The Existence Of Solutions For The Planar Schr(?)dinger-Poisson Equations

This dissertation is devoted to studying symmetric and normalized solutions of the Schrodinger-Poisson equations on the plane.We study the existence and regularity of these two types of solutions.For symmetric solutions,we mean the rotational periodic symmetry or mirror symmetry solutions.The so-called normalized solutions are the solutions with the prescribed mass for the Schr(?)dingerPoisson equations.In chapter 1,we briefly introduce our research motivations and state the main results.Chapter 2 concerns with a recall of some preparatory material for this dissertation.In Chapter 3,we study the following planar Schrodinger-Poisson equations where p≥2 is a constant,V(x)and f(x,t)are continuous,mirror symmetric or rotationally periodic functions.We study the existence of the group invariant solutions of the above equations with a new class of nonlocal termφ|u|p-2u and nonlinear term f(x,t).The nonlinear term satisfies a certain monotonicity condition and has critical exponential growth in the Trudinger-Moser sense.Precisely,we construct a Cerami sequence by adopting a version of mountain pass theorem,which in turn leads to the existence of a ground state solution.Our method has two new insights.First,we observe that the integral∫R2∫F2 ln(|x-y|)|u(x)|p|u(y)|p dxdy is always nonpositive if u belongs to a function space,which is different from the three dimensional case.Secondly,we build a new Moser type functions to ensure the boundedness of the Cerami sequence,which further guarantees its compactness.Furthermore,when the nonlinear term f(x,t)is an odd function for t∈R,we use the Krasnoselski genus theory to prove the existence of infinitely many solutions.Finally,by replacing the monotonicity condition with Ambrosetti-Rabinowitz condition,our approach works also for the subcritical growth case.In Chapter 4,we study the following planar normalized Schrodinger-Poisson equations where 2≤p<q,γ∈R\{0},a ∈ R and c>0 are constants.Here λ∈R appears as a Lagrange parameter and is part of the unknowns.We have first investigated the existence of a single classical nontrivial solution to the above equation by means of restricted sets,provided that the parameters γ,a,p,q,c satisfy certain types of conditions.On the other hand,we have also studied the existence of multiple nontrivial classical solutions for the above five parameters satisfying certain conditions by using minimax techniques.In the study of multiple solutions,we use the expression of functional differentials on restricted manifolds to prove the boundedness of the corresponding Langrange multipliers of the Palais-Smale sequence.Therefore,we can obtain the compactness of the Palais-Smale sequence under certain conditions.In the last chapter,we present some related problems that we will consider in the near future.