Weighted Robin eigenvalue problems and nonlinear elliptic equations with general growth in the gradient

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 $$\lambda_{1,f,\gamma}(\Omega)= \inf_{\psi\in H^{1},\;\int_{\Omega}f\psi^{2}=1}\left\{\int_{\Omega}|\nabla \psi|^{2}dx+\gamma\int_{\partial\Omega}\psi^{2}\right\}.$$