A multiplicity result in finite-dimensional vector spaces

Abstract

Let $f_1,\dots ,f_n$ be n ($n\ge 2$) continuous real-valued functions on $\mathbf{R}$ such that $\underset{|t|\to +\mathrm{\infty }}{lim}\frac{{\int }_0^t{f}_k(s)\mathrm{d}s}{t^2}=-\mathrm{\infty }$for all $k=1,\dots ,n$. This sole condition is far from ensuring the existence of multiple solutions for the classical problem $\{\begin{array}{l}-(x_{k+1}-2x_k+x_{k-1})=f_k(x_k)k=1,\dots ,n,\\ x_0=x_{n+1}=0.\end{array}$However, as a by-product of a much more general result, we get the following: for each $\rho \in \mathbf{R}$ and for each $i=1,\dots ,n$, there exists $(\lambda ,\mu )\in {\mathbf{R}}^2$ such that the problem $\{\begin{array}{l}-\rho (x_{k+1}-2x_k+x_{k-1})=f_k(x_k)k=1,\dots ,n,k\ne i\\ -\rho (x_{i+1}-2x_i+x_{i-1})=f_i(x_i)+\lambda x_i+\mu \\ x_0=x_{n+1}=0\end{array}$has at least three solutions.

Keywords:

• Quadratic forms
• inner products
• minimax theorems
• difference equations
• variational methods
• multiplicity of solutions

Mathematics Subject Classifications:

• 15A63
• 39A27

Acknowledgments

The author wishes to thank the referees and the managing editor for their comments and remarks.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author has been supported by PRIN 2022BCFHN2 “Advanced theoretical aspects in PDEs and their applications”, by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by the Università degli Studi di Catania, PIACERI 2020-2022, Linea di intervento 2, Progetto “MAFANE”.