Existence And Multiplicity Of Solutions For Elliptic Type Equations With Nonlocal Terms

In this doctoral dissertation,we mainly study existence and multiplicity of solutions for some nonlocal elliptic partial differential equations.This thesis covers the Schrodinger-Bopp-Podolsky equation,the fractional order p-Kirchhoff equation and the Kirchhoff type system of equations.The main contents are as follows.Firstly,we introduce the historical background,research significance and current development of the problem studied in this thesis,and give the necessary preliminary knowledge for this paper.Then,we deal with the following Schrodinger-Bopp-Podolsky system:where ε>0 and f is a continuous,superlinear and subcritical nonlinearity.By using perturbation approach and the method of invariant sets of descending flow incorporated with minimax arguments,we prove the existence and multiplicity of sign-changing solutions of system.Moreover,the asymptotic behavior of sign-changing solutions is also established.Furthermore,we study the fractional p-Kirchhoff equation:(a+b∫RN∫RN(|u(x)-u(y)|p)/(|x-y|N+ps)dxdy)(-Δ)psu-μ|u|p-2u=|u|q-2u,x ∈ RN,where(-Δ)ps is the fractional p-Laplacian operator,a and b are strictly positive real numbers,s ∈(0,1),1<p<N/s,and p<q<ps*-2 with ps*=Np/(N-ps).By using the variational method,we prove the existence and uniqueness of global minimum or mountain pass type critical points on the Lp-normalized manifold S(c):={u∈Ws,p(RN):∫RN|u|pdx=cp}.In addition,we study the Kirchhoff type system:the combined effect of sign-changing weight functions and concave-convex nonlinearities of nontrivial solutions for Kirchhoff type system involving subcritical Sobolev exponents.With the help of the Nehari manifold,we prove that the system admits at least two nontrivial solutions when the pair of parameters(λ,μ)belongs to a suitable subset of R2.Finally,a brief summary and outlook of the work in this dissertation are given.