The Existence And Behavior Of Solutions To Some Elliptic Systems

In this thesis, we mainly deal with the existence and behavior of the solutions to some elliptic systems by using variational method. We divide it into five Chapter.In Chapter One, we summarize the background of the related problems and state the main results of the present thesis.In Chapter Two, we establish a relationship between the following elliptic sys-tem and its corresponding single elliptic problem, where λ∈R, βi>0,μi<0,qi> 0,1<pi+qi=2p+1 for i=1,2, Ω(?) RN(N≥1) can be a bounded or unbounded domain. By using this fact, we can obtain many results on the existence, non-existence and uniqueness of classical vector solutions to this system via the related single, elliptic problem.The Chapter Three, we study the foljowing fractional nonlinear Schrodinger system where 0< s<1,μ1>0, μ2> 0,1< p< 2s*/2,2s*=+∞ for N≤ 2s and 2s*=2N/(N-2s) for N>2s, and β∈R is a coupling constant. We investigate the existence and non-degeneracy of proportional positive vector solutions for the above system in some ranges of μ1,μ2, p,β.We also prove that the least energy vector solutions must be proportional and unique under some additional assumptions.In Chapter Four, we consider the following coupled fractional nonlinear Schrodinger system in RN where N≥2,0< s< 1,1<p<N/N-2s,μ1>0,μ2>0 and β∈R R is a coupling constant. We prove that this system has infinitely many non-radial positive solutions under some additional conditions on P(x), Q(x), p and β. More precisely, we will show that for the attractive case, it has infinitely many non-radial positive synchronized vector solutions, and for the repulsive case, infinitely many non-radial positive segregated vector solutions can be found, where we assume that P(x) and Q(x) satisfy some algebraic decay at infinity.In Chapter Five, we are interested in the ground states of the following M-coupled semilinear system We extend the characterization results in [21] and partial results in [22] to more general cases. What’s more important, we give a new characterization of the ground states, which provides a more convenient way to find or check a ground state, and we also obtain an important result on the number of the ground states, which may be the first result studying not only the positive ground state but also semi-trivial ground state and implies that the positive ground state is unique for some special cases.