I have a question about trying 'odd' like extension to obtain some sign changing solutions of an elliptic equation.
Lets first consider the 1 dimensional problem. To solve a sign changing solution of $ -\Delta u = |u|^{p-1} u$ in $ 1<r<3$ with $ u(1)=u(3)=0$ we can just find a positive solution on $ (1,2)$ and then take an odd reflection across $ r=2$.
In higher dimension the first order term screws up the odd reflection, but it still seems one can play the same game. We consider the same problem on the annulus $ 1<r<3$ and we first solve for a positive solution on $1<r<R$ where $ R$ is a parameter. We then solve for a positive solution on $ (R,3)$ and we flip this solution. To get a solution on the full annulus we probably just need the first derivative to be continuous at $R$. By taking $ R$ close to $ 1$ and close to $ 3$ it appears we can use some sort of intermediate value property to get the desired result.
So I guess it appears you can try a few things with this. (1) One could try and look for sign changing solutions on the exterior of say $B_1$. Again $R$ would be a parameter and then this would determine at iteration $ R_k(R)$ (depending on $R$) which are increasing in $k$. If $ R_k \rightarrow \infty$ then we should have a sign changing solution on the exterior of a ball. Also if we can do this with various $R$ we find a whole family.
(2) One could try the same game on say $B_1$. So now take $ 0<R<1$ (maybe it has to be larger than 1/2) and try and play the same game. If $ R_k(R) \searrow 0$ then we'd have a solution on $B_1$ (or maybe just on the punctured ball).
So I guess my question is: whether this approach work and has been tried and whether it really gives anything or.
