| Title:
|
Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problem via variational techniques (English) |
| Author:
|
Mokhtari, A. |
| Author:
|
Moussaoui, T. |
| Author:
|
O’Regan, D. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
51 |
| Issue:
|
3 |
| Year:
|
2015 |
| Pages:
|
163-173 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper discusses the existence and multiplicity of solutions for a class of $p(x)$-Kirchhoff type problems with Dirichlet boundary data of the following form \[ {\left\rbrace \begin{array}{ll} -\Big (a+b\int _{\Omega }\frac{1}{p(x)}|\nabla u|^{p(x)}\; dx\Big )\textrm{div}\big (|\nabla u|^{p(x)-2 } \nabla u\big )= f(x,u)\,, & in \quad \Omega \\[6pt] u=0 & on \quad \partial \Omega\,, \end{array}\right.} \] where $\Omega $ is a smooth open subset of $\mathbb{R}^N$ and $p\in C(\overline{\Omega })$ with $N <p^-= \inf _{x\in \Omega } p(x)\le p^+= \sup _{x\in \Omega } p(x)<+\infty $, $a$, $b$ are positive constants and $f\colon \overline{\Omega }\times \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory. (English) |
| Keyword:
|
existence results |
| Keyword:
|
genus theory |
| Keyword:
|
nonlocal problems Kirchhoff equation |
| Keyword:
|
critical point theory |
| MSC:
|
34B27 |
| MSC:
|
35B05 |
| MSC:
|
35J60 |
| idZBL:
|
Zbl 06487028 |
| idMR:
|
MR3397269 |
| DOI:
|
10.5817/AM2015-3-163 |
| . |
| Date available:
|
2015-09-09T09:46:37Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144427 |
| . |
| Reference:
|
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