In this article, we study the following nonlinear Neumann boundary-value problem  diva(x,ru)þjujp(x)2 u1⁄4f in , @u 1⁄4 0 on @, where  is a
@
smooth bounded open domain in RN, N  3, @u is the outer unit normal @
derivative on @, div a(x, ru) a p(x)-Laplace type operator. We prove the existence and uniqueness of a weak solution for f 2 L( p)0 (), the existence and uniqueness of an entropy solution for L1-data f independent of u and the existence of weak solutions for f dependent on u. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

generalized Lebesgue and Sobolev spaces weak solution entropy solution p(x)-Laplace operator