Qualitative Analysis Of Solutions To Some Systems Of Fractional Elliptic Equations And Integral Equations

This thesis mainly studies the qualitative analysis of solutions to some sys-tems of fractional elliptic equations and integral equations.Our main results are concerned with the maximum principles for elliptic equation involving Laplacian or fractional Laplacian(hereafter referred to as(fractional)Laplacian)with ze-ro order term and first order term,symmetry and monotonicity property for the nonnegative solutions to the nonlinear system of integral equations,monotonicity and uniqueness for the solutions to the system involving fractional Laplacian.The main contents are the following:Chapter 1 first provides an introduction to the background of this thesis,including the research progress in the study of the fractional Laplacian and the qualitative analysis of the solutions to the equations and system of integral form.Then it briefly describes the problems to be solved and the core methods to be used:the method of moving plane and sliding method.Chapter 2 studies maximum principles for Laplacian or fractional Laplacian with zero order term and first order term.We first consider maximum principles for the Schrodinger operator-?+c(x),where c(x)is a given potential,and show that the maximum principle holds when c(x)satisfies the critical integrability condition,i.e.c(x)?Lp(B1),p=n/2.On the hand,the strong maximum principle holds requires p>n/2.In particular,we present a counterexample to illustrate that no matter how small ?c?/Lp(B1)is,the critical integrability on c(x)is not enough to ensure the strong maximum principle to hold.This gives a partial answer to an open problem proposed by Bertsch,Smarrazzo and Tesei in[A note on the strong maximum principle,J.Differential Equations,259(2015),pp.4356-4375].As compared to the previous case,we obtain that the maximum principle and strong maximum principle both hold for Laplacian with first order term in the critical case.Moreover,we extend some of the results above to fractional Laplacian and study maximum principles for fractional Laplacian.In particular,we remove the lower semi-continuous on u(x)and present a weaker condition to prove the maximum principle for fractional superharmonic functionsChapter 3 discusses symmetry and monotonicity property for the nonnegative solutions to the nonlinear system of integral equations.Considering the properties of nonnegative solutions when the nonlinear terms respectively satisfy the following three cases:nonlinear terms are homogeneous of degree,nonlinear terms can be represented as the sum form of the functions which are homogeneous of degree,nonlinear terms satisfy the general monotonicity condition.For these three cases,we use the method of moving plane in an integral form to prove the symmetry and monotonicity property for nonnegative solutions to the nonlinear system of integral equations.Here,the monotonicity condition for nonlinear terms contains the critical and subcritical homogenous degree as special cases.Moreover,the homogeneous degree can be different.Due to our condition here is more general,the more delicate estimate method is needed to deal with this difficultyChapter 4 investigates the monotonicity of solutions for system involving frac-tional Laplacian with nonlinear terms fi(x,u1,u2,...,um,?ui),i=1,…,m in?.Here ? is a bounded domain or an unbounded domain in Rn which is convex in x1 direction.Based on the sliding method,we show that the solutions ui(x)are monotone increasing with respect to x1 in ?,and then obtain the uniqueness of the solutions.The appearance of ?ui(x)in the nonlinear terms brings in some new difficulties,we need establish some pointwise estimates of(-?)swi0(x)for some constructed function wi0(x)at the minimum point,1?i0?m.In addition,we introduce a new iteration method to deal with the system.