Existence Of Weak Solutions For A Class Of Elliptic Equations With Perturbations

With the development of the objects studied in mathematical physics,the application of PDE is more extensive.P.Marcellini was the first people to study the nonlinear variational problem with double phase structure.And it has been widely concerned and studied deeply in the past ten years.Recently,on the background of image processing,nonlinear hydrodynamics and elasticity,equations with perturbations has become a part of the research on partial differential equations.A lot of researches have been done on the existence,uniqueness and regularity of solutions to it.In the framework of Musielak-Orlicz-Sobolev space,we mainly discuss the existence of weak solutions of the following double phase elliptic equation with perturbations:-div(|?u|p-2 ?u+a(x)|?u|q-2 ?u)+|u|p-2 u+a(x)|u|q-2 u=f(x,u)+h(x),x?RN where 1<p<q<N,N?2,0?a(x)? L1(RN),h(x)?Lq'(RN).By using critical point theory,we study the critical points of the energy functional related to the above equation.Then we get the existence of weak solutions to the equation.The followings are the main contents of this dissertation.(1)By using Ekeland variational principle,we get a local minimizer for the energy functional,which has a negative energy.This local minimizer is a critical point for the energy functional,then it is a non-trivial weak solution for the equation.(2)By using Mountain Pass lemma,we obtain a(PS)c sequence {un} for the energy functional.As RN is unbounded,the embedding W1,H(RN)?Lr(RN)is only continuous but not compact.Although the energy functional may not satisfy(PS)c condition for some energy value c,we obtain a non-trivial weak solution by concentration compactness principle from Lions.(3)We prove that the above two weak solutions have different energy values.Thus,the equation has at least two different non-trivial weak solutions.