Degenerate singular Kirchhoff problems in Musielak-Orlicz spaces

TL;DR

This paper establishes the existence of multiple solutions for complex Kirchhoff problems with non-homogeneous operators and diverse reaction terms in Musielak-Orlicz spaces, using advanced variational methods.

Contribution

It introduces a very general framework for Kirchhoff problems with unbalanced growth, providing shorter proofs and covering a wide class of differential operators.

Findings

Proved existence of at least two solutions under broad conditions.

Applicable to various operators like p-Laplacian, (p,q)-Laplacian, and double phase.

Simplified and more general proof techniques.

Abstract

In this paper we study quasilinear elliptic Kirchhoff equations driven by a non-homogeneous operator with unbalanced growth and right-hand sides that consist of sub-linear, possibly singular, and super-linear reaction terms. Under very general assumptions we prove the existence of at least two solutions for such problems by using the fibering method along with an appropriate splitting of the associated Nehari manifold. In contrast to other works our treatment is very general, with much easier and shorter proofs as it was done in the literature before. Furthermore, the results presented in this paper cover a large class of second-order differential operators like the $p$-Laplacian, the $(p,q)$-Laplacian, the double phase operator, and the logarithmic double phase operator.