Existence and multiplicity of solutions for a p ( x ) -Kirchhoff type problem via variational techniques

A. Mokhtari; Toufik Moussaoui; D. O’Regan

Archivum Mathematicum (2015)

  • Volume: 051, Issue: 3, page 163-173
  • ISSN: 0044-8753

Abstract

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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 - a + b ∫ Ω 1 p ( x ) | ∇ u | p ( x ) d x div ( | ∇ u | p ( x ) - 2 ∇ u ) = f ( x , u ) , i n Ω u = 0 o n ∂ Ω , where Ω is a smooth open subset of ℝ N and p ∈ C ( Ω ¯ ) with N < p - = inf x ∈ Ω p ( x ) ≤ p + = sup x ∈ Ω p ( x ) < + ∞ , a , b are positive constants and f : Ω ¯ × ℝ → ℝ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

How to cite

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Mokhtari, A., Moussaoui, Toufik, and O’Regan, D.. "Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problem via variational techniques." Archivum Mathematicum 051.3 (2015): 163-173. <http://eudml.org/doc/271562>.

@article{Mokhtari2015,
abstract = {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.},
author = {Mokhtari, A., Moussaoui, Toufik, O’Regan, D.},
journal = {Archivum Mathematicum},
keywords = {existence results; genus theory; nonlocal problems Kirchhoff equation; critical point theory},
language = {eng},
number = {3},
pages = {163-173},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problem via variational techniques},
url = {http://eudml.org/doc/271562},
volume = {051},
year = {2015},
}

TY - JOUR
AU - Mokhtari, A.
AU - Moussaoui, Toufik
AU - O’Regan, D.
TI - Existence and multiplicity of solutions for a $p(x)$-Kirchhoff type problem via variational techniques
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 3
SP - 163
EP - 173
AB - 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.
LA - eng
KW - existence results; genus theory; nonlocal problems Kirchhoff equation; critical point theory
UR - http://eudml.org/doc/271562
ER -

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