INFINITELY MANY HOMOCLINIC ORBITS OF SECOND-ORDER $p$-LAPLACIAN SYSTEMS

Abstract

In this paper, we give several new sufficient conditions for the existence of infinitely many homoclinic orbits of the second-order ordinary $p$-Laplacian system $$ \frac{d}{dt} \left(|\dot{u}(t)|^{p-2} \dot{u}(t)\right) - a(t) |u(t)|^{p-2} u(t) + \nabla W(t,u(t)) = 0, $$ where $p \gt 1, \ t\in {\mathbb{R}},\ u\in {\mathbb{R}}^{N}$, $a\in C({\mathbb{R}}, {\mathbb{R}})$ and $W\in C^{1}({\mathbb{R}}\times {\mathbb{R}}^{N}, {\mathbb{R}})$ are no periodic in $t$, which greatly improve the known results due to Rabinowitz and Willem.