We prove an existence result for Robin boundary value problems modeled on \[ \begin{ cases} \Delta u + |\nabla u|^2 + \lambda f(x) = 0 & \text{ in } \Omega \\ \frac{ \partial u}{ \partial \nu} + \beta u = 0 & \text{ on } \partial\Omega \end{ cases} \] where $\Omega$ is a bounded, sufficiently smooth open set in $\mathbb R^N$, $f(x)$ belongs to the Marcinkiewicz space $M^{ \frac N2}$ and { $\beta>0$}, under a smallness assumption on the datum $\lambda$. In order to study such problem, we will show several properties of the weighted, singular Robin eigenvalue problem \[ \lambda_{ 1,f,\gamma}(\Omega)= \inf_{ \psi\in H^{ 1},\;\int_{ \Omega}f\psi^{ 2}=1}\left\{ \int_{ \Omega}|\nabla \psi|^{ 2}dx+\gamma\int_{ \partial\Omega}\psi^{ 2}\right\}. \]
Categories: math.AP