Study On Layer Solutions Of A Class Of Non-autonomous Equations Involving The Fractional Laplacian

In this thesis we study the layer solution of a class of non-autonomous elliptic equations involving the fractional Laplacian. The fractional power of the Laplacian is a nonlocal elliptic operator which appears in many physical situations dealing with long-range or anomalous diffusions. We obtain the existence and asymptot-ic estimates of layer solutions of this class of nonlinear elliptic equations. The asymptotic behavior of layer solutions as the fractional power tends to one is also studied. During this process, we obtain a classical local elliptic equation.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 nonlocal elliptic equation where (-△)s (s∈(0,1)) is the fractional Laplacian which is a peseudo-differential operator can be defined as Here Cn,s is a positive constant depending only on n and s, P.V. stands for the Cauchy principle value, b is a positive periodic function and f is an Allen-Cahn type nonlinearity. Due to the "nonlocal" property of this type elliptic operator, we need to be careful dealing with "solutions" of its.In this chapter, we firstly consider f is odd and nonnegative in (0,1), and b is even. Converting this nonlocal differential equation to a degenerate local elliptic equation, we can use the classical local elliptic approach to discuss our problem. On the other hand, this converting would pay price to adding a spatial variable, and we also need to be careful of the "degeneracy" of the operator. In this case we use two different approaches to obtain layer solutions.1. Via the classical variation method, we prove that for every s≥1/2, there exists a layer solution which is a bounded solution with values varying from-1at-∞to1at∞. To obtain the limit behavior at infinity, we introduce a Liouville theorem which is right for s≥1/2but has no conclusion for s<1/2. This is the reason why we only settle the existence of layer solutions for s≥1/2in this part. Clearly our equation is non-autonomous which explicitly depends on x, the sliding method cannot be used and solutions have no monotonicity, and furthermore solutions are not unique.2. Via the layer solution of the autonomous equation, we successfully con-struct a supersolution and a subsolution to our problem in (0,∞), further by the monotone iteration method we prove the existence of layer solutions for every s∈(0,1).In chapter three, we secondly consider more general inhomogeneous term f and more general periodic function b, i.e.., f and b have no symmetry assumptions. We still use the variation method to construct the layer solution. In this part we consider the nonlocal energy functional involving the Gagliardo seminorm. Here an estimate of energy of auxiliary function is quite involved. We show the nonlocal energy of the local minimizer is finite if s>1/2, while which is infinite if s≤1/2. For s>1/2, we construct a sequence of minimizers in the growing bounded interval, and prove that, up to a subsequence, they would converge to the desired layer solution which is also a local minimizer. For s<1/2, due to the infinite of nonlocal energy, the desired limit behavior at infinity of solutions we can not obtain, there is no any existence result regarding to layer solutions, which is one direction that we struggle for future.In chapter four, we show asymptotic estimates at infinity for layer solutions, which is a consequence by comparing with the layer solution of the autonomous equation at infinity. We also investigate the limit behavior of layer solutions as the fractional power tends to1, during this process a local elliptic equation is presented.