# Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces

• Volume: 113, Issue: 3, page 283-294
• ISSN: 0066-2216

top

## Abstract

top
We consider Kirchhoff type problems of the form ⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω where $\Omega \subset {ℝ}^{N}$, N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, $\rho \left(u\right)={\int }_{\Omega }\left(\Phi \left(|\nabla u|\right)+\Phi \left(|u|\right)\right)dx$, M: [0,∞) → ℝ is a continuous function, $K\in {L}^{\infty }\left(\Omega \right)$, and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.

## How to cite

top

Nguyen Thanh Chung. "Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces." Annales Polonici Mathematici 113.3 (2015): 283-294. <http://eudml.org/doc/286230>.

@article{NguyenThanhChung2015,
abstract = {We consider Kirchhoff type problems of the form ⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω where $Ω ⊂ ℝ^\{N\}$, N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, $ρ(u)= ∫_\{Ω\} (Φ(|∇u|) + Φ(|u|))dx$, M: [0,∞) → ℝ is a continuous function, $K ∈ L^\{∞\}(Ω)$, and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.},
author = {Nguyen Thanh Chung},
journal = {Annales Polonici Mathematici},
keywords = {Kirchhoff type problems; Neumann boundary condition; Orlicz-Sobolev spaces; mountain pass theorem},
language = {eng},
number = {3},
pages = {283-294},
title = {Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces},
url = {http://eudml.org/doc/286230},
volume = {113},
year = {2015},
}

TY - JOUR
AU - Nguyen Thanh Chung
TI - Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces
JO - Annales Polonici Mathematici
PY - 2015
VL - 113
IS - 3
SP - 283
EP - 294
AB - We consider Kirchhoff type problems of the form ⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω ⎨ ⎩ ∂u/∂ν = 0 on ∂Ω where $Ω ⊂ ℝ^{N}$, N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, $ρ(u)= ∫_{Ω} (Φ(|∇u|) + Φ(|u|))dx$, M: [0,∞) → ℝ is a continuous function, $K ∈ L^{∞}(Ω)$, and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.
LA - eng
KW - Kirchhoff type problems; Neumann boundary condition; Orlicz-Sobolev spaces; mountain pass theorem
UR - http://eudml.org/doc/286230
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.