<abstract xmlns="http://www.w3.org/1999/xhtml">We consider the perturbed nonlinear boundary condition problem<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2017-0015_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><mi>Δ</mi><mi>p</mi></msub><mi>u</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mrow><mi>λ</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>u</mi></mrow><mo>)</mo></mrow><mi> </mi><mtext>in</mtext><mi> </mi><mi>Ω</mi></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mrow><mo>|</mo><mrow><mo>∇</mo><mi>u</mi></mrow><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>.</mo><mi>ν</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow><mi>λ</mi><mi>ρ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mi> </mi><mtext>on</mtext><mi> </mi><mi>Γ</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>$$\left\{ {\matrix{ { - \Delta _p u} \hfill &amp; = \hfill &amp; {\left| u \right|^{p - 2} u + f\left( {\lambda ,x,u} \right)\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill &amp; = \hfill &amp; {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$</tex-math></alternatives></disp-formula>Using the Sobolev trace embedding and the duality mapping defined on <italic>W</italic>1,<italic>p</italic>(Ω), we prove that this problem bifurcates from the principal eigenvalue <italic>λ</italic>1 of the eigenvalue problem<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2017-0015_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>-</mo><msub><mi>Δ</mi><mi>p</mi></msub><mi>u</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mi> </mi><mtext>in</mtext><mi> </mi><mi>Ω</mi></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mrow><mo>|</mo><mrow><mo>∇</mo><mi>u</mi></mrow><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>.</mo><mi>ν</mi></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mrow><mi>λ</mi><mi>ρ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mi> </mi><mtext>on</mtext><mi> </mi><mi>Γ</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>$$\left\{ {\matrix{ { - \Delta _p u} \hfill &amp; = \hfill &amp; {\left| u \right|^{p - 2} u\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill &amp; = \hfill &amp; {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$</tex-math></alternatives></disp-formula></abstract>

We consider the perturbed nonlinear boundary condition problem{-Δpu=|u|p-2u+f(λ,x,u) in Ω|∇u|p-2∇u.ν=λρ(x)|u|p-2u on Γ.$$\left\{ {\matrix{  { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u + f\left( {\lambda ,x,u} \right)\,{\rm{in}}\,\Omega } \hfill \cr  {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$Using the Sobolev trace embedding and the duality mapping defined on W1,p(Ω), we prove that this problem bifurcates from the principal eigenvalue λ1 of the eigenvalue problem{-Δpu=|u|p-2u in Ω|∇u|p-2∇u.ν=λρ(x)|u|p-2u on Γ.$$\left\{ {\matrix{  { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u\,{\rm{in}}\,\Omega } \hfill \cr  {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$

A bifurcation result involving Sobolev trace embedding and the duality mapping of <italic>W</italic>1<italic>,p</italic>

Al-Imam Mohammad Ibn Saud Islamic University (IMSIU)

College of Science, Department of Mathematics and Statistics

Moroccan Journal of Pure and Applied Analysis

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

A bifurcation result involving Sobolev trace embedding and the duality mapping of 

We consider the perturbed nonlinear boundary condition problem{-Δpu=|u|p-2u+f(λ,x,u) in Ω|∇u|p-2∇u.ν=λρ(x)|u|p-2u on Γ.$$\left\{ {\matrix{  { - \Delta _p u}...

A bifurcation result involving Sobolev trace embedding and the duality mapping of W¹^,p

Published Online: Jul 28, 2018

Page range: 186 - 198

Received: Mar 01, 2017

Accepted: Jun 25, 2018

DOI: https://doi.org/10.1515/mjpaa-2017-0015

Keywords
-Laplacian operator, nonlinear boundary conditions, Sobolev trace embedding, principal eigenvalue, duality mapping, topological degree, bifurcation phenomenon

© 2018 Abdelouahed El Khalil, published by Sciendo

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

A bifurcation result involving Sobolev trace embedding and the duality mapping of W1,p

Published Online: Jul 28, 2018

Page range: 186 - 198

Received: Mar 01, 2017

Accepted: Jun 25, 2018

DOI: https://doi.org/10.1515/mjpaa-2017-0015

Keywords-Laplacian operator, nonlinear boundary conditions, Sobolev trace embedding, principal eigenvalue, duality mapping, topological degree, bifurcation phenomenon

© 2018 Abdelouahed El Khalil, published by Sciendo

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

A bifurcation result involving Sobolev trace embedding and the duality mapping of W¹^,p

Keywords
-Laplacian operator, nonlinear boundary conditions, Sobolev trace embedding, principal eigenvalue, duality mapping, topological degree, bifurcation phenomenon