Existence Of Positive Solutions For A Discontinuous Problems With Critical Sobolev Exponents Via Variational Approach

In1997,Alves employed variational techniques to study the existence and multiplicity of positive solutions of the problem-△mu=λh(x)uq+Um*-1, x∈RN.where u≥0,u≠0,u∈D1,m(RN),入>0,2≤m<N,0<g<m-1, h(x)is both nonnegative and integrable on RN.In this paper,we consider the equation-△pu=λh(x)H(u-a)uq+up*-1,x∈RN,(1)where1<p<N,0≤q<p*-1,H is the Heaviside function,i.e.H(t)=0if t≤0,H(t)=1ift>0.We obtain existence and multiplicity of solutions via the Mountain Pass Geometry,the Ekeland Variational Principle,and the Mini-max Principle by distinguishing the cases0≤q≤p-1and p一1<q<p*-1.