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

In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2022-0016_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mi>p</mi></msub><mi>u</mi><mo>=</mo><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow></mtd><mtd columnalign="left"><mrow><mtext>in</mtext><mi> </mi><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mrow><mo>∇</mo><mi>u</mi></mrow> <mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mfrac><mrow><mo>∂</mo><mi>u</mi></mrow><mrow><mo>∂</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mi>λ</mi><msubsup><mrow><mrow><mrow><mo>‖</mo> <mi>u</mi> <mo>‖</mo></mrow></mrow></mrow><mrow><mi>q</mi><mo>,</mo><mo>∂</mo><mi mathvariant="normal">Ω</mi></mrow><mrow><mi>p</mi><mo>-</mo><mi>q</mi></mrow></msubsup><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>q</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow></mtd><mtd columnalign="left"><mrow><mtext>on</mtext><mi> </mi><mo>∂</mo><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{{{\Delta _p}u = {{\left| u \right|}^{p - 2}}u} \hfill &amp; {{\rm{in}}\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}{{\partial u} \over {\partial v}} = \lambda \left\| u \right\|_{q,\partial \Omega }^{p - q}{{\left| u \right|}^{q - 2}}u} \hfill &amp; {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right.</tex-math>
</alternatives>
</disp-formula>
where Ω is a smooth bounded domain in ℝ<italic>N</italic>(<italic>N</italic> ≥ 1), λ is a real number which plays the role of eigenvalue and the unknowns <italic>u</italic> ∈ <italic>W</italic>1,<italic>p</italic>(Ω). Using the Ljusterneck-Shnirelmann theory on <italic>C</italic>1 manifold and Sobolev trace embedding we prove the existence of an increasing sequence positive of eigenvalues (λ<italic>k</italic>)<italic>k</italic>≥1, for the above problem. We then establish that the first eigenvalue is simple and isolated.
</abstract>

In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form



{ Δpu=| u |p-2uin Ω,| ∇u |p-2∂u∂v=λ‖ u ‖q,∂Ωp-q| u |q-2uon ∂Ω, 
\left\{ {\matrix{{{\Delta _p}u = {{\left| u \right|}^{p - 2}}u} \hfill & {{\rm{in}}\,\Omega ,} \hfill  \cr {{{\left| {\nabla u} \right|}^{p - 2}}{{\partial u} \over {\partial v}} = \lambda \left\| u \right\|_{q,\partial \Omega }^{p - q}{{\left| u \right|}^{q - 2}}u} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill  \cr  } } \right.


where Ω is a smooth bounded domain in ℝN(N ≥ 1), λ is a real number which plays the role of eigenvalue and the unknowns u ∈ W1,p(Ω). Using the Ljusterneck-Shnirelmann theory on C1 manifold and Sobolev trace embedding we prove the existence of an increasing sequence positive of eigenvalues (λk)k≥1, for the above problem. We then establish that the first eigenvalue is simple and isolated.

Steklov problems for the <italic>p</italic>−Laplace operator involving <italic>Lq</italic>-norm

Department of Mathematics, Faculty of Sciences and Technologies Moulay Ismail University of Meknes

College of Science Al Imam Mohammad Ibn Saud Islamic University (IMSIU)

Faculty of Sciences Dhar El Mahraz University Sidi Mohamed Ben Abdellah

Moroccan Journal of Pure and Applied Analysis

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

Steklov problems for the −Laplace operator involving -norm

In this paper, we are concerned with the study of the spectrum for the nonlinear Steklov problem of the form



{ Δpu=| u |p-2uin Ω,| ∇u |p-2∂u∂v=λ‖ u...

Steklov problems for the p−Laplace operator involving L^q-norm

Publicado en línea: 28 may 2022

Páginas: 228 - 243

Recibido: 17 sept 2021

Aceptado: 22 feb 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0016

Palabras clave
-Laplace operator, Nonlinear eigenvalue problem, Boundary value problems, Sobolev trace embedding, manifold, Ljusternick-Schnirelmann theory

© 2022 My Driss Morchid Alaoui et al., published by Sciendo

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

Steklov problems for the p−Laplace operator involving Lq-norm

Publicado en línea: 28 may 2022

Páginas: 228 - 243

Recibido: 17 sept 2021

Aceptado: 22 feb 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0016

Palabras clave-Laplace operator, Nonlinear eigenvalue problem, Boundary value problems, Sobolev trace embedding, manifold, Ljusternick-Schnirelmann theory

© 2022 My Driss Morchid Alaoui et al., published by Sciendo

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

Steklov problems for the p−Laplace operator involving L^q-norm

Palabras clave
-Laplace operator, Nonlinear eigenvalue problem, Boundary value problems, Sobolev trace embedding, manifold, Ljusternick-Schnirelmann theory