Qualitative Properties Of The Solutions Of Elliptic Equations With Magnetic Fields

This dissertation is devoted to the study of the following two types of elliptic equations with magnetic fields:the magnetic Schrodinger equations and the critical magnetic Choquard equations,and we prove the existence,multiplicity and concentration of the solution by variational methods.In chapter 1,we introduce the problem backgrounds and recent developments of the elliptic equations with magnetic fields.Then we recall some preliminary knowledge and state the main results obtained in the thesis.In Chapter 2,we investigate the existence of the following nonlinear Schrodinger equation with magnetic fields and the Neumann boundary condition:#12 where Ω is a domain in RN with C1 boundary,λ is a real parameter,v is the outward normal vector field.N≥3,2≤p≤2*,where 2*=2N/(N-2)is the critical Sobolev exponent.We prove the existence of solutions for the above problem by variational methods.In Chapter 3,we investigate the existence,multiplicity and concentration of solutions for the following magnetic Choquard equation#12 where ε>0 is a small parameter,N≥3,0<μ<N,2μ*=(2N-μ)/(N-2)is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality,V(x):RN→R is an electric potential,the magnetic potential A(x):RN→RN is a continuous vector function,f(t):R+→R is a subcritical nonlinear term,F is the primitive function of f.Under suitable assumptions for the electric potential V(x)and nonlinearity f(t),we prove the multiplicity and concentration properties of nontrivial solutions of the above problem by the variational methods,the penalization techniques and the Ljusternik-Schnirelmann theory.In Chapter 4,we investigate the existence,multiplicity and concentration of solutions for the following fractional magnetic Choquard equation#12 where ε>0 is a small parameter,N≥3,0<μ<N,s∈(0,1),2(μs)*=(2N-μ)/(N-2s),is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality and fractional Laplace operator.The magnetic fractional Laplacian-Δ)(A/ε)s u is defined,up to a normalization constant,for all u∈Cc∞(RN,C)by setting#12 Under the similar assumptions of the electric potential V(x),the magnetic potential A(x),the nonlinear term f(t)in Chapter 3,we apply the same the method to discusse the multiplicity and concentration properties of nontrivial solutions of the above problem.