Multiplicity Results for the px-Laplacian Equation with Singular Nonlinearities and Nonlinear Neumann Boundary Condition

We investigate the singular Neumann problem involving the $p(x)$ -Laplace operator: $({P}_{\lambda })\{-{\mathrm{\Delta }}_{p(x)}u+|u{|}^{p(x)-\mathrm{2}}u$ $=\mathrm{1}/{u}^{\delta (x)}+f(x,u)$ , in $\mathrm{\Omega }; u>\mathrm{0}, \text{in} \mathrm{\Omega }; {|\nabla u|}^{p(x)-\mathrm{2}}\partial u/\partial \nu =\lambda {u}^{q(x)}, \text{on} \partial \mathrm{\Omega }\}$ , where $\mathrm{\Omega }\subset {\mathbb{R}}^{N}(N\ge \mathrm{2})$ is a bounded domain with ${C}^{\mathrm{2}}$ 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 $\mathrm{\Lambda }\in (\mathrm{0},\mathrm{\infty })$ such that problem $({P}_{\lambda })$ has two solutions for $\lambda \in (\mathrm{0},\mathrm{\Lambda }),$ one solution for $\lambda =\mathrm{\Lambda }$ , and no solutions for $\lambda >\mathrm{\Lambda }$ .