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Volume 40, Issue 4
Eigenvalue Problem for a Class of Quasilinear Elliptic Operators with Mixed Boundary Value Condition in a Variable Exponent Sobolev Space

Junichi Aramaki

DOI: 10.4208/cmr.2024-0039

Commun. Math. Res., 40 (2024), pp. 437-481.

Published online: 2024-12

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  • Abstract

In this paper, we consider an eigenvalue problem for a class of nonlinear elliptic operators containing $p(·)$-Laplacian and 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 another part of the boundary. Using the Ljusternik-Schnirelmann variational method, we show the existence of infinitely many positive eigenvalues of the equation. Furthermore, under some conditions, we derive that the infimum of the set of all the eigenvalues becomes zero or remains to be positive.

  • Keywords

Eigenvalue problem, $p(·)$-Laplacian type equation, mean curvature operator, mixed boundary value problem.

  • AMS Subject Headings

49R50, 35A01, 35J62, 35J57

  • Copyright

COPYRIGHT: © Global Science Press

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  • BibTex
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  • TXT
@Article{CMR-40-437, author = {Aramaki , Junichi}, title = {Eigenvalue Problem for a Class of Quasilinear Elliptic Operators with Mixed Boundary Value Condition in a Variable Exponent Sobolev Space}, journal = {Communications in Mathematical Research }, year = {2024}, volume = {40}, number = {4}, pages = {437--481}, abstract = {

In this paper, we consider an eigenvalue problem for a class of nonlinear elliptic operators containing $p(·)$-Laplacian and 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 another part of the boundary. Using the Ljusternik-Schnirelmann variational method, we show the existence of infinitely many positive eigenvalues of the equation. Furthermore, under some conditions, we derive that the infimum of the set of all the eigenvalues becomes zero or remains to be positive.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2024-0039}, url = {http://global-sci.org/intro/article_detail/cmr/23701.html} }
TY - JOUR T1 - Eigenvalue Problem for a Class of Quasilinear Elliptic Operators with Mixed Boundary Value Condition in a Variable Exponent Sobolev Space AU - Aramaki , Junichi JO - Communications in Mathematical Research VL - 4 SP - 437 EP - 481 PY - 2024 DA - 2024/12 SN - 40 DO - http://doi.org/10.4208/cmr.2024-0039 UR - https://global-sci.org/intro/article_detail/cmr/23701.html KW - Eigenvalue problem, $p(·)$-Laplacian type equation, mean curvature operator, mixed boundary value problem. AB -

In this paper, we consider an eigenvalue problem for a class of nonlinear elliptic operators containing $p(·)$-Laplacian and 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 another part of the boundary. Using the Ljusternik-Schnirelmann variational method, we show the existence of infinitely many positive eigenvalues of the equation. Furthermore, under some conditions, we derive that the infimum of the set of all the eigenvalues becomes zero or remains to be positive.

Aramaki , Junichi. (2024). Eigenvalue Problem for a Class of Quasilinear Elliptic Operators with Mixed Boundary Value Condition in a Variable Exponent Sobolev Space. Communications in Mathematical Research . 40 (4). 437-481. doi:10.4208/cmr.2024-0039
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