Ground state solutions and asymptotics for discrete Kirchhoff problems

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.  For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/77/abstr.html