Geodesic
Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problem via variational techniques
Archivum mathematicum, Tome 51 (2015) no. 3, pp. 163-173 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $, $a$, $b$ 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.
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 $, $a$, $b$ 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.
DOI : 10.5817/AM2015-3-163
Classification : 34B27, 35B05, 35J60
Keywords: existence results; genus theory; nonlocal problems Kirchhoff equation; critical point theory 
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     author = {Mokhtari, A. and Moussaoui, T. and O{\textquoteright}Regan, D.},
     title = {Existence and multiplicity of solutions for a $p(x)${-Kirchhoff} type problem via variational techniques},
     journal = {Archivum mathematicum},
     pages = {163--173},
     year = {2015},
     volume = {51},
     number = {3},
     doi = {10.5817/AM2015-3-163},
     mrnumber = {3397269},
     zbl = {06487028},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2015-3-163/}
}
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Mokhtari, A.; Moussaoui, T.; O’Regan, D. Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problem via variational techniques. Archivum mathematicum, Tome 51 (2015) no. 3, pp. 163-173. doi: 10.5817/AM2015-3-163

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