Existence of solutions for 1-laplacian problems with singular first order terms

2025/02/05 by Francesco Balducci, Balducci, Francesco · 1 citation
Mathematics · #Nonlinear Differential Equations Analysis #Differential Equations and Numerical Methods #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2502.03050

Abstract

We prove existence of solutions to the following problem \begincases -Δ₁ u +g(u)|Du|=h(u)f amp; in Ω,
u=0 amp; on ∂Ω, \endcases where Ω⊂ ℝ^N, with N≥2, is an open and bounded set with Lipschitz boundary, g is a continuous and positive function which possibly blows up at the origin and bounded at infinity and h is a continuous and nonnegative function bounded at infinity (possibly blowing up at the origin) and finally 0 ≤ f ∈ L^N(Ω). As a by-product, this paper extends the results found where g is a continuous and bounded function. We investigate the interplay between g and h in order to have existence of solutions.

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