# On the regularity of an elliptic PDE involving a power of gradient

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 :-)