Existence or nonexistence of semilinear Poisson equation?

Question

Suppose that $\Omega\subset\mathbb{R}^n$ is a bounded domain with smooth boundary $\partial\Omega$. Consider the following semilinear Poisson equation with prescribed Dirichlet data on the boundary: \begin{equation} \begin{cases} %\vspace{3mm} -\Delta \phi=\frac{1}{\phi}\quad \text{in}\quad\hspace{2.5mm} \Omega,\\ \hspace{7mm}\phi=0 \hspace{6mm} \text{on}\quad \partial\Omega. \end{cases} \end{equation}

How to investigate existence or nonexistence of solutions to this problem?

Answer 1

The problems of this type are called Singular elliptic equations. There is a big variety of methods how you can obtain solutions. Try to look here and here.