Three solutions for a double phase variable exponent Kirchhoff problem

TL;DR

This paper investigates a complex double-phase variable-exponent Kirchhoff problem, establishing the existence of at least three solutions by developing new regularity results and a novel vector inequality for the energy functional.

Contribution

It introduces a new vector inequality and regularity analysis for a double-phase Kirchhoff problem, enabling the application of critical point theory to find multiple solutions.

Findings

Proved the existence of at least three solutions.

Developed a new vector inequality for energy functional analysis.

Established regularity and compactness properties of the energy functional.

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.