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

TL;DR

This paper establishes existence results for nonlinear elliptic Robin boundary value problems with general growth in the gradient, using properties of weighted Robin eigenvalues and small data assumptions.

Contribution

It introduces new existence results for elliptic equations with gradient growth and analyzes weighted Robin eigenvalue problems with singular weights.

Findings

Existence of solutions under small data conditions

Properties of weighted Robin eigenvalues established

Extension to general growth in gradient cases

Abstract

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