Nontrivial Solutions Of Non-local Elliptic Equations With Resonance

In this paper, nontrivial solutions of non-local elliptic equations with resonance is discussed by means of variational method of nonlinear functional analysis, the critical point theory, especially Morse theory, as well as the calculations of the critical groups.where Ω(?)Rn(n≥2) is a bounded domain with smooth boundary (?)Ω,LK is the non-local elliptic operator defined by: there exist θ>0and s∈(0,1) such that f:Ω×R→R is a Caratheodory function that satisfies the subcritical growth condition: This paper is composed of four chapters.In the chapter one, the background and the method of the study for non-local elliptic equations with resonance, the significance of study and main results of this paper are presented. Using variational method gives the problem(1.2.1) sufficient conditions of the existence of a non-trivial solution in the four case and the existence of two sufficient conditions of non-trivial solutions.In the chapter two, the basic lemmas and definitions about critical point theory and the energy functional of the problem (1.2.1) are introduced.In the chapter three, we give the variational structure of the problem (1.2.1), and the energy functional and the corresponding all linear feature values.In the chapter four, by using related lemma and theory, the main results are proved by using Morse theory and the calculations of the critical groups.