Study On Solutions Of Two Types Of Nonlinear Elliptic Partial Differential Equations

In this thesis, we study the solutions of two types of nonlinear elliptic partial differ-ential equations. The first kind is singular elliptic equation coming from the equilibrium states in thin films. We get the existence and uniqueness of classical radial solution to this class of singular equation with a given initial value, and the existence of singular radial solutions(singularity occurs at the origin). The oscillation and the limit behaviour at infinity of these radial solutions are also studied. In particular, the research of singular radial solutions is of great significance in the coatings industry. The second type is the Allen-Cahn equation with Neumann boundary condition in a bounded domain in R2. We study its interior layer solutions.In chapter one, the background and main results are briefly presented. Several notations and some basic theorems used in the thesis, and the outline of this work are also given in this chapter.In chapter two, we consider the classical radial solutions of the following singular elliptic equation where r=|x|. Due to the negative exponent of the nonlinearity in this equation, we need to be careful in discussing "solutions " to this problem. To begin with, we give the definition of solutions to this equation. If a nonnegative continuous function u(?) 0 satisfy this equation in the open set{x∈RN:u(x)> 0}, then we call such u as a solution to this equation. In this chapter, we obtain that for anyη> 0, there exists a unique classical radial solution u(r) to this equation, which satisfies u(0)=η, furthermore, u(r) is oscillatory and has a limit at∞. During the consideration of limit behaviour, we discuss and obtain the same limit result of solution u(r), for three possible cases of u(r).In chapter three, we first check the growth rate of singular solutions near the origin. Afterwards, a direct application of contraction mapping principle yields the existence of the singular radial soluitons to the previous singular problem. Similarly as in the chapter two, we can get the oscillation and limit result of the singular radial solutions.In chapter four, we study the interior layer solutions to the following Allen-Cahn equation with Neumann boundary condition We get the result that there exist two interior layer solutions with opposite direction, by applying infinite-dimensional Liapunov-Schmidt reduction. The key is to achieve one layer solution, since the other one can be obtained similarly. In this chapter, we need to construct repeatedly approximation to the solution so as to improve the accuracy step by step, and a fixed point argument is also used repeatedly. We transform the local problem into similar nonlocal one, by introducing a smooth cut-off function.We will follow the following main steps. First of all, we construct approximation repeatedly in order to reduce error. Secondly, we apply Taylor's expansion about the equation to be considered, and get a linear operator. Afterwards we prove the linear operator in the corresponding projected problem to be invertible, furthermore, the in-verse operator is bounded. Thirdly, act the previous inverse operator and a projection operator on two sides of the Taylor's expansion, then the nonlinear projected problem is transformed into a fixed point problem. Now we can prove the nonlinear projected problem posses a unique solution, by applying fixed point theorem. Finally, we need to verify that the solution of the projected problem is actually the solution of the original problem.