Existence and non-existence of solutions for a p(x)-biharmonic problem
In this article, we study the following problem with Navier boundary conditions $$\displaylines{ \Delta (|\Delta u|^{p(x)-2}\Delta u)+|u|^{p(x)-2}u =\lambda |u|^{q(x)-2}u +\mu|u|^{\gamma(x)-2}u\quad \text{in } \Omega,\cr u=\Delta u=0 \quad \text{on } \partial\Omega. }$$ where $\Omega$ is a b...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-06-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/158/abstr.html |
| _version_ | 1818766533755666432 |
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| author | Ghasem A. Afrouzi Maryam Mirzapour Nguyen Thanh Chung |
| author_facet | Ghasem A. Afrouzi Maryam Mirzapour Nguyen Thanh Chung |
| author_sort | Ghasem A. Afrouzi |
| collection | DOAJ |
| description | In this article, we study the following problem with Navier boundary
conditions
$$\displaylines{
\Delta (|\Delta u|^{p(x)-2}\Delta u)+|u|^{p(x)-2}u
=\lambda |u|^{q(x)-2}u +\mu|u|^{\gamma(x)-2}u\quad \text{in } \Omega,\cr
u=\Delta u=0 \quad \text{on } \partial\Omega.
}$$
where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary
$\partial \Omega$, $N\geq1$. $p(x),q(x)$ and $\gamma(x)$ are continuous
functions on $\overline{\Omega}$, $\lambda$ and $\mu$ are parameters.
Using variational methods, we establish some existence and non-existence
results of solutions for this problem. |
| first_indexed | 2024-12-18T08:35:30Z |
| format | Article |
| id | doaj.art-0ae1cb6e1f374abe97455fb1f05cf0d9 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-18T08:35:30Z |
| publishDate | 2015-06-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-0ae1cb6e1f374abe97455fb1f05cf0d92022-12-21T21:14:20ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912015-06-012015158,18Existence and non-existence of solutions for a p(x)-biharmonic problemGhasem A. Afrouzi0Maryam Mirzapour1Nguyen Thanh Chung2 Univ. of Mazandaran, Babolsar, Iran Univ. of Mazandaran, Babolsar, Iran Quang Binh Univ., Dong Hoi, Vietnam In this article, we study the following problem with Navier boundary conditions $$\displaylines{ \Delta (|\Delta u|^{p(x)-2}\Delta u)+|u|^{p(x)-2}u =\lambda |u|^{q(x)-2}u +\mu|u|^{\gamma(x)-2}u\quad \text{in } \Omega,\cr u=\Delta u=0 \quad \text{on } \partial\Omega. }$$ where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary $\partial \Omega$, $N\geq1$. $p(x),q(x)$ and $\gamma(x)$ are continuous functions on $\overline{\Omega}$, $\lambda$ and $\mu$ are parameters. Using variational methods, we establish some existence and non-existence results of solutions for this problem.http://ejde.math.txstate.edu/Volumes/2015/158/abstr.htmlp(x)-Biharmonicvariable exponentcritical pointsminimum principlefountain theoremdual fountain theorem |
| spellingShingle | Ghasem A. Afrouzi Maryam Mirzapour Nguyen Thanh Chung Existence and non-existence of solutions for a p(x)-biharmonic problem Electronic Journal of Differential Equations p(x)-Biharmonic variable exponent critical points minimum principle fountain theorem dual fountain theorem |
| title | Existence and non-existence of solutions for a p(x)-biharmonic problem |
| title_full | Existence and non-existence of solutions for a p(x)-biharmonic problem |
| title_fullStr | Existence and non-existence of solutions for a p(x)-biharmonic problem |
| title_full_unstemmed | Existence and non-existence of solutions for a p(x)-biharmonic problem |
| title_short | Existence and non-existence of solutions for a p(x)-biharmonic problem |
| title_sort | existence and non existence of solutions for a p x biharmonic problem |
| topic | p(x)-Biharmonic variable exponent critical points minimum principle fountain theorem dual fountain theorem |
| url | http://ejde.math.txstate.edu/Volumes/2015/158/abstr.html |
| work_keys_str_mv | AT ghasemaafrouzi existenceandnonexistenceofsolutionsforapxbiharmonicproblem AT maryammirzapour existenceandnonexistenceofsolutionsforapxbiharmonicproblem AT nguyenthanhchung existenceandnonexistenceofsolutionsforapxbiharmonicproblem |