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 slightly 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 the 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.