Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass

Yanghuan Hu,Haidong Liu,Mingjie Wang and Mengjia Xu

College of Data Science,Jiaxing University,Jiaxing 314001,China.

Abstract.Consider the Kirchhoff type equation

Key words:Kirchhoff type equation,zero mass,mountain pass approach.

1 Introduction

This paper is concerned with the Kirchhoff type equation

where a>0,b≥0,0<µ

The so called “zero mass” case(that is,roughly speaking,when f′(0)=0)is of particular interest in the current paper.In such a case,(1.1)is closely related to Yang-Mills equations.In the context of semilinear elliptic equation

Kirchhoff type problem has also attracted much attention in recent years.Many existence and multiplicity results have been established by classical variational methods(see,for example,[9,11,15,17,22]).However,“zero mass” case is relatively less studied.In[4],Azzollini considered the Kirchhoff type equation

and proved that(1.3)has a positive solution if g is a(possibly “zero mass”)Berestycki-Lions nonlinearity(see[4,Definition 0.2]or[8]).In[16],Li,Li and Shi studied(1.3)with M(t)=a+bt and g(u)replaced by K(x)g(u).They established two existence results for the problem if b≥0 is suitably small and g is supercritical at the origin and superlinear and subcritical at infinity.

In[2],Alves and Yang considered(1.1)with b=0 and N=3.They proved the existence of a nontrivial solution if f is a “zero mass” Berestycki-Lions type nonlinearity(in fact,they made an additional assumption on the nonlinearity:f(t)t−F(t)≥0 for all t∈R).We also would like to refer interested readers to[10]for(1.1)with b=0and f being the critical pure power nonlinearity.Motivated by above works,we consider(1.1)and assume that f∈C(R,R)satisfies

Our main result is as follows.

Theorem 1.1.Let a>0,b≥0 and 0<µ

Remark 1.2.Theorem 1.1 is a complement of the results in[4,16],where only local nonlinearity is taken into account.Moreover,we do not assume that b≥0 is suitably small and the nonlinearity is superlinear at infinity as in[16].Theorem 1.1 is also a generalization of[2,Theorem 1.1]even for the particular case b=0,because we do not assume that f satisfies f(t)t−F(t)≥0 for all t∈R.

Throughout this paper,we use ‖·‖ and|·|pto denote the usual norms in D1,2(RN)and Lp(RN)with p≥1,respectively.We also set D1,2r(RN)={u∈D1,2(RN)|u(x)=u(|x|)}andThe symbol on(1)means a quantity tending to 0 as n→∞.The letters C,Cjand Cεstand for positive constants which may take different values at different places.

2 Proof of Theorem 1.1

which implies that‖un‖→‖u‖ as n→∞.This fact together with weak convergence shows that un→ u inThen Φ′(u)=0 and Φ(u)=c,From which we see that u is a nontrivial spherically symmetric solution of problem(1.1).

Acknowledgments

We would like to thank the anonymous referees for their valuable comments.This work is supported by ZJNSF(LY21A010020),NSFC(11701220,11926334,11926335)and SRT project of Jiaxing University(8517193282).