The Existence And Multiplicity Of Solutions To Elliptic Equations With Non-local Terms

In this thesis,we use the variational methods,Nehari manifold and theory of fixed point index to study the existence and multiplicity of positive solutions for two classes of elliptic equations involving nonlocal term.The thesis consists of five parts.In Chapter 1,we introduce the physical and biological background and research status on elliptic equations involving nonlocal term,give the development of the related topics in the domestic and overseas at the present time.Then we list some necessary lemmas and theorems.Finally,we simply present the structure of the thesis.In Chapter 2,we study the following Kirchhoff-type elliptic equation with a weakly singular term(?)where ? is a bounded domain in RN and parameter ?>0.For above-mentioned equation,most of the literatures considering is the case of G'(s)= s.In this chapter,we study a more general case,that is function G(s)with smoothness and monotonicity.We found the existence or multiplicity of positive solutions de-pend on the growth of the relationship between f and G.Roughly speaking,when|f(x,u)|?|u|p-1(1?p<2*-1),G(||u||2)?||u||2q,by using variational and per-turbation methods,we deal with the cases of p>2q(see Theorem 2.1),p<2q(see Theorem 2.2)and p = 2q(see Theorem 2.3),and obtain the existence and multiplicity of positive solutions,which improve and extend the results of paper[84].In Chapter 3,we deal with the Kirchhoff-type elliptic equations with a general singular term such as(?)Due to the lack of smoothness(or even continuity)of the energy functional for the above equation,the known results are mainly concentrated in the case of ?<1.As far as we know,there are almost no result on the case of ??1 except[1].In this chapter,we consider the case of ?= 1 and prove that the equation has unique positive solution if ?>0 by using the fixed point theorem and approximation method,which improves the results of[79].In Chapter 4,we discuss the p-Kirchhoff-type elliptic equations involving a gen-eral nonlinearity(?)where ?pu =div(|?u|p-2?u),and ? is a bounded domain of RN.Moreover,0<r<1<p<q<p*and M(s)= asp-1 +b(a,b>0),f,g?C(?)are nonnegative and nontrivial functions.By using Nehari manifold method,we assert that there are at least two positive solutions when p2<q<p*and ? is positive and small enough.And there exists at least one positive solution with p2 = q<p*and a,?>0.Furthermore,at least two positive solutions can be achieved if a,?>0 are positive and small enough.These results generalize and improve the results in[80].In Chapter 5,we consider another nonlocal elliptic problem(?)where ?(?)RN(N?1)is a bounded domain with smooth boundary(?),?>0,a:[0,+?)?(0,+?)and f:? x(0,+?)are functions satisfied some suitable conditions.We study the equation in a general bounded domain or an annal domain of Q.when f(x,u)is continuous or singular at u = 0.Using the theory of fixed point index,we obtain the multiplicity of positive solutions.The main differences between this section and the previous literatures are:(1)we get the multiplicity of positive solutions,not only the existence of positive solutions,when ? is bounded domain and a(t)is bounded;(2)we remove the monotonicity of f(|x|,u)in u and obtain multiplicity of the positive solutions when ? is a bounded annular domain in RN and a(t)is unbounded.