Infinitely Many Small Energy Solutions to Schrödinger-Kirchhoff Type Problems Involving the Fractional <inline-formula><math display="inline"><semantics><mrow><mi>r</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></semantics></math></inline-formula>-Laplacian in <inline-formula><math display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></mrow></semantics></math></inline-formula>

Abstract

This paper is concerned with the existence result of a sequence of infinitely many small energy solutions to the fractional r(·)-Laplacian equations of Kirchhoff–Schrödinger type with concave–convex nonlinearities when the convex term does not require the Ambrosetti–Rabinowitz condition. The aim of the present paper, under suitable assumptions on a nonlinear term, is to discuss the multiplicity result of non-trivial solutions by using the dual fountain theorem as the main tool.

Keywords

• fractional <i>r</i>(<i>x</i>)-Laplacian
• variable exponent fractional Sobolev spaces
• weak solution
• dual fountain theorem