I have a question which looks like some sort of inverse problem.  

Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$).  

Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) we would like to find some $ f \ge 0$ in $B$ (sufficiently regular, say $ f \in L^q(B) $ for some $q>N$) such that $v$ satisfies the following: 


$\Delta v(x)=f(x)$ in $B$ with $ v=0$ on $ \partial B$ and $ x \cdot \nabla v(x)=h(x)$ on $ |x|=1$. 


In the case of $h=C>0$ (constant) one sees they can just explicitly write out a solution. In the case of $h(x)=1 +\epsilon g(x)$ ($g$ fixed $ \epsilon $ small they can also do it). 


Any comments would be greatly appreciated. 



finding subharmonic function on the ball with both Dirichlet and Neumann boundaries prescribed
It is possible to do this. Here is a sketch of the construction:


1) Let $w = (|x|-1)h(x/|x|)$. Then $w$ satisfies the desired boundary conditions, and is smooth away from the origin with $\Delta w = (n-1)h > 0$ on $\partial B\_1$. The idea is to find an appropriate extension of $w$ from a neighborhood of the boundary to the interior.


2) Let $v\_0 = \max\{w, \, c\_0(|x|^2 - 1) - c\_0\}$. For $c\_0 > 0$ small, $v\_0 = w$ near the boundary and $v\_0$ is the quadratic on a set $E$ whose boundary is a radial graph just inside of $\partial B\_1$. Furthermore, $\Delta v\_0 > c\_1 > 0$ in the distributional sense.


3) Now let $\eta$ be a smooth cutoff function that is $1$ near $\partial B\_1$ and $0$ in a neighborhood of $E$. Take 
$$v = \eta v\_0 + (1-\eta) v\_{\epsilon},$$
where $v\_{\epsilon}$ is a mollification of $v\_0$ (so $\Delta v\_{\epsilon} > c\_1$). Then $v = w$ near $\partial B\_1$. Since $v\_0 = w$ is smooth where the derivatives of $\eta$ are supported, it is straightforward to check that $\Delta v$ is smooth and positive in $B\_1$ for $\epsilon$ small, completing the construction.


