On sequences of homoclinic solutions for fractional discrete $ p $-Laplacian equations

Abstract

<abstract><p>In this paper, we consider the following discrete fractional $ p $-Laplacian equations:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} (-\Delta_{1})^{s}_{p}u(a)+V(a)|u(a)|^{p-2}u(a) = \lambda f(a, u(a)), \; \mbox{in}\ \mathbb{Z}, \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ \lambda $ is the parameter and $ f(a, u(a)) $ satisfies no symmetry assumption. As a result, a specific positive parameter interval is determined by some requirements for the nonlinear term near zero, and then infinitely many homoclinic solutions are obtained by using a special version of Ricceri's variational principle.</p></abstract>

On homoclinic solutions of nonlinear Laplacian partial difference equations with a parameter

“…Nonlinear difference equations, which appear as discrete counterparts and as numerical solutions of differential equations, are used in many fields to model diverse phenomena [20]. Since 2003, Guo and Yu [6] first applied the variational method to difference equations, there have emerged many results, which involve homoclinic solutions [14,8], heteroclinic solutions [9], solutions of boundary value problems [3,19] and so on.…”