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Suppose $\Omega$ is a bounded domain in $\mathbb{R}^N$ and consider the following Dirichlet boundary condition: $$ -\Delta u=\frac{f(x)}{u^\delta}\text{ in }\Omega, u>0\text{ in }\Omega; $$ where $\delta>0$ and $f(x)$ is a continuous function in $\Omega$. Since $u^{-\delta}\to \infty$ near $u=0$ on $\partial\Omega$. Such type of problems are referred to as Singular elliptic problems.

I am trying to understand a physical motivation of such type of problems. When I had gone through literature, such type of problems occurs in the study of viscus fluids, non-Newtonian fluids, to study the dynamics of van daar waals force in thin films.

But did not found a proper derivation of such equations. Can you kindly explain how to derive such models physically.

This will give an intuition on these problems. I have attached below a paper which may be helpful for you: http://www.math.utoronto.ca/mpugh/Prints/thin_films1/thin_films1.pdf

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