R ~ N Type Kirchhoff Equation On The Existence And Multiplicity Of Solutions

In this paper, we firstly study the existence and multiplicity of weak solutions for a class of Kirchhoff type problems in RN. Secondly, we study the existence and multiplicity, of weak solutions for p(x)-Kirchhoff-type problems in RN.Firstly, we consider the following Kirchhoff-type problem where constants a>0, b>0, N=2or3. V(x) satisfies the following condition:(V1) V∈C(RN,R) satisfies infx∈RN V(x)≥a1>0for a constant a1. Moreover, for every M>0. mcas({x∈RN:V(x)≤M})<∞, where meas denotes the Lcbcsguc measure in RN.(V2) V∈C(RN, R) and there exists α<2such that limx�'∞v(x)|x|α-2=∞, where v(x)=inf|ξ|=1V(x)ξ·ξ.Then we can obtain the following theorems.Theorem1Suppose that condition (V1) holds. Assume that f(x. t) satisfies the following condition:(f1) f∈C(RN×R,R) and there exist a2>0,p∈(2,2*) such that|f(x,t)|≤a2(1+|t|p-1), for all (x.t)∈RN×R, where2*=6, if N=3,2*=+∞, if N=2; uniformly for x∈RN; uniformly for x∈RN, where F(x,t):=∫0tf(x, s)ds;(f4) There exist C>0and r∞>0such thattf(x,t)-4F{x,t)>-C|t|2for all x∈RN and t∈R with|t|≥r∞. Then problem (P1) has at least one positive solution.Theorem2Suppose that (V1),(f1)-(f4) hold and/satisfies the following condition (/5) f{x,-t)=-f(x,t), V(x,t)∈RN×R. Then problem (P1) has infinitely many large energy solutions.Theorem3Suppose that (V2),(f1),(f5) hold. Assume that uniformly for all x∈RN; Then problem (P1) has a sequence of nontrivial weak solutions {un} with un�'0as n�'∝.Secondly, we discuss the existence and multiplicity of solutions for the following p(x)-Kirchhoff-type problem in RN: where N>2. p is a function defined on RN, m: R�'R is continuous,f RN×R�'R satisfies Caratheodory conditions, i.e.. f(x,t) is continuous in x for almost every t and measurable in t for all x.The elementary assumptions on p and m are as follows:(p) p is Lipschitz continuous, p∈L∞(RN),1<p#<p#<N;(m0) there exists m0>0such that m(t)≥m0for all t>0;(m1) there is0<μ <1such that M(t)≥μtm{t) for all t≥0, where M(t)=∫0tm(s)ds.For the functions b(x) and q(x), we suppose that they satisfy the following conditions:(b0) b(x)≥0, b≠0and b∈C(RN,R);(b1) b(x)≥0, and b∈Lτ(x)(RN), where r∈L∞(RN), r≥1; (q) q∈L∝(RN) and1≤q#<q#<(p*)#where p*(x):=Np(Np(x))/N-p(x)Now we state our main results.Theorem4Suppose that m satisfies(m0).(m1) and f satisfies the following conditions:(g1)|f(x,t)|<b(x)|t|q(x)-1,V(x,t)∈RN×R, where b satisfies (b1) and r in (b1) satisfies p(x)<s(x):=r(x)q(x)/r(x)-1≤p*, for a.e. x∈RN, q satisfies (q)(g2) there exists δ>0such that f(x, t)≥b0(x)tq0(x)-1for x∈RN and0<t<δ, where b0satisfies condition (b0) and q0satisfies (q) with q0#<p#. Let q#<p#in (g1), then problem (P2) has a nontrivial solution.Theorem5Suppose that m satisfies (m0),(m1), f satisfies (gl) with q#>p#, and(g3) there is a positive constant α>p#/μ such that0<αF(x, t)<tf(x,t),(?)x∈RN, t≠0. Then problem (P2) has a nontrivial solution.Theorem6Suppose that m satisfies (m0),(m1), f satisfies (gl) with q#>p#,(g3) and (gA) f(x,-t)=-f(x, t), for a.e. x∈RN and t e R. Then problem (P2) has a sequence of high energy solutions.Theorem7Suppose that m satisfies (m0),(m1), f satisfies (g1).(g2) and (g4). Then problem (P2) has a sequence of nontrivial negative energy solutions converging to0.