Research On The Existence And Multiplicity Of Solutions For A Class Of Nonlocal Problems

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(?)Ω|▽|2dx)=λm(/u)+f(x,u)=μh(x),x∈Ω(MB)where Ω(?)RN(N≥1)is a bounded domain with smooth boundary or Ω=RN,m(u),(?)(x,u),h(x)h(x)are continual functions,λ>0 and μ ≥ 0 are parameters.Firstly,we consider(MB)with m(u)= |u|p-2u and f(x,u)= μ=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 Ω is an open ball with the 0-Dirichlet’s boundary condition.Secondly,we research(MB)near resonance,here m(u)=-bu3 and μ = 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 Ω = R4,f(x,u)= 0 and λm(u)= u3.Applying the achieving functions of the best Sobolev embedding constant,multiple positive solutions are obtained when μ = Q.And there exist at least two weakly positive solutions for(MB)by the Mountain Pass Theorem,Ekeland’s variational principle and contradiction when μ>0 small enough.