Two-bubble Solutions For Super-critical Elliptic Problems In Bounded Domains

In this paper,we consider the existence of two-bubble solutions for the following super-critical elliptic problems:(?)where?is a bounded domain with smooth boundary in R~N,N?3,K is a positive bounded smooth function in?,and?is a positive parameter.By means of finite dimen-sional reduction and variational method,we prove that there are two-bubble solutions when the parameter is sufficiently small.In the research process,we assume that the function K has two different strict local maximum points or strict local minimum points in?,and these two points need to satisfy certain conditions.We select a appropriate weighted space and construct an approximate solution,thus we obtain the asymptotic expansion of a reduced functional and related estimates.Then we make an asymptotic analysis of the reduced functional and prove that the critical point of the finite energy functional corresponds to the critical point of the above problem energy functional.Finally,according to the properties of function K and Brouwer fixed point theorem,we find a critical point of the reduced functional and a solution of the above problem.And we obtain the asymptotic analysis of the solution when the parameter tends to zero.In addition,we require to ensure that the errors in the research process will not affect the results.