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

2025/12/23 by Francesco Della Pietra, Della Pietra, Francesco, Giuseppina di Blasio +3
Mathematics · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Advanced Harmonic Analysis Research

paper · doi:10.48550/arxiv.2512.20192

Abstract

We prove an existence result for Robin boundary value problems modeled on \begincases Δu + |∇ u|² + λf(x) = 0 amp; in Ω
(∂ u)/(∂ ν) + βu = 0 amp; on ∂Ω\endcases 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∈ H¹, ∫Ωfψ²=1}\∫_Ω|∇ ψ|²dx+γ∫_∂Ωψ²\.

Citations

• Blow-up solutions for some nonlinear elliptic equations involving a\n Finsler-Laplacian

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