Existence and multiplicity of solutions for a fractional p -Laplacian problem of Kirchhoff type via Krasnoselskii’s genus

Ghania Benhamida; Toufik Moussaoui

Mathematica Bohemica (2018)

  • Volume: 143, Issue: 2, page 189-200
  • ISSN: 0862-7959

Abstract

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We use the genus theory to prove the existence and multiplicity of solutions for the fractional p -Kirchhoff problem - M ∫ Q | u ( x ) - u ( y ) | p | x - y | N + p s d x d y p - 1 ( - Δ ) p s u = λ h ( x , u ) in Ω , u = 0 on ℝ N ∖ Ω , where Ω is an open bounded smooth domain of ℝ N , p > 1 , N > p s with s ∈ ( 0 , 1 ) fixed, Q = ℝ 2 N ∖ ( C Ω × C Ω ) , λ > 0 is a numerical parameter, M and h are continuous functions.

How to cite

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Benhamida, Ghania, and Moussaoui, Toufik. "Existence and multiplicity of solutions for a fractional $p$-Laplacian problem of Kirchhoff type via Krasnoselskii’s genus." Mathematica Bohemica 143.2 (2018): 189-200. <http://eudml.org/doc/294212>.

@article{Benhamida2018,
abstract = {We use the genus theory to prove the existence and multiplicity of solutions for the fractional $p$-Kirchhoff problem \[ \{\left\lbrace \begin\{array\}\{ll\} \displaystyle -\biggl [M \biggl (\int \_\{Q\}\dfrac\{\vert u(x)-u(y)\vert ^\{p\}\}\{\vert x-y \vert ^\{N+ps\}\} \{\rm d\}x \{\rm d\}y\biggr )\biggr ]^\{p-1\} (-\Delta )\_\{p\}^\{s\}u=\lambda h(x,u) \quad \text\{in\}\ \Omega , \\ u=0 \quad \text\{on\}\ \mathbb \{R\}^N \setminus \Omega , \end\{array\}\right.\} \] where $\Omega $ is an open bounded smooth domain of $\mathbb \{R\}^N$, $p>1$, $N>ps$ with $s\in (0,1)$ fixed, $Q = \mathbb \{R\}^\{2N\}\setminus (C\Omega \times C\Omega )$, $\lambda > 0$ is a numerical parameter, $M$ and $h$ are continuous functions.},
author = {Benhamida, Ghania, Moussaoui, Toufik},
journal = {Mathematica Bohemica},
keywords = {existence results; genus theory; fractional $p$-Kirchhoff problem},
language = {eng},
number = {2},
pages = {189-200},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence and multiplicity of solutions for a fractional $p$-Laplacian problem of Kirchhoff type via Krasnoselskii’s genus},
url = {http://eudml.org/doc/294212},
volume = {143},
year = {2018},
}

TY - JOUR
AU - Benhamida, Ghania
AU - Moussaoui, Toufik
TI - Existence and multiplicity of solutions for a fractional $p$-Laplacian problem of Kirchhoff type via Krasnoselskii’s genus
JO - Mathematica Bohemica
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 143
IS - 2
SP - 189
EP - 200
AB - We use the genus theory to prove the existence and multiplicity of solutions for the fractional $p$-Kirchhoff problem \[ {\left\lbrace \begin{array}{ll} \displaystyle -\biggl [M \biggl (\int _{Q}\dfrac{\vert u(x)-u(y)\vert ^{p}}{\vert x-y \vert ^{N+ps}} {\rm d}x {\rm d}y\biggr )\biggr ]^{p-1} (-\Delta )_{p}^{s}u=\lambda h(x,u) \quad \text{in}\ \Omega , \\ u=0 \quad \text{on}\ \mathbb {R}^N \setminus \Omega , \end{array}\right.} \] where $\Omega $ is an open bounded smooth domain of $\mathbb {R}^N$, $p>1$, $N>ps$ with $s\in (0,1)$ fixed, $Q = \mathbb {R}^{2N}\setminus (C\Omega \times C\Omega )$, $\lambda > 0$ is a numerical parameter, $M$ and $h$ are continuous functions.
LA - eng
KW - existence results; genus theory; fractional $p$-Kirchhoff problem
UR - http://eudml.org/doc/294212
ER -

References

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