The first non-trivial Fučík curve for the sublinear Dirichlet Laplacian
Abstract
Let Ω⊂ℝN be a bounded open connected set, and let q∈(1,2). We consider the sublinear q-eigenvalue problem for the Dirichlet Laplacian −Δu=λ|u|Lq(Ω)2−q|u|q−2u, u∈ H10(Ω)∖{0}, whose first eigenvalue λ1(Ω) is simple, with eigenfunctions of constant sign. The definition of a second eigenvalue is not unique. Following the approach introduced by Molle and Passaseo for the linear Laplacian, we construct the first non-trivial curve of the corresponding Fučík spectrum, and show that it has a unique diagonal point $(γ(Ω),γ(Ω))$. The Fučík-selected second value λ2F(Ω)=21−2/qγ(Ω) is strictly larger than λ1(Ω), admits a two-phase variational characterisation, and satisfies a sharp Hong-Krahn-Szegő inequality: among sets of given measure it is minimised by the union of two equal balls. The selected solution at the diagonal point is a classical (2,q)-eigenfunction exactly when its nodal Lq-masses are balanced. We finally show that on balls in dimensions N≥2 every Fučík-selected solution is foliated Schwarz symmetric.
Related
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- On Faber-Krahn inequality for Dirichlet Fractional-Logarithmic Laplacian
- Arithmetic classification of the Dirichlet eigenvalues for a one-dimensional nonlocal p-Laplacian problem
- Weighted heat traces of the Dirichlet Laplacian on Lipschitz domains
- Explicit sharp gradient stability of the Lp-Poincaré inequality and fundamental gaps for the Dirichlet p-Laplacian operator
BibTeX
@misc{alhelou2026first,
title = {{The first non-trivial Fučík curve for the sublinear Dirichlet Laplacian}},
author = {Elie Alhelou and Gisella Croce and Giovanni Pisante},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03971},
eprint = {2611.03971},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03971},
}
RIS
TY - UNPB
TI - The first non-trivial Fučík curve for the sublinear Dirichlet Laplacian
AU - Elie Alhelou
AU - Gisella Croce
AU - Giovanni Pisante
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03971
UR - https://arxiv.org/abs/2611.03971
AB - Let Ω⊂ℝ^N be a bounded open connected set, and let q∈(1,2). We consider the sublinear q-eigenvalue problem for the Dirichlet Laplacian −Δu=λ|u|_L^q(Ω)^2−q|u|^q−2u, u∈ H^1_0(Ω)∖{0}, whose first eigenvalue λ_1(Ω) is simple, with eigenfunctions of constant sign. The definition of a second eigenvalue is not unique. Following the approach introduced by Molle and Passaseo for the linear Laplacian, we construct the first non-trivial curve of the corresponding Fučík spectrum, and show that it has a unique diagonal point $(γ(Ω),γ(Ω))$. The Fučík-selected second value λ_2^F(Ω)=2^1−2/qγ(Ω) is strictly larger than λ_1(Ω), admits a two-phase variational characterisation, and satisfies a sharp Hong-Krahn-Szegő inequality: among sets of given measure it is minimised by the union of two equal balls. The selected solution at the diagonal point is a classical (2,q)-eigenfunction exactly when its nodal L^q-masses are balanced. We finally show that on balls in dimensions N≥2 every Fučík-selected solution is foliated Schwarz symmetric.
ER -