On nonlocal p(x)p(x)-Laplacian Dirichlet problems

This paper deals with the nonlocal p(x)p(x)-Laplacian Dirichlet problems with non-variational form −A(u)Δp(x)u(x)=B(u)f(x,u(x)) in Ω;  u∣∂Ω=0,−A(u)Δp(x)u(x)=B(u)f(x,u(x)) in Ω;  u∣∂Ω=0, and with variational form −a(∫Ω|∇u|p(x)p(x)dx)Δp(x)u(x)=b(∫ΩF(x,u)dx)f(x,u(x)) in Ω;  u∣∂Ω=0, where F(x,t)=∫0tf(x,s)ds, and aa is allowed to be singular at zero. Using (S+)(S+) mapping theory and the variational method, some results on existence and multiplicity for the problems are obtained under weaker hypotheses. Our results are also new even for the case when p(x)≡pp(x)≡p is a constant.