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

Abstract

AbstractGiven a Musielak–Orlicz function$$\varphi (x,s):\Omega \times [0,\infty )\rightarrow {\mathbb R}$$φ(x,s):Ω×[0,∞)→Ron a bounded regular domain$$\Omega \subset {\mathbb R}^n$$Ω⊂Rnand a continuous function$$M:[0,\infty )\rightarrow (0,\infty )$$M:[0,∞)→(0,∞), we show that the eigenvalue problem for the elliptic Kirchhoff’s equation$$-M\left( \int \limits _{\Omega }\varphi (x,|\nabla u(x)|)\textrm{d}x\right) \text {div}\left( \frac{\partial \varphi }{\partial s}(x,|\nabla u(x)|)\frac{\nabla u(x)}{|\nabla u(x)|}\right) =\lambda \frac{\partial \varphi }{\partial s}(x,|u(x)|)\frac{u(x)}{|u(x)|} $$-M∫Ωφ(x,|∇u(x)|)dxdiv∂φ∂s(x,|∇u(x)|)∇u(x)|∇u(x)|=λ∂φ∂s(x,|u(x)|)u(x)|u(x)|has infinitely many solutions in the Sobolev space$$W_0^{1,\varphi }(\Omega )$$W01,φ(Ω). No conditions on$$\varphi $$φare required beyond those that guarantee the compactness of the Sobolev embedding theorem.