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

Abstract: We prove an existence result for Robin boundary value problems modeled on [ \begin{cases} Δu + |\nabla u|^2 + λf(x) = 0 & \text{in } Ω \ \frac{\partial u}{\partial ν} + βu = 0 & \text{on } \partialΩ\end{cases} ] where $Ω$ is a bounded, sufficiently smooth open set in $\mathbb R^N$, $f(x)$ belongs to the Marcinkiewicz space $M^{\frac N2}$ and {$β>0$}, under a smallness assumption on the datum $λ$. In order to study such problem, we will show several properties of the weighted, singular Robin eigenvalue problem [ λ{1,f,γ}(Ω)= \inf{ψ\in H^{1},\;\int_Ωfψ^{2}=1}\left{\int_Ω|\nabla ψ|^{2}dx+γ\int_{\partialΩ}ψ^{2}\right}. ]