Mohammed Belahcen, Siham El Habib, Najib Tsouli
Abstract:
While superlinear discrete Kirchhoff problems have been extensively studied
via variational minimax methods, the sublinear regime \(\sigma\in(1,2)\),
where the Kirchhoff energy dominates the reaction term, remains essentially
unexplored. Also the asymptotic behavior of ground states as \(\lambda\to0^+\)
has not been investigated even in the continuous setting. We address this
gap for the discrete Kirchhoff boundary value problem
$$\displaylines{
-m\Big(\sum_{j=0}^{N}|\delta y_j|^2\Big)\Delta_d y_j
=\lambda h_j(y_j), \quad j=1,\ldots,N,\cr
y_0=y_{N+1}=0.
}$$
Combining coercive minimization, constant-sign truncation, Krasnosel'skii
genus theory, and scaling arguments, we establish existence of a nontrivial
ground state for every \(\lambda >0\), two constant-sign solutions, and \(N\)
pairs of nontrivial solutions under an evenness assumption. We further
derive the sharp scaling \(\inf\mathcal J_\lambda\asymp-\lambda^{\frac{2}{2-\sigma}}\),
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.
Submitted July 16, 2026. Published September 30, 2026.
Math Subject Classifications: 39A10, 35J60, 49J40.
Key Words: Asymptotic profile; convergence rate; discrete Kirchhoff problem; ground states; variational methods.
DOI: 10.58997/ejde.2026.77
Show me the PDF file (452 KB), TEX file for this article.
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Mohammed Belahcen Department of Mathematics LAMAO Laboratory, Faculty of Sciences Mohammed I University 60000 Oujda, Morocco email: mohammed.belahcen.d24@ump.ac.ma |
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Siham El Habib Department of Mathematics LAMAO Laboratory, Faculty of Sciences Mohammed I University 60000 Oujda, Morocco email: s.elhabib@ump.ac.ma |
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Najib Tsouli Department of Mathematics LAMAO Laboratory, Faculty of Sciences Mohammed I University 60000 Oujda, Morocco email: n.tsouli@ump.ac.ma |
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