Say I have a PDE of the form
$(1-\phi(x))\Delta u(x) + \phi(x) (f(x)-u(x)) = 0,\quad x\in\Omega$
with homogenous Neumann boundary conditions on the boundary $\partial\Omega$ and for some function $\phi : \Omega \rightarrow [0,1]$. For the sake of simplicity we can assume that $\Omega$ is a bounded domain with smooth boundary and that $\phi$ and $f$ are smooth as well.
I know that there exists a strong solution of this PDE if $1 > c_0 \geqslant \phi(x)$ for all $x\in\Omega$, because the differential operator is uniformly elliptic then. This is the case the is dealt with in most textbooks. If $\phi(x)$ is allowed to reach the value 1, then I force the solution to be equal to $f$ at these places and basically I obtain a mixed boundary value problem with non homogenous Dirichlet conditions and Neumann boundary conditions.
What can I say about solutions in that setting?
My feeling is that I should still have strong solutions in that case if $f$ is well behaved enough, but I can't find any clear references in the Literature. I also would expect problems to occur if regions where $\phi$ equals 1 touch the boundary of $\Omega$ and thus causing conflicting boundary conditions.
Any references where such a problem is discussed in detail are welcome.