Ground state solutions and asymptotics for discrete Kirchhoff problems
- 1Mohammed I Univ., Oujda, Morocco
Abstract
While superlinear discrete Kirchhoff problems have been extensively studied via variational minimax methods, the sublinear regime σ∈(1,2), where the Kirchhoff energy dominates the reaction term, remains essentially unexplored. Also the asymptotic behavior of ground states as λ→0+ has not been investigated even in the continuous setting. We address this gap for the discrete Kirchhoff boundary value problem displaylines −m(∑j=0N|δ yj|2)Δd yj =λ hj(yj), j=1,…,N,cr y0=yN+1=0. Combining coercive minimization, constant-sign truncation, Krasnosel'skii genus theory, and scaling arguments, we establish existence of a nontrivial ground state for every λ >0, two constant-sign solutions, and N pairs of nontrivial solutions under an evenness assumption. We further derive the sharp scaling inf𝒥λasymp−λ2/(2−σ), show that normalized ground states converge to minimizers of an explicit limit functional, and obtain, to our knowledge, the first quantitative convergence rate for this class of problems. Newton-continuation computations confirm the predicted scaling exponents to high precision.
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BibTeX
@article{belahcen2026ground,
title = {{Ground state solutions and asymptotics for discrete Kirchhoff problems}},
author = {Mohammed Belahcen and Siham El Habib and Najib Tsouli},
journal = {Electronic Journal of Differential Equations},
year = {2026},
month = sep,
url = {http://ejde.math.txstate.edu/Volumes/2026/77/abstr.html},
note = {CC BY},
}
RIS
TY - JOUR
TI - Ground state solutions and asymptotics for discrete Kirchhoff problems
AU - Mohammed Belahcen
AU - Siham El Habib
AU - Najib Tsouli
PY - 2026
DA - 2026/09/01
JO - Electronic Journal of Differential Equations
UR - http://ejde.math.txstate.edu/Volumes/2026/77/abstr.html
AB - While superlinear discrete Kirchhoff problems have been extensively studied via variational minimax methods, the sublinear regime σ∈(1,2), where the Kirchhoff energy dominates the reaction term, remains essentially unexplored. Also the asymptotic behavior of ground states as λ→0^+ has not been investigated even in the continuous setting. We address this gap for the discrete Kirchhoff boundary value problem displaylines −m(∑_j=0^N|δ y_j|^2)Δ_d y_j =λ h_j(y_j), j=1,…,N,cr y_0=y_N+1=0. Combining coercive minimization, constant-sign truncation, Krasnosel'skii genus theory, and scaling arguments, we establish existence of a nontrivial ground state for every λ >0, two constant-sign solutions, and N pairs of nontrivial solutions under an evenness assumption. We further derive the sharp scaling inf𝒥_λasymp−λ^2/(2−σ), show that normalized ground states converge to minimizers of an explicit limit functional, and obtain, to our knowledge, the first quantitative convergence rate for this class of problems. Newton-continuation computations confirm the predicted scaling exponents to high precision.
SN - 1072-6691
ER -