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

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