Existence Of Changing Sign Solutions For A Class Of Fractional Laplace Equations

In recent decades,the fractional Laplace equation has been fully developed.The fractional Laplace operator is a generalization of the classical Laplace operator,inheriting some important properties of the classical Laplace operator.However,unlike the classical Laplace operator,the fractional Laplace operator is a nonlocal quasi-differential operator,so it can explain the anomalous diffusion phenomenon well.At present,fractional Laplace equations have played an important role in the fields of chemistry,physics,engineering,and finance.In this thesis,we mainly discusse the existence of the changing sign solution of the fractional Laplacian equation as follow???where s??0,1?,? is a bounded C^1,1 domain in R^N.h:R¹?R¹,is defined as follow???where,?>0,?>0,a,d>?₁,?₁represent the first eigenvalue of?-??,l₁?u?/u is strictly increasing on?0,??,limu?0l₁?u?/u=0,limu?+??l₁?u?/2?=+?,l₂?-u?satisfies the same conditions as l₁?u?.Firstly,in this thesis we introduce the research background and analyzes the research status at home and abroad.Then the definition of the fractional Sobolev space and its basic properties and other basic definitions and lemmas which are often used in the fractional Laplace equation are introduced.Finally,we use upper and lower solutions method,the maximal principle,and the relevant conclusions of the critical point theory to study the conditions that the parameters?a,d?which make the above equation have changing solutions satisfy.