Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation \begin{equation} -\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0. \end{equation} Here the force $f \in L^2$, and $div u = \sum_j \partial u/\partial x_j $. Is there any solutions of the equation in $H^1$? And how to prove it. Thanks for any hints and references.

Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation \begin{equation} -\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0. \end{equation} Here the force $f \in L^2$, and $div u = \sum_j \partial u/\partial x_j $. Is there any solutions of the equation in $H^1$? And how to prove it. Thanks for any hints and references.

\textbf{Edit}: In fact, the $1-d$ equation we considered is the stationary equation of the BBM equation $$ u_t - u_{txx} - u_{xx} + u^2u_x = f. $$