A New Abstract Critical Point Theorem On Locally Convex Topological Linear Spaces And Its Applications

As we all know,nonlinear analysis is an extremely important branch of mathematics,which takes nonlinear differential equations as the main research object,and has developed a set of profound and systematic mathematical theories,the most important of which is critical point theory,and many important problems are related to critical point theory,such as geometric analysis,Hamilton system and symplectic geometry.In this dissertation,we prove a new abstract critical point theorem on locally convex topological linear spaces,and discuss the existence of nonlinear Dirac equations and the existence of minimal energy solutions of nonlinear Hamilton-type elliptic equations.The main contents of this article are arranged as follows:The first chapter introduces the relevant research background.In Chapter 2,we prove a new abstract critical point theorem by using topological degree theory and deformation lemma on locally convex topological linear spaces,which is very important for dealing with strongly indefinite problems.In Chapter 3,by establishing the appropriate variational framework,we study the nonlinear Dirac equation on Nehari-Pankov manifold.where x=(x1,x2,x3)∈R3,u(x)∈C4,and (?)k=(?)/(?)xk,f:R3 × R+→R;and R+:=[0,∞),p>0 is a positive constant,α1,α2,α3 and β are 4 × 4 Pauli-Dirac matrices(in 2×2 blocks),we obtain the existence of the ground state solution.In Chpter 4,by establishing the appropriate variational framework,we study nonlinear elliptic systems in Hamiltonian form on Nehari-Pankov manifold.where N≥3,V∈C(RN,R)are 1-periodic,0(?)σ(-Δ+V)and f,g∈C(RN ×R,R),we obtain the existence of the ground state solution.