On the generalized Fredholm alternative for the p-Laplacian with resonant subhomogeneous terms
Abstract
Let 1 ≤ q < p and let λ1 be the first eigenvalue of the p-Laplacian in a bounded domain Ω. We study the energy functional Eλ(u)=1/p(∫Ω|∇ u|p dx −λ∫Ω|u|p dx)−ℱ(u), u∈ W01,p(Ω), where ℱ is positively q-homogeneous and vanishes along the first eigenspace. Assuming a suitable relation between ℱ and the κ-th power of the principal part of Eλ1 near this eigenspace, we describe the behavior of Eλ1 according to the relations pκ<q, pκ=q, or pκ>q. In particular, the functional is unbounded from below in the first case, and has a negative infimum in the last case. We then study how sufficiently small q-homogeneous perturbations of ℱ influence the geometry of Eλ. In this way, we describe assumptions guaranteeing the existence of three critical points in a left neighborhood of λ1 and two critical points in a right neighborhood of λ1, which indicates an S-shaped structure of the solution set. The results are applied to double-phase functionals and the nonlinear Fredholm alternative.
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BibTeX
@misc{bobkov2026generalized,
title = {{On the generalized Fredholm alternative for the $p$-Laplacian with resonant subhomogeneous terms}},
author = {Vladimir Bobkov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25768},
eprint = {2609.25768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25768},
}
RIS
TY - UNPB
TI - On the generalized Fredholm alternative for the p-Laplacian with resonant subhomogeneous terms
AU - Vladimir Bobkov
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25768
UR - https://arxiv.org/abs/2609.25768
AB - Let 1 ≤ q < p and let λ_1 be the first eigenvalue of the p-Laplacian in a bounded domain Ω. We study the energy functional E_λ(u)=1/p(∫_Ω|∇ u|^p dx −λ∫_Ω|u|^p dx)−ℱ(u), u∈ W_0^1,p(Ω), where ℱ is positively q-homogeneous and vanishes along the first eigenspace. Assuming a suitable relation between ℱ and the κ-th power of the principal part of E_λ_1 near this eigenspace, we describe the behavior of E_λ_1 according to the relations pκ<q, pκ=q, or pκ>q. In particular, the functional is unbounded from below in the first case, and has a negative infimum in the last case. We then study how sufficiently small q-homogeneous perturbations of ℱ influence the geometry of E_λ. In this way, we describe assumptions guaranteeing the existence of three critical points in a left neighborhood of λ_1 and two critical points in a right neighborhood of λ_1, which indicates an S-shaped structure of the solution set. The results are applied to double-phase functionals and the nonlinear Fredholm alternative.
ER -