The Existence Of Several Types Of Nonlinear Differential Equations

With the development of modern physics and applied mathematics, it de-mands the mathematical ability of analyzing and controling the objective phe-nomena toward to the overall high and precision level, which made the results of the nonlinear analysis was accumulated, and gradually formed an important branch subject of the present analysis mathematics-Nonlinear functional analy-sis. Nonlinear functional analysis is a research discipline in analysis mathematics both to have the profound theory and to have the widespread application. It takes the nonlinear problems appearing in mathematics and the natural sciences as background to establish some general theories and methods to handle nonlinear problem. Because it can commendably explain all kinds of natural phenomenal, coupled with the widely application in the realistic production and life, it has received highly attention of the domestic and foreign mathematics and natural science field in recent years.The differential equation problem of nonlinear stems from the applied math-ematics, the physics, the cybernetics and each kind of application discipline. It is an important kind of question in the differential equations, it is one of most active domains of functional analysis studies at present, is also the hot spot which has been discussed in recent years. So it attachs more and more attention.In this paper, we use Variational methods, minimax theory, as well as critical methods, to study the existence of positive solutions for several kinds of nonlinear differential equation.The thesis is divided into three chapters according to contents.In chapter 1, we introduce some fundamental knowledge and source of theory.In chapter 2, We concerns the existence of solutions for the following schroding er equation with a quadratic term-△u+V(x)u=f(x,u) (2.1.1) where V(x)∈C(RN, R) and is positive for all x∈RN, f(x, u)∈C(RN x R, R) is a positive function. Under another valid assumption for V and f, we obtain a new criterion for guaranteeing that (1,1) has one solution at least by use of a standard minimizing argument in critical point theory. In chapter 3, we consider the the existence of homoclinic solutions for the following second-order non-autonomous Hamiltonian system: z[t)-L(t)z+VH(t,z)=0(3.1.1) where L(t)∈C(R, RN) and is a symmetric and positive definite matrix for all t∈R, H(t, z)∈C1(Rx RN, R) is a positive continuous function,▽H(t, z) satisfies the subcritical condition:|▽H(t,z)|≤(|t|p+1) for some 2 3,2*=+∞if N=1,2. Adopting some another reasonable assumption for H and L, we obtain a new criterion for guaranteeing that (3.1.1) has one solution at least by use of a standard minimax methods in critical point theory.