Research On Ground State Solutions And Nodal Solutions Of Nonlocal Problems

The theories of variational method and its application in differential equations is one of the heat problems in nonlinear functional analysis,these problems have important applications in analysis and partial differential equations and other fields,and many scholars have carried out extensive researches for it.This dissertation mainly studies the existence of ground state solutions and sign-changing solutions for some nonlocal elliptic equations.In Chapter 1,we mainly review the research background about variational method,research development at home and abroad,research results of this dissertation and the basic knowledge of critical point theory used in this dissertation.In Chapter 2,we consider the existence of least energy nodal solutions and ground state solutions,energy doubling property of solutions for the following critical problem-(a+b ∫R3|▽7u|2dx)Δu+V(x)u=|u|4u+kf(u),x ∈ R3,where the nonlinear function f∈ C(R,R).By the constrained variational method,for each b>0,we obtain a least energy nodal solution ub and a ground state solution vb for this problem when k>>1.The nodal solution is sign-changing solution.In Chapter 3,we consider the existence of ground state nodal solutions and ground state solutions,energy doubling property of solutions for the following fractional critical problem where a,b,κ are positive parameters,α∈3/4，1),β∈(0，1）and 2α*=6/3-2α,(-△)αstands for the fractional Laplacian.By the constrained variational method,for each b>0,we obtain a ground state nodal solution ub and a ground state solution vb of above problem when k>>1,where the nonlinear function f:R3 × R → R is a Caratheodory function.In Chapter 4,we study the existence of sign-changing solutions and energy doubling property of solutions for the following N-Laplacian equation of Kirchhoff type in RN We apply the constraint minimization arguments to establish the existence of sign-changing solutions and ground state solutions for above problem.Our results extend existing results to the N-Laplacian equation of Kirchhoff type with logarithmic and exponential nonlinearities.In Chapter 5,the existence and multiplicity of homoclinic solutions are considered for the following fractional discrete equation:(-Δ1)su(n)+V(n)u(n)=f(n,u(n))for n ∈ Z,where(-Δ1)s denotes the fractional discrete Laplacian,the sequence V(n)is the potential,and f(n,u)is a sequence of functions.Under some conditions on V and f,the existence of ground state sign-changing homoclinic solution u1 and ground state homoclinic solution u0 are obtained.Moreover,it is proved that the energy of u1 is more than twice of the energy of u0 and so that u0≠u1.We also study the multiplicity of solutions in case of concave-convex nonlinearity.To the best of our knowledge,this is the first attempt in the literature on the multiplicity of homoclinic solutions for fractional discrete Schrodinger type equation.