The eigenvalue problem for Kirchhoff-type operators in Musielak–Orlicz spaces

Abstract

Given a Musielak–Orlicz function \(\varphi (x,s):\Omega \times [0,\infty )\rightarrow {\mathbb R}\) on a bounded regular domain \(\Omega \subset {\mathbb R}^n\) and a continuous function \(M:[0,\infty )\rightarrow (0,\infty )\), we show that the eigenvalue problem for the elliptic Kirchhoff’s equation \(-M\left( \int \limits _{\Omega }\varphi (x,|\nabla u(x)|)\textrm{d}x\right) \text {div}\left( \frac{\partial \varphi }{\partial s}(x,|\nabla u(x)|)\frac{\nabla u(x)}{|\nabla u(x)|}\right) =\lambda \frac{\partial \varphi }{\partial s}(x,|u(x)|)\frac{u(x)}{|u(x)|} \) has infinitely many solutions in the Sobolev space \(W_0^{1,\varphi }(\Omega )\). No conditions on \(\varphi \) are required beyond those that guarantee the compactness of the Sobolev embedding theorem.