Resonance And Near Resonance Problems Of Elliptic Equations With Nonlocal Operators

Three classes of elliptic equations are considered in this dissertation, in which both of Kirchhoff type elliptic equation and fractional elliptic equation include non-local operators. Based on critical theory and analysis technologies, some existence and multiplicity of elliptic equations with resonance and near resonance are ob-tained. The thesis is made up by four chapter.In chapter Ⅰ, a brief introduction is given to the physical, historic and applica-tion background for all investigated problems. Besides, some preliminaries are also introduced in this chapter.In chapter Ⅱ, we deal with the following Kirchhoff problem where Ω(?)RN(N=1,2,3) is a bounded domain with a smooth boundary (?)Ω, a≥0, b>0 are real numbers, and f:Ω×R→R is a Caratheodory function with the subcritical growth. For the case that f(x,t)=μu3+g(x,u)+h(x), the term g is sublinear and the parameter μ is sufficiently close to μ1(μ1 denotes the first nonlinear eigenvalue) from the left, existence of three solutions is found by Ekeland’s Variational Principle and the Mountain Pass Theorem. For the case that the quotient 4F(x,t)/bt4 stays between μk and μk+1 allowing for resonance with μk+1 at infinity, solutions are obtained by applying the G-Linking Theorem, where F(x, t)=（?)01 f(x, s)ds is the primitive function of f. In addition, when the quotient 4F(x,t)/bt4 stays between μ1 and μ’2 allowing for resonance with μ’2 at infinity, a nontriv-ial solution is found by using the classical Linking Theorem and argument of the characterization of μ’2.In chapter Ⅲ, fractional elliptic equation Ω(?)RN is a bounded domain with a smooth boundary (?)Ω, p∈(1,+∞),s∈(0,1) and ps<N. (-△)ps denotes fractional p-Laplacian operator, defined as On the analogy of the results in chapter Ⅱ, existence of three solutions is established for the case that λ approaches the principle eigenvalue λ1 from the left. For p=2, we deal with near resonance that λ is near to a non-principle eigenvalue of the fractional Laplacian from above and below respectively. In the case that λ is near to a non-principle eigenvalue from below, two solutions are found by the standard linking theorem and shown to be distinct since they lie at different levels. However, the situation changes for the other one, due to the asymmetrical characteristic of the quadratic part of the functional in the sense that the resonant subspace is part of the (finite-dimensional) negative subspace of quadratic part while it is in the (infinite-dimensional) positive in the other case. In order to discuss the case for λ approaches eigenvalue from above similar to the other case, we confine problems to finite dimensional space. Then we take limit on the dimension of such problems and prove that the limit is actually the critical point we looking for.In chapter IV, we discuss resonance problems of p-Laplacian with respect to the Fucik spectrum and obtain the existence of solutions by using Linking Theorem.