Existence and multiplicity of solutions for a Schrödinger–Kirchhoff type equation involving the fractional $$p\left( .,.\right)$$-Laplacian operator in $${\mathbb {R}}^{N}$$

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In this paper, by using variational approach, Mountain Pass Theorem and Krasnoselskii’s genus theory, we show the existence and multiplicity of solutions for a Schrodinger–Kirchhoff type equation involving the fractional $$p\left( .,.\right)$$ -Laplacian in fractional Sobolev space with variable exponent. We also establish a Bartsch–Wang type compact embedding theorem for fractional Sobolev space with variable exponent.