<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

Online veröffentlicht: 28. Juli 2018

Seitenbereich: 186 - 198

Eingereicht: 01. März 2017

Akzeptiert: 25. Juni 2018

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

Schlüsselwörter
-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

Online veröffentlicht: 28. Juli 2018

Seitenbereich: 186 - 198

Eingereicht: 01. März 2017

Akzeptiert: 25. Juni 2018

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

Schlüsselwörter-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

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