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

Abstract

We investigate the singular Neumann problem involving the $p(x)$-Laplace operator: $\left(P_{\lambda }\right)\{-{\mathrm{\Delta }}_{p\left(x\right)}u+|u{|}^{p\left(x\right)-\mathrm{2}}u$ $=\mathrm{1}/u^{\delta \left(x\right)}+f\left(x,u\right)$, in $\mathrm{\Omega };\hspace{0.17em}\hspace{0.17em}u>\mathrm{0},\hspace{0.17em}\hspace{0.17em}\text{in}\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega };\hspace{0.17em}\hspace{0.17em}{\left|\nabla u\right|}^{p\left(x\right)-\mathrm{2}}\partial u/\partial \nu =\lambda u^{q\left(x\right)},\hspace{0.17em}\hspace{0.17em}\text{on}\hspace{0.17em}\hspace{0.17em}\partial \mathrm{\Omega }\}$, where $\mathrm{\Omega }\subset {\mathbb{R}}^N\left(N\ge \mathrm{2}\right)$ is a bounded domain with $C^{\mathrm{2}}$ boundary, $\lambda$ is a positive parameter, and $p\left(x\right),$$q\left(x\right),$$\delta \left(x\right)$, and $f\left(x,u\right)$ are assumed to satisfy assumptions (H0)–(H5) in the Introduction. Using some variational techniques, we show the existence of a number $\mathrm{\Lambda }\in \left(\mathrm{0},\mathrm{\infty }\right)$ such that problem $\left(P_{\lambda }\right)$ has two solutions for $\lambda \in \left(\mathrm{0},\mathrm{\Lambda }\right),$ one solution for $\lambda =\mathrm{\Lambda }$, and no solutions for $\lambda >\mathrm{\Lambda }$.