I got some problem when proving a priori estimate for the solution. For $\lambda>0$ big enough and $\kappa>0$ very small, consider the following elliptic equation: $$ \lambda u-\Delta u(x)=f(x)+\kappa|\nabla u(x)|^2, $$ where $f\in L^p(R^n)$ with $p>n/2$. To solve the above equation, we only need to show a priori estimate, that is, one can assume that there exists a $u\in W^{2,p}$ satisfies the above eqaution, and we need to show that $$ \|u\|_{2,p}\leq C\|f\|_p\,\,(or,\,\, C\|f\|_p^2) $$ For this, we have $$ \|u\|_{2,p}\leq C\|f+\kappa|\nabla u|^2\|_p\leq C\|f\|_p+C\kappa\||\nabla u|^2\|_p. $$ Since $u\in W^{2,p}$ and $p>n/2$, by Sobolev embedding theorem, we have $\nabla u\in W^{1,p}\hookrightarrow L^q(R^n)$ with $q>2p$. Thus, $|\nabla u|^2\in L^p(R^n)$.
But the question is: we could only have $$ \|u\|_{2,p}\leq C\|f\|_p+C\kappa\|\nabla u\|_{2p}^2\leq C\|f\|_p+C\kappa\|u\|_{2,p}^2. $$ Is there any possible to handel this problem? Many thanks!