In this paper,nonlocal problems are studied by the variational methods and the analysis techniques.More precisely,we consider the existence and multiplicity of solutions for the following nonlocal problem-(a-b(?)?|?|2dx)=?m(/u)+f(x,u)=?h(x),x??(MB)where ?(?)RN(N?1)is a bounded domain with smooth boundary or ?=RN,m(u),(?)(x,u),h(x)h(x)are continual functions,?>0 and ? ? 0 are parameters.Firstly,we consider(MB)with m(u)= |u|p-2u and f(x,u)= ?=0.The unique positive solution is got for p G(2,4](N =1,2,3)and p ?(2,2N/N-2)(N?4)by the Mountain Pass Theorem and contradiction when ? is an open ball with the 0-Dirichlet's boundary condition.Secondly,we research(MB)near resonance,here m(u)=-bu3 and ? = 0.It admits a weakly solution by the Mountain Pass Theorem when f(x,u)satisfies some condition.Combining with Ekeland's variational principle and a Mountain Pass Theorem,we obtain that there exist at least three weakly solutions.Lastly,we study(MB)involving critical exponent,here ? = R4,f(x,u)= 0 and ?m(u)= u3.Applying the achieving functions of the best Sobolev embedding constant,multiple positive solutions are obtained when ? = Q.And there exist at least two weakly positive solutions for(MB)by the Mountain Pass Theorem,Ekeland's variational principle and contradiction when ?>0 small enough. |