Behaviour of solutions to p -Laplacian with Robin boundary conditions as p goes to 1

2023/01/26 by Francesco Della Pietra, Francescantonio Oliva, Sergio Segura de León

paper · doi:10.1017/prm.2022.92

Abstract

We study the asymptotic behaviour, as p→ 1⁺ , of the solutions of the following inhomogeneous Robin boundary value problem: P \begincases -Δ_p u_p = f in Ω,
|∇ u_p|^p-2∇ u_p⋅ ν +λ |u_p|^p-2u_p = g on ∂Ω, \endcases where Ω is a bounded domain in \mathbb R^N with sufficiently smooth boundary, ν is its unit outward normal vector and Δ _p v is the p -Laplacian operator with p>1 . The data f∈ L^N,∞ (Ω ) (which denotes the Marcinkiewicz space) and λ, g are bounded functions defined on ∂ Ω with λ ≥ 0 . We find the threshold below which the family of p –solutions goes to 0 and above which this family blows up. As a second interest we deal with the 1 -Laplacian problem formally arising by taking p→ 1⁺ in (P).

Related

• Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1
• Some Results on the 1-Laplacian Elliptic Problems with Singularities and Robin Boundary Conditions
• On the behaviour of the first eigenvalue of the p-Laplacian with Robin\n boundary conditions as p goes to 1
• On the solutions to p-Poisson equation with Robin boundary conditions when p goes to +∞
• On the principal eigenvalue of a Robin problem with a large parameter