Multiplicity Of Solutions For Two Classes Of Elliptic Systems With Critical Exponents

The research on the multiplicity and other properties of the solutions of elliptic system is of great significance both in theory and reality.This paper mainly studies two types of elliptic equations,namely:the multiple solutions for a critical biharmonic equation involving singular nonlinearity and a fractional p-q-Laplacian equation with sign-changing weight functions and concave-convex nonlinearities by making use of Ekeland's variational principle in this paper.This paper studies the following fourth-order nonlinear elliptic equations Through the Nehari manifold and Ekeland's variational principle,we prove the problem(1)has at least two positive solutions when the singularity coefficient ? belongs to a certain subset.Further the upper bound of the singularity coefficient ? is estimated accurately.At the same time,we consider the multiple solutions of fractional p-q-Laplacian equation with sign-changing weight functions and concave-convex nonlinearities By using the Ekeland's variational principle,the Nehari manifold and some analysis methods,two nontrivial solutions of the problem(2)is obtained.