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

In this paper, we study the following nonlinear eigenvalue problem:
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2022-0013_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 mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>=</mo><mo>…</mo><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><mo>=</mo><mn>0</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>o</mi><mi>n</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 {u = \Delta u = \ldots {\Delta ^{2p - 1}}u = 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>N ≥</italic> 1, <italic>p</italic> ∈ ℕ*, <italic>m</italic> ∈ <italic>L∞</italic> (Ω), <italic>µ</italic>{<italic>x</italic> ∈ Ω: <italic>m</italic>(<italic>x</italic>) > 0} ≠ 0, and Δ2<italic>pu</italic> := Δ (Δ...(Δ<italic>u</italic>)), 2<italic>p</italic> times the operator Δ.
Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.
</abstract>

In this paper, we study the following nonlinear eigenvalue problem:



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


Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, N ≥ 1, p ∈ ℕ*, m ∈ L∞ (Ω), µ{x ∈ Ω: m(x) > 0} ≠ 0, and Δ2pu := Δ (Δ...(Δu)), 2p times the operator Δ.
Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.

Existence of solutions for 4p-order PDES

Mathematics and Computer Science Department

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"}

In this paper, we study the following nonlinear eigenvalue problem:



{ Δ2pu=λm(x)u   in  Ω,u=Δu=…Δ2p−1u=0    on  ∂Ω. 
\left\{ {\matrix{   {{\Delta ^{2p}}u =...

Existence of solutions for 4p-order PDES

Publicado en línea: 28 may 2022

Páginas: 179 - 190

Recibido: 28 ago 2021

Aceptado: 15 ene 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0013

Palabras clave
Fourth order PDEs, Eigenvalue problems

© 2022 F. 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

Publicado en línea: 28 may 2022

Páginas: 179 - 190

Recibido: 28 ago 2021

Aceptado: 15 ene 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0013

Palabras claveFourth order PDEs, Eigenvalue problems

© 2022 F. Moradi et al., published by Sciendo

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

Palabras clave
Fourth order PDEs, Eigenvalue problems