The Existence Of The Solutions For Some Nonlinear Elliptic Equations

In this thesis,we mainly deal with the problems on the existence of solutions for a ?(2)SHG system.There are three chapters in this thesis.In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.In Chapter Two,we consider about the solutions for the following ?(2)SHG system where 2?N<6,?>0 and ?>?.We establish the non-degeneracy of this problem.With the knowledge of the non-degeneracy of this system,we construct many non-radial symmetric synchronized positive solutions,by utilizing Liapunov-Schmidt reduction.The Chapter Three is concerned with the following ?(2)SHG(Second Harmonic Generation)system in RN(2?N<6),where the potentials P(x),Q(x)are continuous functions satisfying suitable decay assumptions,but without any symmetry properties,? is a positive constant,? and? are some parameters.We mainly use the Liapunov-Schmidt reduction method.There are two main difficulties.Firstly,we need to show that the maximum points will not go to infinity.This is guaranteed by the slow decay assumption.Secondly,we have to detect the difference in the energy when the spikes move to the boundary of the configuration space.A crucial estimate will be given in a Lemma,in which we prove that the accumulated error can be controlled from step m to step(m + 1).In Chapter four,by an approximating argument,we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth provided N>6s,?>0,0<s<1,2s*=2N/N-2s,a>0 is a constant and ? is an open bounded domain in RN which corntains the origin.