Some Researches On Nonlinear Elliptic Equations And Systems

This thesis consists five parts.In the first part, we consider four classes of elliptic boundary value problem: Steklov boundaryvalue problem, No-?ux boundary value problem, Neumann boundary value problem and Robinboundary value problem. We assume that the boundary conditions are nonlinear and with indefiniteweight. Using the critical point theorem over cones and the perturbation method for eigenvalueproblem, we obtain the existence of nontrivial solutions.In the second part, we consider a p-Laplace equation on the whole space. Under a very generalnonlinearity and the condition that the potential has a lower bound (may be sign-changing), weobtain the existence of nontrivial solutions for this problem.In the third part, we consider a class of coupled Schr(o|¨)dinger systems. We get the existenceof nontrivial solutions assuming that the potential is sign-changing. We also consider the radiallysymmetrical case.In the fourth part, we consider a class of Schr(o|¨)dinger-Korteweg-de Vries systems and get theexistence of solutions with both of the two components nonzero.In the five part, we consider a class of discrete Schr(o|¨)dinger systems. We prove the existenceof solutions with two nonzero components. We also consider the symmetrical case. In this case,we first prove a compactness result, then prove the existence of nontrivial radially symmetricalsolutions.