On a Brezis-Oswald-type result for degenerate Kirchhoff problems

Stefano Biagi; Eugenio Vecchi

doi:10.3934/dcds.2023122

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2024, Volume 44, Issue 3: 702-717. Doi: 10.3934/dcds.2023122

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On a Brezis-Oswald-type result for degenerate Kirchhoff problems

Stefano Biagi^1, , and
Eugenio Vecchi^2,

1.
Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy
2.
Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

^*Corresponding author: Stefano Biagi
^*Corresponding author: Stefano Biagi

Received: June 2023

Revised: October 2023

Early access: October 2023

Published: March 2024

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Abstract

In the present note we establish an almost-optimal solvability result for Kirchhoff-type problems of the following form

$ \begin{cases} -M\big(\|\nabla u\|^2_{L^2(\Omega)}\big)\Delta u = f(x, u) & \text{in } \Omega , \\ u \geq 0, \, u\not\equiv 0 & \text{in } \Omega , \\ u = 0 & \text{on } \partial \Omega . \end{cases} $

where $ f $ has sublinear growth and $ M $ is a non-decreasing map with $ M(0)\geq 0 $. Our approach is purely variational, and the result we obtain is resemblant to the one established by Brezis and Oswald (Nonlinear Anal., 1986) for sublinear elliptic equations.

Keywords:

Kirchhoff equation,
existence and uniqueness of weak solutions,
optimal solvability.

Mathematics Subject Classification: Primary: 35A01, 35R11.

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References

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