Semilinear Elliptic Systems And Schr(?)dinger Equations

In this paper,we are concerned with the existence of solutions of semilinear ellipticsystems on bounded domain and the nonlinear Schro¨dinger equations on RN.In chapter two, we consider the semilinear elliptic systems of the formwhere ? ? RN is a smooth bounded domain, N≥3,λandμare nonnegative numbers.When the nonlinearities are superlinear, by using Linking theorem, blow up argumentand Liouville type theorem, we discuss whenλandμsatisfies 0≤λμ< 1 andλμ> 1separately, the existence of nonnegative and nontrivial solutions of the elliptic systems.In chapter three, we consider the semilinear elliptic systems of the formwhich is similar to the chapter one, where ? ? RN is a smooth bounded domain,N≥3,λandμare nonnegative numbers. But our nonlinearities are asymptoticallylinear. a.e. ? l,m∈(0,∞), such thatuniformly with respect to x∈?. By using Linking theorem, we prove systems has apositive solution if 0≤λμ< 1 andλ1 < mλ+μl+√2(1(m?λλμ?)μl)2+4ml, whereλ1 is the firsteigenvalue of (??,H01(?)). Especially, systems has a least positive energy solution.In chapter four, we study the Schro¨dinger equation:where f does not satisfy (AR) condition. By using symmetric Mountain pass theorem,we prove the existence of infinitely many solutions if f is odd.In chapter five, we study the Schro¨dinger equation with magnetic field: where Vλ(x) and Q(x) are both continuous functions and changing sign on RN, N≥3,1 <γ< 2? andγ= 2. By using Nehari manifold and fibrering maps, we discuss inthe case that 1 <γ< 2 or 2 <γ< 2?, the relationship of the situation ofλand thenumber of solutions.