On nonlocal p(x)-Laplacian Dirichlet problems

This paper deals with the nonlocal p(x)-Laplacian Dirichlet problems with non-variational form -A(u)Delta(p(x))u(x) = B(u)f(x, u(x)) in Omega; u vertical bar(partial derivative Omega) = 0, and with variational form -a(integral(Omega)vertical bar del u vertical bar(p(x))/p(x) dx) Delta(p(x))u(x) = b (integral F-Omega(x, u)dx) f(x, u (x)) in Omega; u vertical bar(partial derivative Omega) = 0, where F(x, t) = integral(t)(0) f(x, s)ds, and a is allowed to be singular at zero. Using (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) equivalent to p is a constant. (C) 2009 Elsevier Ltd. All rights reserved.