I've been studying the following singular PDE
$$
\mathrm{div}\left(\left(1+\frac{|\nabla g|}{|\nabla f|}\right)\nabla f\right)=0$$


in $\Omega \subset \mathbb{R}^{2}$. 
Do you know any reference, where this kind of singularity has been studied?



Solution singular PDE
Assume for simplicity that $g$ is linear. Then the integrand $|\nabla f|^2 + const.|\nabla f|$ is smooth and uniformly convex (as a function of $\nabla f$) except at the origin. It is known that the minimizers of such functionals are at least $C^1$-regular.


Indeed, if $f$ minimizes $\int F(\nabla f)$ and $F$ is smooth and uniformly convex away from some bounded set $E$, then it is known that for $x \in \Omega$ the gradients $\nabla f(B\_r(x))$ localize as $r \rightarrow 0$ either to the convex hull $K\_E$ of $E$, or to a point outside $K\_E$. See the result of Colombo-Figalli here:


<http://cvgmt.sns.it/media/doc/paper/1996/c1deg_final.pdf> 


In your case $E = K\_E$ is a single point, so $f$ is $C^1$.


Addendum: The idea of the above result is that ellipticity kicks in as soon as $|\nabla f| > 0$. The argument is based on applying linear estimates to each derivative $\partial\_kf$ of $f$. These solve the linearized Euler-Lagrange equation $\text{div}(D^2F(\nabla f) \nabla \partial\_k f) = 0$, which is uniformly elliptic when $\partial\_k f > 0$. The De Giorgi-Nash-Moser theorem gives continuity of solutions to such equations.


When $|\nabla g|$ is not constant I expect the result is still true provided e.g. $g \in C^{1,1}$. Indeed, for the more general case $F(\nabla f,\,x)$ the equation for $\partial\_kf$ has the right hand side $-\partial\_i(F\_{p\_ix\_k}(\nabla f,\,x))$. If $f\_k > 0$ then uniformly elliptic estimates give continuity of solutions provided $F\_{px}$ is e.g. bounded (or even $L^{\gamma}$ for some $\gamma > 2$, since you are in $\mathbb{R}^2$). In your case $|F\_{px}| = |\nabla |\nabla g||$, so if $|D^2g|$ is bounded (or $g \in W^{2,\,\gamma}$ for some $\gamma > 2$) it seems similar arguments may work.


