$N$-Laplacian Equation with a Nonlinear Neumann Boundary Condition and a Singular Nonlinearity

In this work, we investigate the existence, nonexistence, multiplicity of weak solution for the following singular Neumann problem:
\begin{equation*} (\mathrm{P}_{\mu,\lambda})\qquad \begin{cases} - \Delta_N u +|u|^{N-2}u =\mu g(u) + h(x,u)e^{bu^{N/(N-1)}} &\text{in }\Omega, \\[2mm] u>0 & \text{in }\Omega, \\[2mm] |\nabla u|^{N-2} \dfrac{\partial u}{\partial\nu}= \lambda\psi |u|^{q-1}u &\text{on }\partial\Omega, \end{cases} \end{equation*}
where $\Omega\subset\mathbb{R}^N,$ $N\geq 2$ be a bounded smooth domain, $\Delta_N u = \nabla\cdot (|\nabla u|^{N-2}\nabla u)$ denotes the $N$-Laplace operator, $\mu,\lambda>0,$ $0<\delta<1$ and $b>0$ is a constant. Here $h(x,u)$ is a $C^{1}(\overline{\Omega}\times \mathbb{R})$ having superlinear growth at infinity and $g(u)\simeq u^{-\delta}$. Using the sub-supersolution method and the variational method, under appropriate assumptions on $g$ and $h,$ we show that there exists a region $\mathcal{R}\subset \{(\mu,\lambda)\colon\mu,\lambda>0\}$ bounded by the graph of a map $\Lambda$ such that $(P_{\mu,\lambda})$ admits at least two solutions for all $(\mu,\lambda) \in \mathcal{R},$ at least one solution for $(\mu,\lambda)\in \partial\mathcal{R}$ and no solution for all $(\mu,\lambda)$ outside $\overline{\mathcal{R}}.$