Let $\Omega \subset \mathbb{R}^N$ be an unbounded domain with nonempty boundary.
Let $$\mathcal{D}^{1,2}(\mathbb{R}^N) = \{ u \in L^{2^*}(\mathbb{R}^N) : \nabla u \in L^2(\mathbb{R}^N ; \mathbb{R}^N) \}.$$ We denote $\mathcal{D}_0^{1,2}(\Omega)$ the closure of $\mathcal{D}(\Omega) = C_c^\infty(\Omega)$ in $\mathcal{D}^{1,2}(\mathbb{R}^N)$.
Let $u \in \mathcal{D}_0^{1,2}(\Omega)$ be a (weak) solution of $$ \begin{aligned} -\Delta u + au &= \lambda \lvert u \rvert^{2^*-2}u &&\text{in} \quad \Omega, \\ u &= 0 &&\text{on} \quad \partial \Omega, \end{aligned} $$ where $a \in C^1(\Omega)$ and $\lambda \in \mathbb{R}$.
Using elliptic regularity theory, I managed to prove (I think) that $$u \in C_{\mathrm{loc}}^{2,\alpha}(\Omega) \quad \forall \alpha \in ]0,1[.$$ In particular, $u \in C^2(\Omega)$.
Proof: Observe that $u \in W_{\mathrm{loc}}^{1,2}(\Omega)$. By the Brezis-Kato theorem, $u \in L_{\mathrm{loc}}^q(\Omega)$ for all $q \in ]1,\infty[$. Hence $-au +\lambda \lvert u \rvert^{2^*-2}u \in L_{\mathrm{loc}}^q(\Omega)$. Using the Calderón-Zygmund inequality, $u \in W_{\mathrm{loc}}^{2,q}(\Omega)$ for all $q \in ]1,\infty[$. Morrey's inequality implies that, on every open ball $B$ in $\Omega$, there holds $u \in C^{1,\alpha}(\overline{B})$ for all $\alpha \in ]0,1[$. For each compact set $K \subset \Omega$, we can find a finite open covering of $K$ by open balls in $\Omega$. We deduce that $u \in C_{\mathrm{loc}}^{1,\alpha}(\Omega)$ for all $\alpha \in ]0,1[$. In particular, $u \in C^1(\Omega)$. Consequently, $-au + \lambda \lvert u \rvert^{2^*-2}u \in W_{\mathrm{loc}}^{1,q}(\Omega)$ for all $q \in ]1,\infty[$. By theorem 9.19 in Gilbarg and Trudinger, $u \in W_{\mathrm{loc}}^{3,q}(\Omega)$ for all $q \in ]1,\infty[$. Using Morrey's inequality once again, we get $u \in C_{\mathrm{loc}}^{2,\alpha}(\Omega)$.
Unfortunately, this isn't enough. Is it possible to extend the $C^2$-regularity of $u$ to the boundary of $\Omega$ so that $u \in C^2(\overline{\Omega})$?