Study Of Multi-layers Solutions For A Class Of One-dimensional P-Laplace Equation With Double Well Potential

This paper is a promotion of the studies for one-dimensional Allen-Cahn equa-tion. When Laplace operator is replaced by p-Laplace operator, the correspondingconclusions also can be drawn. In this paper we discuss the following issues, thefirst of all is the existence and uniqueness of the solution to the following problemwhere F(u) = (1 ? u2)p,and p > 2,u∈W1,p(R), and also u (0) > 0(or u (0) < 0).Next, to the following problem, we make an estimate of the solution and obtain anestimate of the derivative of the solution.where h(x)∈C[0,1];h(x) > 0 (x∈(0,1)). We also discuss the zero distributionof the solutions.This paper is composed of four chapters. In the first chapter we mainly intro-duce the background and the current development of Allen-Cahn equations, and weintroduce the main works of this paper. At the same time, some basic knowledgeto be used in this paper will be introduced. In the second chapter, we use calculusmethod to prove the existence and uniqueness of the solution to the first problem.When we use the Calculus method, we can also obtain the specific expression ofsolution. In this paper, the focus is in the third chapter and the fourth chapter. Inthe third chapter, we primarily make an estimate of the solution and obtain an es-timate of the derivative of the solution, where the solution is of the second problem.The solution is very close to±1 in the regions which the solution is not equal tozero for su?ciently smallε> 0. In the fourth chapter, we focus on the problem ofthe distribution of the zeros. There is a one-one correspondence between the zerosand the layers of the solutions.By the discussion we can conclude that the zeros ofthe solutions can only exist in a small neighborhood of the extremum points of h(x). At most a single layer can appear near each local minimum point of h(x),and the multi-layers can appear near the local maximum point of h(x).