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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 $u \in C^2(\overline{\Omega})$?

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  • $\begingroup$ Not in general. Do you have some particular boundary conditions (Dirichlet, Neumann, Robin, etc.)? $\endgroup$ – Jose27 Apr 15 '15 at 23:32
  • $\begingroup$ BTW, could you expand on how you got the $C^{2,\alpha}$ result? As far as I know the best you can do with an $f\in L^\infty$ is $u\in C^{1,\alpha}$ (see Gilbarg-Trudinger Exercise 4.8) so the result looks a little weird. $\endgroup$ – Jose27 Apr 15 '15 at 23:39
  • $\begingroup$ We may assume that $u = 0$ on $\partial \Omega$. I have included the proof of the $C_{\mathrm{loc}}^{2,\alpha}$-regularity result in my original post. This result is mainly due to the specific form of the function $f$. $\endgroup$ – Gatz' Apr 16 '15 at 9:46
  • $\begingroup$ The answer is yes, by essentially the same arguments ($L^p$ estimates), adapted to the boundary. See Thm. 9.13 in Gilbarg-Trudinger and notice that $u^{2^*-1}\in L^q$ for $q>1$ small. $\endgroup$ – Jose27 Apr 16 '15 at 21:25
  • $\begingroup$ @Jose27 Thanks. To apply Theorem 9.13, $u$ must be a strong solution: twice weakly differentiable satisfying the eq. a.e. in $\Omega$. Assuming it to be true, the theorem asserts that $u \in W_{\mathrm{loc}}^{2,q}(\overline{\Omega})$ for every $q \in ]1,\infty[$. However, I don't see how to use Morrey's inequality to get $C_{\mathrm{loc}}^{2,\alpha}(\overline{\Omega})$-regularity since for $K \subset \overline{\Omega}$ compact intersecting the boundary of $\Omega$, we cannot cover $K$ by open balls in $\Omega$. Morrey's inequality needs some regularity. $\endgroup$ – Gatz' Apr 23 '15 at 19:01

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