Existence And Multiplicity Of Solutions For Some Differential Equations

In this paper, we mainly consider two problems. Firstly, we concern with the existence of multiple solutions of following Kirchhoff-Schr¨odinger-Poisson equations:where constants a > 0, b ≥ 0, and λ ≥ 0. When f has sublinear growth in u and also only defined for u small. we obtain infinitely many solutions under certain assumption that V do not has a positive lower bound. The mainy technique we use in this paper is the symmetric mountain pass theorem estabished by Kajikiva.Next, we concern with the existence and multiplicity of solutions of following Schr¨odingerPoisson systems with radial potentials. It obtained by variational methods.Secondly, we are interested in the Schr¨odinger-Poisson system where λ > 0,V and Q are radial functions, which can be vanishing or coercive at ∞. With assumptions on f just in a neighbourhood of the origin, existence and multiplicity of nontrivial radial solutions are obtained via variational methods. In particular, if f is suplinear near the origin, we obtain infinitely many solutions for any λ > 0.