Multiplicity Results for the p(x)-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

We investigate the singular Neumann problem involving the p(x)-Laplace operator: (P-lambda){-Delta(p(x))u + vertical bar u vertical bar(p(x)-2) u = 1/u(delta(x)) + f(x,u), in Omega; u > 0, in Omega; vertical bar del u vertical bar(p(x)-2)partial derivative u/partial derivative v = lambda u(q(x)) on partial derivative Omega}, where Omega subset of R-N (N >= 2) is a bounded domain with C-2 boundary, lambda is a positive parameter, and p(x), q(x), delta(x), and f(x,u) are assumed to satisfy assumptions (H0)-(H5) in the Introduction. Using some variational techniques, we show the existence of a number Lambda is an element of (0,infinity) such that problem (P-lambda) has two solutions for lambda is an element of (0, Lambda), one solution for lambda = Lambda, and no solutions for lambda > Lambda.