<abstract xmlns="http://www.w3.org/1999/xhtml">We study the following nonlinear eigenvalue problem with nonlinear Robin boundary condition<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2019-0020_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>-</mo><msub><mrow><mi>Δ</mi></mrow><mi>p</mi></msub><mi>u</mi><mo>=</mo><mi>λ</mi><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><mi> </mi><mi> </mi><mi> </mi><mi>i</mi><mi>n</mi><mi> </mi><mi> </mi><mi>Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><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><mo>∇</mo><mi>u</mi><mo>.</mo><mi>v</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><mo>=</mo><mn>0</mn><mi> </mi><mi> </mi><mi> </mi><mi>o</mi><mi>n</mi><mi> </mi><mi> </mi><mi>Γ</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>\left\{ {\matrix{ { - {\Delta _p}u = \lambda {{\left| u \right|}^{p - 2}}u\,\,\,in\,\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u.v + {{\left| u \right|}^{p - 2}}u = 0\,\,\,on\,\,\Gamma .} \hfill \cr } } \right.</tex-math></alternatives></disp-formula>We successfully investigate the existence at least of one nondecreasing sequence of positive eigenvalues λ<italic>n</italic>↗∞. To this end we endow <italic>W</italic>1,<italic>p</italic>(Ω) with a norm invoking the trace and use the duality mapping on <italic>W</italic>1,<italic>p</italic> (Ω) to apply mini-max arguments on <italic>C</italic>1-manifold.</abstract>

We study the following nonlinear eigenvalue problem with nonlinear Robin boundary condition{-Δpu=λ|u|p-2u   in  Ω,|∇u|p-2∇u.v+|u|p-2u=0   on  Γ.\left\{ {\matrix{   { - {\Delta _p}u = \lambda {{\left| u \right|}^{p - 2}}u\,\,\,in\,\,\Omega ,} \hfill  \cr    {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u.v + {{\left| u \right|}^{p - 2}}u = 0\,\,\,on\,\,\Gamma .} \hfill  \cr  } } \right.We successfully investigate the existence at least of one nondecreasing sequence of positive eigenvalues λn↗∞. To this end we endow W1,p(Ω) with a norm invoking the trace and use the duality mapping on W1,p (Ω) to apply mini-max arguments on C1-manifold.

On the spectrum of Robin boundary <italic>p</italic>-Laplacian problem

Imam Mohammad Ibn Saud Islamic University (IMSIU)

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On the spectrum of Robin boundary -Laplacian problem

We study the following nonlinear eigenvalue problem with nonlinear Robin boundary condition{-Δpu=λ|u|p-2u   in  Ω,|∇u|p-2∇u.v+|u|p-2u=0   on  Γ.\left\{...

On the spectrum of Robin boundary p-Laplacian problem

Published Online: Jan 24, 2020

Page range: 279 - 293

Received: Nov 09, 2019

Accepted: Dec 29, 2019

DOI: https://doi.org/10.2478/mjpaa-2019-0020

Keywords
Trace embedding theorems, duality mapping, mini-max, -manifold, Spectrum of -Laplacian operator, nonlinear Robin boundary conditions

© 2019 Abdelouahed El Khalil, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On the spectrum of Robin boundary p-Laplacian problem

Published Online: Jan 24, 2020

Page range: 279 - 293

Received: Nov 09, 2019

Accepted: Dec 29, 2019

DOI: https://doi.org/10.2478/mjpaa-2019-0020

KeywordsTrace embedding theorems, duality mapping, mini-max, -manifold, Spectrum of -Laplacian operator, nonlinear Robin boundary conditions

© 2019 Abdelouahed El Khalil, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Trace embedding theorems, duality mapping, mini-max, -manifold, Spectrum of -Laplacian operator, nonlinear Robin boundary conditions