Three Solutions for a Double-Phase Variable-Exponent Kirchhoff Problem

Mustafa Avci ()
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Mustafa Avci: Faculty of Science and Technology, Applied Mathematics, Athabasca University, Athabasca, AB T9S 3A3, Canada

Mathematics, 2025, vol. 13, issue 15, 1-20

Abstract: In this article, we study a double-phase variable-exponent Kirchhoff problem and show the existence of at least three solutions. The proposed model, as a generalization of the Kirchhoff equation, is interesting since it is driven by a double-phase operator that governs anisotropic and heterogeneous diffusion associated with the energy functional, as well as encapsulating two different types of elliptic behavior within the same framework. To tackle the problem, we obtain regularity results for the corresponding energy functional, which makes the problem suitable for the application of a well-known critical point result by Bonanno and Marano. We introduce an n -dimensional vector inequality, not covered in the literature, which provides a key auxiliary tool for establishing essential regularity properties of the energy functional such as C 1 -smoothness, the ( S + ) -condition, and sequential weak lower semicontinuity.

Keywords: double-phase variable-exponent problem; p( x )-Kirchhoff problem; critical point theory; Musielak–Orlicz Sobolev space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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