The Research On The Phase Transitions And Vortex Solutions To Several Nonlinear Elliptic Equations

In this thesis,we mainly use the variational approach and the Lyapunov-Schmidt reduction method to study the existence and stability of vortices and phase transitions solutions for Ginzburg-Landau system,p-Ginzburg-Landau model and inhomogeneous Allen-Cahn equation.This thesis consists of four chapters:In Chapter 1,we mainly present the backgrounds and the recent developments of Ginzburg-Landau and Allen-Cahn equations and give a brief introduction to the main results of this thesis.In Chapter 2,we consider the stability of radial vortex solutions of a coupled Ginzburg-Landau system.For the coupled Ginzburg-Landau system in R2 with following constraints for the constant coefficients A+,A->0,B2<A+A-,t+,t->0,the radially symmetric solution w(x)=(w+,w-):R2?C2 of degree pair(1,1)was given by A.Alama and Q.Gao in J.Differential Equations 255(2013),3564-3591.We will study its linearized operator L around w and prove the following non-degeneracy result under one more assumption B<0:the kernel of L is spanned by the functions(?)and(?)in a natural Hilbert space.As an application of this non-degeneracy result,a solvability theory for the linearized operator L will be given.Next,in Chapter 3,given p>2 for the following coupled p-Ginzburg-Landau model in R2with the constraints A+,A->0,A02<A+A-and t+,t->0,we consider the existence of symmetric vortex solutions u(x)=(Up+(r)ein+?,Up-(r)ein-?)with given degree(n+,n-)? Z2,and prove the uaiqueness and regularity results for the vortex profile(Up+,Up-)under more constraint of the parameters.Moreover,we also establish the stability result for second variation of the energy around this vortex profile when we consider the perturbations in a space of radial functions.In Chapter 4 we consider the inhomogeneous Allen-Cahn equation?2?u+ V(y)(1-u2)u=0 in ?,(?)on(?),where ? is a bounded domain in R2 with smooth boundary(?)and V(x)is a positive smooth function,?>0 is a small parameter,v denotes the unit outward normal of(?).For any fixed integer N>2,we will show the existence of a clustered solution u? with N-transition layers near(?)with mutual distance O(?|ln ?|),provided that the generalized mean curvature H of(?)is positive and ? stays away from a discrete set of values at which resonance occurs.Our result is an extension of those(with dimension two)by A.Malchiodi,W.-M.Ni,J.Wei in Pacific J.Math.(Vol.229,2007,no.2,447-468)and A.Malchiodi,J.Wei in J.Fixed Point Theory Appl.(Vol.1,2007,no.2,305-336).