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 $\hat{I}\text{c◯}$ is a bounded, sufficiently smooth open set in ${\mathbb{R}}^N$, $f(x)$ belongs to the Marcinkiewicz space $M^{\frac{N}{2}}$ and {${\hat{I}}^2>0$}, under a smallness assumption on the datum $\hat{I}»$. In order to study such problem, we will show several properties of the weighted, singular Robin eigenvalue problem \[ λ_{1,f,γ}(Ω)= \inf_{ψ\in H^{1},\;\int_Ωfψ^{2}=1}\left\{\int_Ω|\nabla ψ|^{2}dx+γ\int_{\partialΩ}ψ^{2}\right\}. \]