The Nehari Manifold For Nonlocal Elliptic Operators Involving Exponential Nonlinearities

In this paper,we consider the following nonlocal problem with exponential nonlin-earitywhere Lk is general non-local integrodifferential operators,? is a smooth open bounded set in Rn,s ?(0,1),n=ps,p?2,10?r+1<p,?>0,f ?Lp-r-1/p(?)andis positive on some positive measure subset of ?.Here the function h is a general exponential nonlinearity satisfying some conditions.We mainly study the multiplicity of solutions of the problem.The paper is divided into three chapters.In the first chapter,we explain the research background and status of the problem,we also give some assumptions about the exponential nonlinear term h.In the second chapter,we introduce the corresponding operator,function space,and also give some basic concepts and lemmas which will be used in the later chapters.In the third chapter,we introduce Nehari manifold and fibering maps.By analysing the relationship between them,we obtain multiplicity results.To do this,we prove that when A is within a certain range,the Nehari manifold can be divided into two disjoint subsets,and the corresponding minimized(PS)sequences on them both exist.And we also prove that the Euler functional satisfies(PS)ccondition in the Nehari manifold.Thus we obtain that there exists ?0>0 such that the problem we study exists at least two nontrivial solutions for??(0,?0).The work is a promotion of the reference[3].