Research On The Existence Of Solutions For The Nonlocal Elliptic Equation And Tfdw Model On Lattice Graph

This thesis studies the existence of solutions of nonlocal elliptic equation and Thomas Fermi Dirac von Weiz(?)cker(TFDW)model on lattice graph Z~N.The nonlocal elliptic equation originates from Hartree minimization process which describes the interaction of electrons with static nuclear.TFDW model is a density functional theory which is an important tool to calculate the electronic structure in condensed matter.The first part of this thesis studies the existence and asymptotical behavior of the ground state solution for the nonlocal elliptic equation with potential terms on lattice graphs Z~N.The main idea is to transform the solution of the nonlocal equation into finding the critical points of the energy functional by the variational method.Firstly,we give a proof of the discrete Brézis-Lieb Lemma for the nonlocal term on the lattice graph.Secondly,as the potential function is periodic,the case of weak convergence to zero is excluded by using the constraint method and translation invariance.Then we prove that the sequence is strongly convergent by a contradiction argument,thereby proving the existence of the ground state solution.Finally,as the potential function is confining,the existence of a ground state solution is proved by using Nehari method.Moreover,we obtain the asymptotic properties of the solution which converges to a solution of a corresponding Dirichlet problem.The second part of the thesis investigates the existence and nonexistence of the minimizer of the TFDW model on the lattice graph Z~3.Firstly,the formula of the volume and surface area of a ball on the lattice graph Z~3are given.Then,by establishing a strictly subadditive inequality and using the concentration compactness principle to exclude vanishing and bifurcation.Thus we obtain a strongly convergent subsequence,which proves the existence of a minimizer when the constrained mass m is sufficiently small.Furthermore,by using the contradiction argument,we prove the nonexistence of a minimizer when the constrained mass m is sufficiently large.Finally,we also extend our analysis to Gamow liquid drop model on a subset(?)(?)Z~3.By using subadditive inequality and a contradiction argument,the nonexistence of a minimizer for Gamow model is proved when the constrained volume V is sufficiently large.