Electron. J. Differential Equations, Vol. 2017 (2017), No. 228, pp. 1-12.

Sequences of small homoclinic solutions for difference equations on integers

Robert Steglinski

Abstract:
In this article, we determine a concrete interval of positive parameters $\lambda $, for which we prove the existence of infinitely many homoclinic solutions for a discrete problem
$$ -\Delta \big( a(k)\phi _{p}(\Delta u(k-1))\big) +b(k)\phi_{p}(u(k)) =\lambda f(k,u(k)),\quad k\in \mathbb{Z}, $$
where the nonlinear term $f:\mathbb{Z}\times \mathbb{R} \to \mathbb{R}$ has an appropriate oscillatory behavior at zero. We use both the general variational principle of Ricceri and the direct method introduced by Faraci and Kristaly [11].

Submitted July 18, 2017. Published September 22, 2017.
Math Subject Classifications: 39A10, 47J30, 35B38.
Key Words: Difference equations; discrete p-Laplacian; variational methods; infinitely many solutions.

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Robert Steglinski
Institute of Mathematics
Lodz University of Technology
Wolczanska 215, 90-924 Lodz, Poland
email: robert.steglinski@p.lodz.pl

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