Consider $\Omega \subset \mathbb{R}^N$ a bounded regular domain and the problem $$ (P)\,\,\,\ \begin{cases} -\Delta u = \lambda u - u^p + |\nabla u|^q, \Omega \\ \,\,\,\,\,\,\,\,\, u = 0, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\partial \Omega \end{cases} $$ with $1 < p,q$ and $\lambda > 0$. If $u \in C^{1}(\overline{\Omega})$ is a weak solution of (P), I have to prove that $u \in C^{3,\alpha}(\overline{\Omega})$, for some $\alpha \in (0,1)$.

What I tried: As $u \in C^{1}(\overline{\Omega})$, the function $|\nabla u|^q$ is continous. So, the right side of $(P)$ is a funtion in $L^p(\Omega)$, for every $p>1$, because $\lambda u, u^p$ and $|\nabla u|^q$ are bounded in $\overline{\Omega}$. By elliptic regularity, we have $u \in W^{2,p}(\Omega)$, for every, in particular, $p > N$. So, as in this case $W^{2,p}(\Omega) \subset C^{1,\alpha}(\overline{\Omega})$. Then we have $u \in C^{1,\alpha}(\overline{\Omega})$. Now I'm trying to show that the righ side of $(P)$ is a function in $C^{1,\alpha}(\overline{\Omega})$, so I could use Schauder regularity to conclude that $u \in C^{3,\alpha}(\overline{\Omega})$. However, $u \in C^{1,\alpha}(\overline{\Omega})$ does not even imply $|\nabla u|^q \in C^{0,\alpha}(\overline{\Omega})$.

I don't know if this is the best way to go. Any help would be very welcome :-)



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