The Existence Of Solutions For Kirchhoff-Type Equations With A Steep Potential Well And A Class Of Nonlinear Schr?dinger Equations With A Parameter

In this paper,by applying the variational methods,we investigate the existence of solutions for Kirchhoff-type equations with a steep potential well and a class of nonlinear Schrodinger equations with a parameter.Firstly,in Chapter 2,we consider the nonlinear Kirchhoff-type equation with a steep potential well as follows:（?）where a,b>0 are constants,λ is a positive parameter and the potential V is a steep potential well.Namely,it satisfies the following of conditions:（V₁）V∈C（R³,R）,V（x）≥0 in R³;（V₂）There exists V₀>0 such that the set {V<V₀}:={x∈R³ | V（x）<V₀} is nonempty and has finite measure;（V₃）（?）:=V^-1（0）has nonempty interior.We suppose that the nonlinearity f satisfies certain assumptions.By applying a signchanging Nehari manifold combined with the method of constructing a sign-changing （PS）_c sequence,we claim the existence of least energy sign-changing solutions when λ is large enough.Furthermore,the concentration of least energy sign-changing solutions are proved.Namely,as λ→∞,the least energy sign-changing solutions strongly converge to the least energy sign-changing solutions of the Kirchhoff equation as follows:（?）Secondly,in Chapter 3,based on the relevant results of the equation with a steep potential well,we consider the existence of positive ground state solutions for the nonlinear Schrodinger equations with a parameter as follows:（?） where N≥3,λ is a positive parameter and the nonlinearity f satisfies superlinear at infinity with u.When the potential V does not satisfy that V^-1（0）has nonempty interior,we claim the existence of positive ground state solutions provided that λis large enough by applying variational methods.Furthermore,we also discuss the decay rate of the positive ground state solutions as |x|→∞.