Research On Multiple Solutions Of Nonlocal Problems With Critical Growth

In this paper,we consider two kinds of nonlocal problems with critical exponential growth.By the variational method and the Nehari method,etc,we obtain the existence of multiple solutions for two classes of nonlocal problems.Firstly,we study the following Kirchhoff type equation with critical growth where Ω(?)R3 is a bounded domain with a smooth boundary,Q(x)is continuous sign-changing function and P(x)is positive continuous function on Ω,v is the external normal unit vector,a,b>0,1<q<2,λ>0 is a real parameter.By using the variational method and the concentration compactness principle,the existence of multiple nontrivial solutions of the equation(0.3)are obtained.Secondly,we consider the solvability of the Schrodinger-Poisson system with critical and supercritical growths where Br(?)R3 is a ball,(?),Q(x)is a positive continuous function on Br,0<β<1,6<q<6+2β,λ,μ>0 are real parameters.By the variational method and the Nehari method,we obtain that the system(0.4)has the existence of multiple positive solutions.