Existence Of Multi-Bubble Solutions And Their Properties For Two Types Of Nonlinear Elliptic Equations

In the thesis,we mainly study the existence of multiple bubbling solutions and some corresponding properties of these solutions for a nonlinear Schrodinger-Poisson type system and a Grushin critical problem.The thesis consists of four chapters:In the Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results and notations in the whole thesis.In Chapter Two,we study the following nonlinear Schrodinger-Poisson type system(?)-e2?u +u-?(x)u = Q(x)|u|u,x?R3,-e2?? = u2,x?R3,where e>0 and Q(x)is a positive bounded contiUuous potential on R3 satisfying some suitable conditions.By applying the finite reduction method,we prove that the system has a multi-peak solution.In Chapter Three,we study the following Grushin critical problem-?u(x)=?(x)uN/N-2(x/|y|),u>0,in RN,(?)where x =(y,z)?Rk × K N-k,N ?5,?(x)is positive and periodic in its the k variables(z1,...,zk),1? k<N-2/2.Under some suitable conditions on ?(x)near its critical point,we prove that the problem above has solutions with infinitely many bubbles.Moreover,we also show that the bubbling solutions obtained in our existence result are locally unique.Our result implies that some bubbling solutions preserve the symmetry from the potential ?(x).In Chapter Four,we give some known results and some technical estimates for the two problems.