| Abstract: |
| In the talk I will describe 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, I will show several properties of the weighted, singular
Robin eigenvalue problem \[ \lambda_{1,f,\gamma}(\Omega)= \inf_{\psi\in
H^{1},\;\int_{\Omega}f\psi^{2}=1}\left\{\int_{\Omega}|\nabla
\psi|^{2}dx+\gamma\int_{\partial\Omega}\psi^{2}\right\}.
\] |
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