Degenerate singular Kirchhoff problems in Musielak-Orlicz spaces
Umberto Guarnotta, Patrick Winkert
TL;DR
The paper investigates quasilinear elliptic Kirchhoff equations with a non-homogeneous operator in Musielak–Orlicz spaces, allowing unbalanced growth and nonlinearities that can be singular or super-linear. It develops a variational framework based on the fibering method and a splitting of the Nehari manifold into $\mathcal{N}^+$, $\mathcal{N}^-$, and $\mathcal{N}^0$ to obtain at least two weak solutions with opposite energy signs for small $\lambda>0$. The analysis covers a broad class of second-order operators, including the $p$-Laplacian, $(p,q)$-Laplacian, double-phase, and logarithmic double-phase operators, under general assumptions that avoid Hardy-type inequalities. The results provide shorter, more general proofs and extend the existence theory for Kirchhoff problems with nonstandard growth to models describing materials with complex microstructure.
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.