Discussion On A Class Of Strongly Indefinite Nonlinear Equations

In quantum electrodynamics and relativistic quantum mechanics,Dirac equation is the basic equation,which reveals the interaction between photons and electrons.Photons correspond to electromagnetic fields,and their motion behavior is constrained by Maxwell equations,while electrons correspond to spinor fields,and their motion behavior is constrained by Dirac equation in electromagnetic fields.The linear theory of these two kinds of equations has been developed relatively well,and constitutes the main part of electrodynamics and relativistic quantum mechanics.The nonlinear theory is still developing.Especially in the last two decades,it has made great progress.In this doctoral dissertation,we mainly use variational methods and critical point theory to study the multiplicity of solutions and some special properties of solutions of these two kinds of equations.In Chapter 1,we introduce the background of the problem and the current research situation,and give the main results of this paper.In Chapter 2,we introduce the notations used in this paper and some preparatory knowledge.In Chapter 3,we will study the nonlinear Maxwell equations in R3,where ε(x),μ(x)are the permittivity and magnetic permeability of the material,respectively,and σ(x)is the conductivity of the media.1<q<p/p-1<2<p≤6,Q(x),P(x)∈ C(R3,R+),ξ1,ξ2 are two real numbers.Using variational methods,we prove the existence of infinitely many large energy and small energy cylindrical symmetry solutions which have free divergence.In Chapter 4,we look for solutions of the following critical nonlinear Dirac equation where ε>0 is a small parameter,a>0 is a constant,p ∈(5/2,3),α=(α1,α2,α3)is triplets of matrices,α1,α2,α3 and β are 4 × 4 Pauli-Dirac matrices.The potential V(x)may attain ±a at somewhere or at infinity,K,Q∈C1(R3,R+)are two functions.When ε>0 small,we will prove the existence and concentration of the solutions by using variational methods under some mild assumptions on the potentials V,K and Q.In Chapter 5,we study the following nonlinear boundary value problem where M is a compact Riemannian spin manifold of dimension m≥ 2 and the boundary?M has non-negative mean curvature R,and D is the Dirac-Atiyah-Singer operator.BCHI is the chirality boundary operator.Under some mild assumptions on a,f and g,we obtain the ground state solution of equation with λ=0 and infinitely many large norm solutions by using variational methods.