The Existence Of Solutions For A Class Of Nonlinear Elliptic Equations

In this paper we study a class of n×n system of nonlinear elliptic equations describing the fractional quantum Hall effect.We studied the existence solution of the system under different conditions.We select an appropriate weighted Sobolev space to study the existence of solutions of the equations under the condition of infinite energy.We use variational method to prove the existence of non-topological solutions.In other words,we need to prove the coerciveness and lower semicotinutity of the corresponding functional.Besides,we establish the decay estimates of the configuration functions.Critical point theory usually been used to study various differential equations,and it has been widely used in classical mechanics,field theory and other fields.We apply the mountain path theorem to get the saddle point solution.Namely,we prove that the corresponding functional of the equations satisfies the conditions of PS compactness,and we construct a mountain path structure to obtain the desired result.