In the context of a physics problem on a gauge for a vector potential of a magnetic field I arrive at the following differential equation:
$$ \nabla\cdot\nabla f(\vec{r})={\color{red}-}\left( \frac{1}{2}\vec{B}\times\vec{r} {-} \nabla f(\vec{r}) \right)\cdot\frac{\nabla \rho(\vec{r})}{\rho(\vec{r})}, $$
with $\vec{r}\in\Bbb R^3$ is the general variable the functions depend on, $\rho$ is a given non-negative function $\Bbb R^3\to\Bbb R^+$, $\vec{B}$ a constant non-zero vector and $f$ an unknown scalar function ($f: \Bbb R^3\to\Bbb R$), the $\cdot$ denotes the scalar product of two vectors.
- Do solutions for $f$ exist?
- Is it possible to say something else than the above equation on the form of the solutions $f$? (Especially: is there an analytical solution?)
EDIT:
Rechecking your re-derivation of my starting equation I found I have confused a sign, I have marked the corrected sign in $\color{red}r\color{red}e\color{red}d$ above (it has been $+$ in the original version, the answer is referring to). The overall equation and problem is not affected at all, since it was implied that we can change the constant $\vec{B}$ vector (mag. field) freely and the sign of $f$ as well (scalar gauge term of the vector potential).
The expression I have started of with is actually $$ \nabla\cdot \left[ \left( \frac{1}{2}(\vec{B}\times\vec{r}) + \nabla f(\vec{r}) \right) \rho(\vec{r})\right] = 0 .$$ That is I search for a gauge contribution $\nabla f$ to the vector potential for which the vector field inside the brackets is divergence-less.
Best Answer
Due to the complexity of the topic of the question, it is not possible to expound here all the associated analytic developments, therefore I try to answer by giving precise references and surveying the results presented there.
Yes: to see this, note that, after formally manipulating your equation (perhaps going backward in the steps of your deduction), we get $$ \begin{split} \nabla\cdot\nabla f(\vec{r})&=\left( \frac{1}{2}\vec{B}\times\vec{r} - \nabla f(\vec{r}) \right)\cdot\frac{\nabla \rho(\vec{r})}{\rho(\vec{r})}\\ \rho(\vec{r})\nabla\cdot\nabla f(\vec{r})&=\left( \frac{1}{2}\vec{B}\times\vec{r} - \nabla f(\vec{r}) \right)\cdot\nabla \rho(\vec{r})\\ \rho(\vec{r})\nabla\cdot\nabla f(\vec{r})&=\left( \frac{1}{2}\vec{B}\times\vec{r} - \nabla f(\vec{r}) \right)\cdot\nabla \rho(\vec{r})\\ \rho(\vec{r})\nabla\cdot\nabla f(\vec{r})+\nabla \rho(\vec{r})\cdot\nabla f(\vec{r})&=\frac{1}{2}\vec{B}\times\vec{r}\cdot\nabla \rho(\vec{r})\\ \end{split} $$ i.e. $$ \nabla\cdot\big(\rho(\vec{r})\nabla f(\vec{r})\big)=\frac{1}{2}\vec{B}\times\vec{r}\cdot\nabla \rho(\vec{r}).\label{1}\tag{1} $$ Equation \eqref{1} is a standard linear elliptic equations in divergence form, and as such it admits a fundamental solution (and also a Green function) i.e. a solution of the equation $$ \nabla\cdot\big(\rho(\vec{r})\nabla \mathscr{E}(\vec{r},\vec{s})\big)=\delta(\vec{r}-\vec{s})\label{2}\tag{2} $$ where $\delta$ is the usual Dirac distribution, under fairly general conditions. In particular $\rho(\vec{r})$ is required only to be a bounded and measurable function: moreover, the problem has a solution even if, instead of a scalar function $\rho(\vec{r})$, we have a second-order tensor function $$ \vec{r}\mapsto\overline{\overline{\boldsymbol{\rho}}}(\vec{r})\in\Bbb R^{3\times 3}\quad\vec{r}\in\Bbb R^3 \label{3}\tag{3} $$ with bounded measurable components. This result is due to Walter Littman, Guido Stampacchia and Hans Weinberger: the paper [2] (where the original result is to be found) is not an easy read and even the used notation is not a modern one, while the set of lecture notes [3] is a better readable source (with also an updated notation) but is written in French. A more modern reference is [1], where the authors prove also that the tensor of the coefficients \eqref{3} can be asymmetric.
Yes (with some cautions): for the general case, several properties of the solution to equation \eqref{2}, and these properties "interact" with the ones of the given function $\frac{1}{2}\vec{B}\times\vec{r}\cdot\nabla\rho(\vec{r})$ (have again a look at the items in the bibliography for the details and also at this answer). For example it is known that the quotient between $\mathscr{E}(\vec{r},\vec{s})$ and the fundamental solution of Laplace equation is bounded by two positive constants $K$ and $K^{-1}$. And also you can use $\mathscr{E}(\vec{r},\vec{s})$ to construct a fairly explicit general solution of \eqref{1} as a convolution integral: $$ f(\vec{r})=\int\limits_{\Bbb R^3}\mathscr{E}(\vec{r},\vec{s})\big(\vec{B}\times\vec{s}\cdot\nabla\rho(\vec{s})\big)\mathrm{d}\vec{s} $$ However, it is not always possible to construct explicitly the fundamental solution $\mathscr{E}(\vec{r},\vec{s})$ (in this respect see also this answer) thus, apart from the case $\rho(\vec{r})\equiv\mathrm{const}.$ where \eqref{1} reduce to the Laplace equation $$ \nabla\cdot\nabla f(\vec{r})=0, $$ I am not aware of the existence of any exact (meant as expressed by means of more or less elementary functions) solutions to \eqref{1}.
Bibliography
[1] Michael Grüter and Kjell-Ove Widman (1982), "The Green function for uniformly elliptic equations", Manuscripta Mathematica 37, pp. 303-342 Zbl 0485.35031.
[2] Walter Littman, Hans Weinberger and Guido Stampacchia (1962), "Regular points for elliptic equations with discontinuous coefficients", Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, serie III, Vol. 17, n° 1-2, pp. 43-77, MR161019, Zbl0116.30302.
[3] Guido Stampacchia (1966), "Équations elliptiques du second ordre à coefficients discontinus" (notes du cours donné à la 4me session du Séminaire de mathématiques supérieures de l'Université de Montréal, tenue l'été 1965), (in French), Séminaire de mathématiques supérieures 16, Montréal: Les Presses de l'Université de Montréal, pp. 326, ISBN 0-8405-0052-1, MR0251373, Zbl 0151.15501.