**Stepan Tersian**

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences (BAS), 1113 Sofia, Bulgaria; sterzian@uni-ruse.bg

Received: 17 March 2020; Accepted: 1 April 2020; Published: 2 April 2020

**Abstract:** The existence of infinitely many homoclinic solutions for the fourth-order differential equation *ϕp* (*u* (*t*)) + *w ϕp* (*u* (*t*)) + *<sup>V</sup>*(*t*)*ϕp* (*u* (*t*)) = *<sup>a</sup>*(*t*)*f*(*<sup>t</sup>*, *<sup>u</sup>*(*t*)), *t* ∈ R is studied in the paper. Here *<sup>ϕ</sup>p*(*t*) = |*t*|*<sup>p</sup>*−<sup>2</sup> *t*, *p* ≥ 2, *w* is a constant, *V* and *a* are positive functions, *f* satisfies some extended growth conditions. Homoclinic solutions *u* are such that *u*(*t*) → 0, |*t*| → <sup>∞</sup>, *u* = 0, known in physical models as ground states or pulses. The variational approach is applied based on multiple critical point theorem due to Liu and Wang.

**Keywords:** homoclinic solutions; fourth-order p-Laplacian differential equations; minimization theorem; Clark's theorem
