Research On The Existence And Multiplicity Of Solutions For A Class Of Elliptic Equation With Neumann Boundary Value Problems

In this paper,the existence and multiplicity of solutions for a class of elliptic equation with Neumann boundary value problems are studied by variational methods,truncation techniques,Nehari's methods,etc.Firstly,we consider the existence of solutions for the following type of Kirchhoff-type equations on with Neumann boundary value problems (?) where ?(?)RN is a bounded domain with smooth boundary(N?1),v is the external normal unit vector,a? 0,b?0,?? 0,1 ?q?2 are real parameters.The existence and multiplicity of solutions to the equation(0.4)are obtained by variational methods and truncation techniques.Secondly,we generalize the equation(0.4)to the following Neumann problem(?)where c(x)is a sign-changing potential,a,b?0 and a+b? 0,1?q?2.The existence and multiplicity of solutions to equation(0.5)are also obtained through vari-ational methods and truncation techniques.Finally,we study the following Neumann boundary value problems with singular terms and critical growth(?)where ?(?)RN is a bounded smooth domain(N?3),0<?<1,?>0 are parameter,0<a(x)? L?((?)?)is a non-negative function,and v is the external normal unit vector.The existence of two positive solutions to the equation(0.6)is obtained by variational methods and Nehari's methods.