On the set of eigenvalues for some classes of coercive and noncoercive problems involving (2, p ( x ) )-Laplacian-like operators
We consider a class of double-phase nonlinear eigenvalue problems driven by a ( 2 , ϕ ) -Laplace-like operator: − Δ u − ε div [ ϕ ( x , | ∇ u | ) ∇ u ] = λ ( u + ε ) in a domain Ω , subject to Dirichlet boundary conditions, where Ω is a bounded subset of R N with a smooth boundary. Here, ε > 0...
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| Published in | Electronic journal of qualitative theory of differential equations Vol. 2025; no. 31; pp. 1 - 21 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
University of Szeged
2025
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1417-3875 1417-3875 |
| DOI | 10.14232/ejqtde.2025.1.31 |
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| Abstract | We consider a class of double-phase nonlinear eigenvalue problems driven by a ( 2 , ϕ ) -Laplace-like operator: − Δ u − ε div [ ϕ ( x , | ∇ u | ) ∇ u ] = λ ( u + ε ) in a domain Ω , subject to Dirichlet boundary conditions, where Ω is a bounded subset of R N with a smooth boundary. Here, ε > 0 , and the potential function ϕ exhibits p ( x ) -variable growth.
We establish several results on the existence and concentration of eigenvalues for this problem, focusing on the influence of the growth behavior of the potential function ϕ , specifically through the interaction between the variable growth exponent p ( x ) and the constant growth exponent 2 . The proofs rely on variational arguments based on the Direct Method in the Calculus of Variations, Ekeland's variational principle, and energy estimates. |
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| AbstractList | We consider a class of double-phase nonlinear eigenvalue problems driven by a ( 2 , ϕ ) -Laplace-like operator: − Δ u − ε div [ ϕ ( x , | ∇ u | ) ∇ u ] = λ ( u + ε ) in a domain Ω , subject to Dirichlet boundary conditions, where Ω is a bounded subset of R N with a smooth boundary. Here, ε > 0 , and the potential function ϕ exhibits p ( x ) -variable growth.
We establish several results on the existence and concentration of eigenvalues for this problem, focusing on the influence of the growth behavior of the potential function ϕ , specifically through the interaction between the variable growth exponent p ( x ) and the constant growth exponent 2 . The proofs rely on variational arguments based on the Direct Method in the Calculus of Variations, Ekeland's variational principle, and energy estimates. We consider a class of double-phase nonlinear eigenvalue problems driven by a $(2,\phi)$-Laplace-like operator: $$ -\Delta u - \varepsilon \operatorname{div}\left[\phi(x,|\nabla u|)\nabla u\right] = \lambda(u+\varepsilon) $$ in a domain $\Omega$, subject to Dirichlet boundary conditions, where $\Omega$ is a bounded subset of $\mathbb{R}^N$ with a smooth boundary. Here, $\varepsilon > 0$, and the potential function $\phi$ exhibits $p(x)$-variable growth. We establish several results on the existence and concentration of eigenvalues for this problem, focusing on the influence of the growth behavior of the potential function $\phi$, specifically through the interaction between the variable growth exponent $p(x)$ and the constant growth exponent $2$. The proofs rely on variational arguments based on the Direct Method in the Calculus of Variations, Ekeland's variational principle, and energy estimates. |
| Author | Uța, Vasile |
| Author_xml | – sequence: 1 givenname: Vasile orcidid: 0000-0002-4522-0117 surname: Uța fullname: Uța, Vasile |
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| Cites_doi | 10.1080/17476931003786709 10.1090/S0002-9939-07-08815-6 10.1070/IM1987v029n01ABEH000958 10.14232/ejqtde.2022.1.5 10.1515/crll.2005.2005.584.117 10.1007/s00245-016-9330-z 10.14232/ejqtde.2020.1.28 10.1515/ans-2016-6020 10.1007/s00229-014-0718-2 10.1186/s13662-019-2202-5 10.3934/dcdss.2018015 10.1016/j.crma.2007.10.012 10.1201/b18601 10.3390/sym13040633 10.2478/auom-2019-0015 10.1016/j.na.2014.07.015 10.1007/978-3-642-18363-8 10.1137/050624522 10.3934/cpaa.2005.4.9 10.3934/dcdss.2019026 10.1007/978-3-540-74013-1 10.14232/ejqtde.2020.1.10 10.1007/s00009-022-02097-0 10.14232/ejqtde.2017.1.98 10.1007/s002050000101 10.21136/CMJ.1991.102493 10.1007/s10898-007-9176-7 10.3390/sym13020264 10.1016/j.na.2014.11.002 10.1080/00036810802713826 10.1016/j.aml.2017.05.007 10.1007/s13540-024-00246-8 10.1016/0022-247X(74)90025-0 10.1016/j.na.2009.01.021 10.1186/1687-2770-2013-55 10.1016/j.jmaa.2007.09.015 10.1016/j.na.2018.03.016 10.1515/math-2021-0010 10.1016/j.na.2009.02.117 10.1007/s00009-023-02470-7 10.1016/j.na.2014.11.007 |
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| Title | On the set of eigenvalues for some classes of coercive and noncoercive problems involving (2, p ( x ) )-Laplacian-like operators |
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