Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term
We study the following class of double-phase nonlinear eigenvalue problems $$ -\operatorname{div}\left[\phi(x,|\nabla u|)\nabla u+\psi(x,|\nabla u|)\nabla u\right]=\lambda f(x,u) $$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain from $\mathbb{R}^N$ and the potential funct...
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| Format: | Article |
| Language: | English |
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University of Szeged
2022-01-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=9554 |
| _version_ | 1797830471107739648 |
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| author | Vasile Uța |
| author_facet | Vasile Uța |
| author_sort | Vasile Uța |
| collection | DOAJ |
| description | We study the following class of double-phase nonlinear eigenvalue problems
$$
-\operatorname{div}\left[\phi(x,|\nabla u|)\nabla u+\psi(x,|\nabla u|)\nabla u\right]=\lambda f(x,u)
$$
in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain from $\mathbb{R}^N$ and the potential functions $\phi$ and $\psi$ have $(p_1(x);p_2(x))$ variable growth. The primitive of the reaction term of the problem (the right-hand side) has indefinite sign in the variable $u$ and allows us to study functions with slower growth near $+\infty$, that is, it does not satisfy the Ambrosetti–Rabinowitz condition. Under these hypotheses we prove that for every parameter $\lambda\in \mathbb{R}^*_+$, the problem has an unbounded sequence of weak solutions. The proofs rely on variational arguments based on energy estimates and the use of Fountain Theorem. |
| first_indexed | 2024-04-09T13:36:37Z |
| format | Article |
| id | doaj.art-dc0a6551a2ec4b078c0d7d15c788677c |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:36:37Z |
| publishDate | 2022-01-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-dc0a6551a2ec4b078c0d7d15c788677c2023-05-09T07:53:11ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752022-01-012022512210.14232/ejqtde.2022.1.59554Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction termVasile Uța0University of Craiova, RomaniaWe study the following class of double-phase nonlinear eigenvalue problems $$ -\operatorname{div}\left[\phi(x,|\nabla u|)\nabla u+\psi(x,|\nabla u|)\nabla u\right]=\lambda f(x,u) $$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain from $\mathbb{R}^N$ and the potential functions $\phi$ and $\psi$ have $(p_1(x);p_2(x))$ variable growth. The primitive of the reaction term of the problem (the right-hand side) has indefinite sign in the variable $u$ and allows us to study functions with slower growth near $+\infty$, that is, it does not satisfy the Ambrosetti–Rabinowitz condition. Under these hypotheses we prove that for every parameter $\lambda\in \mathbb{R}^*_+$, the problem has an unbounded sequence of weak solutions. The proofs rely on variational arguments based on energy estimates and the use of Fountain Theorem.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=9554double-phase differential operatorcontinuous spectrumvariable exponentmultiplicity of eigenvaluesinfinitely many solutions |
| spellingShingle | Vasile Uța Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term Electronic Journal of Qualitative Theory of Differential Equations double-phase differential operator continuous spectrum variable exponent multiplicity of eigenvalues infinitely many solutions |
| title | Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term |
| title_full | Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term |
| title_fullStr | Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term |
| title_full_unstemmed | Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term |
| title_short | Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term |
| title_sort | existence and multiplicity of eigenvalues for some double phase problems involving an indefinite sign reaction term |
| topic | double-phase differential operator continuous spectrum variable exponent multiplicity of eigenvalues infinitely many solutions |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=9554 |
| work_keys_str_mv | AT vasileuta existenceandmultiplicityofeigenvaluesforsomedoublephaseproblemsinvolvinganindefinitesignreactionterm |