On multiple solutions to a nonlocal fractional $p(\cdot )$-Laplacian problem with concave–convex nonlinearities

Abstract The aim of this paper is to examine the existence at least two distinct nontrivial solutions a Schrödinger-type problem involving nonlocal fractional $p(\cdot )$ p ( ⋅ ) -Laplacian with concave–convex nonlinearities when, in general, nonlinear term does not satisfy Ambrosetti–Rabinowitz condition. main tools for obtaining result are mountain pass theorem and modified version Ekeland’s variational principle an energy functional compactness condition Palais–Smale type, namely Cerami Also we discuss several results sequence infinitely many our problem. To achieve these results, employ fountain dual as tools.