3
$\begingroup$

I am interested in solving some linear elliptic system like

$$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$ $$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the unit ball in $ R^N$ with $ \phi=\psi=0$ on $ \partial B_1$. We have $ 0<\alpha<\beta$ with $ \alpha+\beta=4$.

My interest is in obtaining solutions in some weighted $L^\infty$ space which allows certain blow up at the origin and in obtaining some estimates on the solutions.

So towards solving this I will use spherical harmonics and we get some ode's like (which appears to be some sort of Euler ode's system)

$$ -a_k''(r) - \frac{(N-1)}{r} a_k'(r) + \frac{ \lambda_k a_k(r)}{r^2} + \frac{C_1 b_k(r)}{r^\beta}=c_k(r) $$

$$ -b_k''(r) - \frac{(N-1)}{r} b_k'(r) + \frac{ \lambda_k b_k(r)}{r^2} + \frac{C_2 a_k(r)}{r^\alpha}=d_k(r). $$

I was attempting to try and `diagonalize' by doing a change of dependent variables and hopefully get some scalar equations which I know how to solve.

But my question is: this system has probably been studied and so I was wondering if anyone has any references for it; or some know approaches to solve.
thanks

Improve this question
$\endgroup$
1
  • $\begingroup$ I see for the homogenous version one can look for solutions of the form $ a(r)= r^{\gamma_1}$ and $b(r)=r^{\gamma_2}$ and it appears one gets four free parameters (which i assume means this is the general solution). Maybe then one can play around with variation of parameters ??? $\endgroup$
    – Math604
    Commented Feb 9, 2019 at 21:12

0

Reset to default

You must log in to answer this question.