On The Studies Of Some Nonlocal Problem

In this thesis,we study the existence and stability of some nonlocal equa-tions.The main contents are organized as following:In chapter 1,we state the background and the main work of this article.In chapter 2,we study the existence of positive solution of following equa-tion.（？）where 0<α<2*-1/2,a>0,b>0,p ∈(0,2*)/{1},Ω(?)Rn(n≥ 2)is a bounded open smooth domain.Under the conditions(A),(H1),(H2)Of section 2.3,we get a priori estimates of positive solution to problem(1)by blow-up method.By making use of these estimates and the continuous method,we further get some existence results for positive solutions to problem(1)when 0<p<1,or 2α+1<p<2*.In chapter 3,we study the existence of concentration solutions of the fol-lowing nonlocal singularly perturbed problem（？）where n≥ 1,1<p<n-2/n-2,and A(s)V(x)satisfy(A),(V1),(V2)of section 3.2.The method will be used is the Lyapunov-Schmidt reduction method.To make this method works well,a non-degeneracy result of the limit equation of(2)will be proved in the section 3.3.Then,we use the non-degeneracy result and Lyapunov-Schmidt reduction method to construct concentration solutions of(2).In chapter 4,we study the orbital stability of the following equation(?)where 1≤n≤4,1<p<1+4/n,0<a,b.Firstly,we structure a Lyapunov function.Then,we get the estimation of linear operator of equation respect to standing waves of(3).Finally,by the conservativeness of Lyapunov function,we get the orbital stability of(3)at standing waves.