I have a basic question regarding rescaling an elliptic system when trying to get apriori estimates.
Consider an elliptic system say of the form
$$-\Delta u(x) = u^{p_1} v^{q_1}, \quad -\Delta v = u^{p_2} v^{q_2}$$ on some bounded domain $\Omega$ in $N$ dimensional euclidean space with zero boundary condition. Suppose one is attempting to prove some apriori estimates for a sequence of positive classical solutions $(u_m,v_m)$. A standard method would be to try rescaling and then apply some known Liouville theorems.
So towards this we assume that $ T_m=\max_\Omega (u_m + v_m) = u_m(x_m ) + v_m(x_m) \rightarrow \infty$ and one could try rescaling something like
$$ \widehat{u}_m(x) = \frac{ u_m(x_m + r_m x)}{T_m}, \quad \widehat{v}_m(x) = \frac{v_m(x_m + r_m x)}{T_m}$$ where $r_m>0$ is chosen later.
When one writes out the rescaled equations they get something like $$-\Delta \widehat{u}_m =(r_m^2 T_m^{p_1+q_1-1}) \widehat{u}_m^{p_1} \widehat{v}_m^{q_1} \quad -\Delta \widehat{v}_m =(r_m^2 T_m^{p_2+q_2-1}) \widehat{u}_m^{p_2} \widehat{v}_m^{q_2}$$.
Now we suppose $p_i,q_i>1$ and lets assume $p_1+q_1> p_2+q_2$. Then we can pick $r_m$ such that $ r_m^2 T_m^{p_1+q_1-1}=1$ but then the other coefficient $ \epsilon_m:=(r_m^2 T_m^{p_2+q_2-1}) \rightarrow 0$ and lets assume we are in the case of where the limiting domain is the full space. Then the limiting equation is
$$ -\Delta \widehat{u} = \widehat{u}^{p_1} \widehat{v}^{q_1}, \quad -\Delta \widehat{v} = 0 $$. Then note that $ \widehat{u}=1$ and $ \widehat{v}=0$ satisfies the limiting equation and there is no contradiction.
QUESTION. There must be some standard way to apply blow up analysis on a simple system like this that gets around this above perceived problem.
