Studies On The Sign-changing Solution To Kirchhoff-Schr?dinger-possion Type Systems

In the present paper,we deal with the existence of sign-changing solutions and property of two classes Kirchhoff-Schrodinger-Poisson system with logarithmic and critical nonlinearity.The first chapter introduces the historical background,research significance and present situation of the Kirchhoff-Schrodinger-Possion equations.The second chapter introduces the existence of sign-changing solutions and property of the following fractional Kirchhoff-Schrodinger-Poisson system with logarithmic and critical nonlinearity,where s ∈(3/4,1),t ∈(0,1),A,a,b>0,4<q<2s*,andΩ is a bounded domain in R3 with Lipschitz boundary.Combining constraint variational methods,topological degree theory and quantitative deformation arguments,we prove that the above problem has a least energy sign-changing solution ub.Moreover,we show that the energy of ub is strictly larger than two times the ground state energy.Finally,we regard b as a parameter and show the convergence property of ub as b→0.The third chapter introduces the existence of sign-changing solutions and property of the following Quasilinear Kirchhoff-Schrodinger-Poisson system with logarithmic and critical nonlinearity,where λ,b>0,a>1/4,4<q<6,V(x)is a smooth potential function and Ω is a bounded domain in R3 with Lipschitz boundary.There are some difficulties in applying variational method directly to the problem because of the quasilinear term ∫Ω u2|▽u|2dx.It appears that finding a suitable space where the corresponding functional has both smoothness and compactness qualities is unattainable.In this chapter,we prove that the problem has a minimal energy sign changing solution and only two nodal domains,and study the energy properties of the sign changing solution by constraint variational method and perturbation method.