<abstract xmlns="http://www.w3.org/1999/xhtml">

In this work, we study the existence of at least one non decreasing sequence of nonnegative eigenvalues for the problem:
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0004_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>{</mo> <mrow><mtable><mtr><mtd><mrow><msup><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn>2</mn><mi>p</mi></mrow></msup><mi>u</mi><mo>=</mo><mi>λ</mi><mi>m</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>u</mi><mi> </mi><mi> </mi><mi> </mi><mi>i</mi><mi>n</mi><mi> </mi><mi> </mi><mi> </mi><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>∂</mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">Δ</mi><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>∂</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mo>…</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mrow><mo>(</mo><mrow><msup><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mn>2</mn><mi>p</mi><mo>-</mo><mn>1</mn></mrow></msup><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>∂</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mi> </mi><mi> </mi><mi> </mi><mi>o</mi><mi>n</mi><mi> </mi><mi> </mi><mi> </mi><mo>∂</mo><mi mathvariant="normal">Ω</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{ {{\Delta ^{2p}}u = \lambda m\left( x \right)u\,\,\,in\,\,\,\Omega ,} \cr {{{\partial u} \over {\partial v}} = {{\partial \left( {\Delta u} \right)} \over {\partial v}} = \ldots = {{\partial \left( {{\Delta ^{2p - 1}}u} \right)} \over {\partial v}} = 0\,\,\,on\,\,\,\partial \Omega .} \cr } } \right.</tex-math>
</alternatives>
</disp-formula>
Where Ω is a bounded domain in ℝ<italic>N</italic> with smooth boundary ∂ Ω, <italic>p</italic> ∈ ℕ*, <italic>m</italic> ∈ <italic>L</italic>∞ (Ω), and Δ2<italic>p</italic><italic>u</italic> := Δ (Δ...( Δ<italic>u</italic>)), 2<italic>p</italic> times the operator Δ.
</abstract>

In this work, we study the existence of at least one non decreasing sequence of nonnegative eigenvalues for the problem:



{ Δ2pu=λm(x)u   in   Ω,∂u∂v=∂(Δu)∂v=…=∂(Δ2p-1u)∂v=0   on   ∂Ω. 
\left\{ {\matrix{   {{\Delta ^{2p}}u = \lambda m\left( x \right)u\,\,\,in\,\,\,\Omega ,}  \cr    {{{\partial u} \over {\partial v}} = {{\partial \left( {\Delta u} \right)} \over {\partial v}} =  \ldots  = {{\partial \left( {{\Delta ^{2p - 1}}u} \right)} \over {\partial v}} = 0\,\,\,on\,\,\,\partial \Omega .}  \cr  } } \right.


Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, p ∈ ℕ*, m ∈ L∞ (Ω), and Δ2pu := Δ (Δ...( Δu)), 2p times the operator Δ.

Existence of solutions for 4p-order PDES with Neumann boundary conditions

Mathematics and Computer Science Department

University Abdelmalk Essaadi, Sciences and Technologies Faculty

University Abdelmalk Essaadi, National School of Applied Sciences

University Mohamed I, Multidisciplinary Faculty

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Existence of solutions for 4p-order PDES with Neumann boundary conditions"}

In this work, we study the existence of at least one non decreasing sequence of nonnegative eigenvalues for the problem:



{...

Existence of solutions for 4p-order PDES with Neumann boundary conditions

Published Online: Feb 01, 2023

Page range: 65 - 74

Received: Sep 30, 2022

Accepted: Jan 16, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0004

Keywords
Fourth order PDEs, Eigenvalue problems

© 2023 N. Moradi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Existence of solutions for 4p-order PDES with Neumann boundary conditions

Published Online: Feb 01, 2023

Page range: 65 - 74

Received: Sep 30, 2022

Accepted: Jan 16, 2023

DOI: https://doi.org/10.2478/mjpaa-2023-0004

KeywordsFourth order PDEs, Eigenvalue problems

© 2023 N. Moradi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Fourth order PDEs, Eigenvalue problems