I'm looking for the stationnary solution of the following equation in 2 dimensions :
$$\nabla^2 f(r)=f(r) $$ $$f(0)=f_c$$ $$f(+\infty)=0 $$ in radial symmetry. $\nabla^2$ corresponds to the Laplacian, here with a radial symmetry : $\nabla^2=d^2/dr^2+(1/r)d/dr$. $f_c$ is a constant. The equation $\nabla^2 f(r)=f(r)$ corresponds to the stationnary equation for the conservation equation $\partial_t f+\nabla.(vf)=-f$, with the velocity $v=-\nabla f$, so I called it the diffusion-consumption equation. But I assume it has an official name (it looks like a Helmotz equation but it is not one). It's a kind of Poisson equation.
It seems that the only solution of $\nabla^2 f(r)=f(r)$ is a sum of Bessel function, but it does not satisfy the boundary conditions.
I'm also ok with taking $f(0)=\infty$. I just want a function that decreases form $r=0$ and reaches $0$ at $r=\infty$.
Moreover, I would be also interested in the same boundary conditions but with the equation : $\nabla^2 f(r)=1 $. Here also the result is $r^2+c_1+c_2 log(r)$ which are not compatible with the boundary conditions.
Since one can argue that the problem is ill posed, could a computer compute numerically the solutions if I impose him those conditions ?