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

Abstract

Abstract Given a Musielak–Orlicz function $$\varphi (x,s):\Omega \times [0,\infty )\rightarrow {\mathbb R}$$ φ ( x , s ) : Ω × [ 0 , ∞ ) → R on a bounded regular domain $$\Omega \subset {\mathbb R}^n$$ Ω ⊂ R n and 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 ∫ Ω φ ( x , | ∇ u ( x ) | ) d x div ∂ 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 $$W_0^{1,\varphi }(\Omega )$$ W 0 1 , φ ( Ω ) . No conditions on $$\varphi $$ φ are required beyond those that guarantee the compactness of the Sobolev embedding theorem.

Keywords

• Variable exponent p-Laplacian
• Sobolev embedding
• Modular spaces
• Musielak–Orlicz spaces
• Variable exponent spaces
• Kirchhoff equations