Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 17, pp. 1-30.

Title: Eigenvalue problems for Kirchhoff-type equations in variable exponent Sobolev spaces

Author: Junichi Aramaki (Tokyo Denki Univ., Saitama, Japan)
  
Abstract:
 In this article, we consider an eigenvalue problem for the Kirchhoff-type 
equation containing p(.)-Laplacian and the mean curvature operator 
with  mixed boundary conditions. More precisely, we are concerned with 
the problem with the Dirichlet condition on a part of the boundary and 
the Steklov boundary condition on an another part of the boundary. 
We show that the eigenvalue problem has infinitely many eigenpairs by 
using the celebrated Ljusternik-Schnirelmann principle in the calculus of 
variation. Moreover, we derive that in a variable exponent Sobolev space,  
there are two cases where the infimum of all eigenvalues is equal to  
zero and is positive.  

Submitted April 21, 2024. Published February 25, 2025.
Math Subject Classifications: 49R50, 35A01, 35J62, 35J57.
Key Words: Eigenvalue problem; Kirchhoff-type operator; p(.)-Laplacian;
  mean curvature operator;  mixed boundary value problem; variable exponent Sobolev space.