Let $ p_m \nearrow \frac{N+2}{N-2}$  and consider the family of elliptic problems 
$$-\Delta u_m(x)=u_m(x)^{p_m}  \quad B  \qquad \quad u_m =0    \quad    \partial B,$$  where $B$ is the unit ball centered at the origin in $ R^N$.    

My interest is in obtaining positive solutions of perturbations of the equation $-\Delta u(x) = u(x)^p$ in $B$  where $ p $ is slighly smaller than $ \frac{N+2}{N-2}$.     So in particular 
I will need to have a good understanding of the linearized operator 
$L_m(\phi)= \Delta \phi + p_m (u_m)^{p_m-1} \phi$ for large $m$.    

(I believe people generally take a slightly different approach where they take a bubble and project it into $H_0^1$ to obtain the correct boundary condition. )



So here is my question.  Are there any suitable $L^\infty$ type spaces maybe with parameters involving $m$ such that $L_m$ is nicely behaved on teh spaces (uniformly in $m$).     I assume one has a kernel to deal with and .... (I realize this question is not at all well poised. Sorry).     Any comments would be greatly appreciated. 

thanks
Craig