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

TL;DR

This work studies Robin boundary value problems with quadratic growth in the gradient and a general weight $f$ in the Marcinkiewicz space $M^{N/2}(\Omega)$. The authors connect solvability to a first weighted Robin eigenvalue $\lambda_{1,f,\gamma}(\Omega)$ and establish a smallness regime $λ < \frac{\lambda_{1,f}(\Omega)}{σ_0}$ under which a weak solution exists, with exponential-type integrability $e^{α|u|}$ in $H^1(\Omega)$. A concentration-compactness framework for weights in $M^{N/2}$ is developed to overcome non-compactness, together with a thorough analysis of $\lambda_{1,f,\gamma}(\Omega)$—its monotonicity, continuity, limits, and differentiability. The results extend the Dirichlet- and Hardy-type analyses to Robin problems with general growth and singular weights, providing robust variational tools and a priori estimates for nonlinear PDEs with gradient-critical terms.

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 \[ λ_{1,f,γ}(Ω)= \inf_{ψ\in H^{1},\;\int_Ωfψ^{2}=1}\left\{\int_Ω|\nabla ψ|^{2}dx+γ\int_{\partialΩ}ψ^{2}\right\}. \]

Theorems & Definitions (21)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Remark 2.3
• Example 1
• Lemma 2.4
• proof
• Proposition 3.1: Simplicity of the first eigenvalue
• proof