New embeddings of some weighted Sobolev spaces with weights a(x)and b(x)are established.The weights a(x)and b(x)can be singular.Some applications of these embeddings to a class of degenerate elliptic problems of the form-div(a(x)?u)=b(x)f(x,u)in?,u=0 on??,where?is a bounded or unbounded domain in RN,N 2,are presented.The main results of this paper also give some generalizations of the well-known Caffarelli-Kohn-Nirenberg inequality.
In this paper,our main purpose is to establish the existence of positive solution of the following system{−△ p(x)u=F(x,u,v),x∈W,−D q(x)v=H(x,u,v),x∈W,u=v=0,x∈∂W,where W=B(0,r)⊂RN or W=B(0,r2)\B(0,r1)⊂RN,00 are parameters,p(x),q(x)are radial symmetric functions,−D p(x)=−div(|∇u|p(x)−2∇u)is called p(x)-Laplacian.We give the existence results and consider the asymptotic behavior of the solutions.In particular,we do not assume any symmetric condition,and we do not assume any sign condition on F(x,0,0)and H(x,0,0)either.
