# Smoothness of critical elliptic problem

I am convinced I have seen results along the lines of: if $$u \ge 0$$ is an $$H_0^1(\Omega)$$ solution of $$-\Delta u = u^{q-1}$$ in $$\Omega$$ with $$u=0$$ on $$\partial \Omega$$ (here $$\Omega$$ is a smooth bounded domain in $$R^N$$ and $$q=2^*$$, then $$u$$ is smooth. Any idea how one proves the regularity result? Since $$q=2^*$$ it appears the standard iteration method fails. Maybe one uses some $$\epsilon$$ regularity approaches (which I don't know). Or maybe I am wrong and the result is false. thanks

The result is true. As you suggest, the starting regularity assumption $$u \in H^1_0$$ is critical in the sense that, if you additionally knew $$u \in L^p$$ with $$p>2^*$$, then you could bootstrap to smoothness (at least, when $$2^*$$ is an integer, as in dimension $$N=3$$, for which $$2^*=6)$$. In the case $$p=2^*$$, the key observation is that $$H^1_0$$ solutions to $$-\Delta u = Vu$$ do satisfy higher integrability in the critical case $$V \in L^{N/2}$$. That is, such solutions belong to all $$L^p$$, $$p < +\infty$$. The idea is to split the potential $$V$$ into a bounded part and a part which is small in $$L^{N/2}$$ and run Moser's iteration. The argument is given in the book Elliptic Partial Differential Equations by Qing Han and Fanghua Lin, see Theorem 4.4, p. 76.