| Schr(?)dinger equation reveals the basic law of matter movement in the atomic world,and it is one of the basic assumptions of quantum mechanics.Kirchhoff equation and Schr(?)dinger coupling system are evolved from Schr(?)dinger equation,which has a profound physical background.In this thesis,the variational method is mainly used to study the existence of nontrivial solutions and Nehari-type ground state solutions of three kinds of Elliptic Equations with critical exponential growth on the plane.The main contents are as follows:Firstly,for the Kirchhoff equation with critical exponential growth in bounded domain:We propose a direct method to estimate the level of energy functional mountain road.Based on this,we prove the existence of Nehari-type ground state solutions and nontrivial solutions of the above problems,and improve and generalize the previous results.Secondly,for the Schr(?)dinger equation with local nonlinear perturbations-Choquard equation:where α∈(0,2),Iα:R2→R is the Riesz potential and f∈C(R,R)is of critical exponential growth in the sense of Trudinger-Moser.The exponent α/2+1 is critical with respect to the Hardy-Littlewood-Sobolev inequality,by using Trudinger-Moser inequality and new analytical techniques,We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the nonlinear term f(u)subcritical case and critical case,respectively.Finally,we consider the following two coupled nonlinear Schr(?)dinger system in R2:where the coupling parameter satisfies 0<λ(x)≤λ0<1 and the reactions f1,f2 have critical exponential growth in the sense of Trudinger-Moser inequality.Using non-Nehari manifold method together with the Lions’s concentration compactness and the Trudinger-Moser inequality,we show that the above system has a Neharitype ground state solution and a nontrivial solution. |
