Research On The Existence Of Solutions For Some Schr(o|¨)odinger Equations With Electromagnetic Fields

In this paper, we mainly study the existence of solutions for some nonlinear Schrodinger equations with electromagnetic fields.In Chapter One, we summarize the background of the related problems and state the main results of the present thesis. We also give some preliminary results and notations used in the whole thesis.In Chapter Two, we consider the nonlinear Schrodinger equation with magnetic fields where A(y)=(A1(y), A2(y),…, AN(y)) is a bounded real-valued vector function on RN and Q(y) is a bounded positive continuous function on RN,2<q<2N/(N-2) if N>3and2<q<+∞if N=1,2. We prove that the equation (E1) has infinitely many non-radial complex-valued solutions under conditions (H1) and (H2). Our main result extends the result of Noussair and Yan given in [60] for the equation (E1) with A(y)=0.In Chapter Three, we are concerned with the existence of multi-bump solutions for a nonlinear Schrodinger equations with electromagnetic fields where2<p<2N/N-2if N>3and2<p<+∞if N=1,2and∈>0is a parameter. a(x) is a positive continuous function on RN, and A(x)=(A1(x),A2(x),..., AN(x)) is such that Aj(x)(j=1,2,..., N) is a real function on R.N. We prove under some suitable conditions that for any positive integer m, there exists∈(m)>0such that, for0<∈<∈(m), the problem (E2) has an m-bump complex-valued solution. As a result, when∈→0, the equation has more and more multi-bump complex-valued solutions. Our main result extends the result of Li, Peng and Wang given in [55] for the equation (E2) with Ao≠0and some weaker symmetric conditions.In Chapter Four, we study the nonlinear Schrodinger equation with non-symmetric electromagnetic fields where A∈(x)=(A∈,1(x), A∈,2(x),…,A∈N(x)) is a magnetic field satisfying that A∈,j(x)(j=1,..., N) is a real C1bounded function on RN and V∈(x) is an elec-tric potential. Both of them satisfy some decay conditions and f(u) is a superlinear nonlinearity satisfying some nondegeneracy condition. Applying two times finite reduction methods and localized energy method, we prove that there exists some∈0>0such that for0<∈<∈0, the above problem has infinitely many complex-valued solutions. Our main result extends part of results of Ao and Wei given in [7] for the equation (E3) with A(x)≡0.