un≤r∞undx≥min⁡a2,38un2-η12ε+dεr∞un,which implies that I is bounded from below on E. Since we have Lemma 2, it follows from a standard argument, we obtain that un-u0→0 as n→∞. Hence I satisfies the (PS) condition.Proof of Theorem1.
By above discussions, we can see thatI is of C1 class and satisfies the (PS) condition. Similar to (28), we obtain that I is bounded from below. Then c=infE⁡I(u) is a critical value of I and there exists u¯ such that I(u¯)=c. Finally, we show that u¯≢0. With θ>0 being small enough, it follows from (f2) and (11) that(29)Iθu~=aθ22∫RN∇u~2dx+bθ44∫RN∇u~2dx2+θ22∫RNVxu~2dx-∫RNFx,θu~dx≤aθ22∫RN∇u~2dx+bθ44∫RN∇u~2dx2+θ22∫RNVxu~2dx-d~θ2∫RNu~2dx+d1θμ∫RNu~μdx≤12c~2a+1-d~θ2+bθ44∫RN∇u~2dx2+d1θμημμc~μ<0.Then we can deduce infE⁡I(u)<0, which implies that u¯≢0.



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*Source: 5364134-2019-04-21.xml*