Analysis demonstrates the solvability of fractional Kirchhoff problems in R^N, highlighting the new functional framework and mathematical properties.
Our primary objective is to study the solvability of two kinds of fractional Kirchhoff double-phase problem involving logarithmic nonlinearity in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> {{R}}N via the variational approach. To address the modulation coefficient in fractional double-phase operators, a new functional framework and the corresponding fractional Musielak-Sobolev space are provided. Among others, the completeness, reflexivity, and uniformly convexity of the space are established, and the continuity, boundedness, and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> ({S}₊) -property of the fractional double-phase operator are given.
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Cheng et al. (2025) studied this question.
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