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

Abstract

Abstract 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 $p(\cdot )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo>)</mml:mo> </mml:math> -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.