<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2017-0003_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>Δ</mo><mtext>(v</mtext><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo><msup><mrow><mrow><mo>|</mo><mrow><mo>Δ</mo><mtext>u</mtext></mrow><mo>|</mo></mrow></mrow><mrow><mtext>r</mtext><mo>−</mo><mn>2</mn></mrow></msup><mo>Δ</mo><mtext>u</mtext><mo stretchy="false">)</mo><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mtext>j</mtext><mo>=</mo><mn>1</mn></mrow><mtext>n</mtext></munderover><mrow><msub><mtext>D</mtext><mtext>j</mtext></msub><mo stretchy="false">[</mo><msub><mtext>w</mtext><mn>1</mn></msub><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo><msub><mtext>𝒜</mtext><mtext>j</mtext></msub><mo stretchy="false">(</mo><mtext>x</mtext><mo>,</mo><mtext>u</mtext><mo>,</mo><mo>∇</mo><mtext>u</mtext><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>+</mo><mtext> </mtext><mtext>b</mtext><mo stretchy="false">(</mo><mtext>x</mtext><mo>,</mo><mtext>u</mtext><mo>,</mo><mo>∇</mo><mtext>u</mtext><mo stretchy="false">)</mo><mtext> </mtext><msub><mtext>w</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo><mo>=</mo><msub><mtext>f</mtext><mn>0</mn></msub><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo><mo>−</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mtext>j</mtext><mo>=</mo><mn>1</mn></mrow><mtext>n</mtext></munderover><mrow><msub><mtext>D</mtext><mtext>j</mtext></msub><msub><mtext>f</mtext><mtext>j</mtext></msub><mo stretchy="false">(</mo><mtext>x</mtext><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>in</mtext><mtext> </mtext><mo>Ω</mo></mrow></mstyle></mrow></mtd></mtr></mtable></math><tex-math>$$\matrix{{\Delta {\rm{(v}}({\rm{x}})\left| {\Delta {\rm{u}}} \right|^{{\rm{r}} - 2} \Delta {\rm{u}}) - \sum\limits_{{\rm{j}} = 1}^{\rm{n}} {{\rm{D}}_{\rm{j}} [{\rm{w}}_1 ({\rm{x}}){\cal{A}}_{\rm{j}} ({\rm{x}},{\rm{u}},\nabla {\rm{u}})]} } \hfill  \cr { + \;{\rm{b}}({\rm{x}},{\rm{u}},\nabla {\rm{u}})\;{\rm{w}}_2 ({\rm{x}}) = {\rm{f}}_0 ({\rm{x}}) - \sum\limits_{{\rm{j}} = 1}^{\rm{n}} {{\rm{D}}_{\rm{j}} {\rm{f}}_{\rm{j}} ({\rm{x}}),\;\;\;\;\;{\rm{in}}\;\Omega } }}$$</tex-math></alternatives></disp-formula>
in the setting of the Weighted Sobolev Spaces.</p></abstract>

In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations
Δ(v(x)|Δu|r−2Δu)−∑j=1nDj[w1(x)𝒜j(x,u,∇u)]+ b(x,u,∇u) w2(x)=f0(x)−∑j=1nDjfj(x),     in Ω$$\matrix{{\Delta {\rm{(v}}({\rm{x}})\left| {\Delta {\rm{u}}} \right|^{{\rm{r}} - 2} \Delta {\rm{u}}) - \sum\limits_{{\rm{j}} = 1}^{\rm{n}} {{\rm{D}}_{\rm{j}} [{\rm{w}}_1 ({\rm{x}}){\cal{A}}_{\rm{j}} ({\rm{x}},{\rm{u}},\nabla {\rm{u}})]} } \hfill  \cr { + \;{\rm{b}}({\rm{x}},{\rm{u}},\nabla {\rm{u}})\;{\rm{w}}_2 ({\rm{x}}) = {\rm{f}}_0 ({\rm{x}}) - \sum\limits_{{\rm{j}} = 1}^{\rm{n}} {{\rm{D}}_{\rm{j}} {\rm{f}}_{\rm{j}} ({\rm{x}}),\;\;\;\;\;{\rm{in}}\;\Omega } }}$$
in the setting of the Weighted Sobolev Spaces.

Existence and uniqueness of solution for a class of nonlinear degenerate elliptic equation in weighted Sobolev spaces

Acta Universitatis Sapientiae, Mathematica

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

In this work we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic...

Existence and uniqueness of solution for a class of nonlinear degenerate elliptic equation in weighted Sobolev spaces

Published Online: Aug 05, 2017

Page range: 26 - 44

Received: May 17, 2016

DOI: https://doi.org/10.1515/ausm-2017-0003

Keywords
degenerate nonlinear elliptic equation, weighted Sobolev spaces

© 2017 Albo Carlos Cavalheiro, published by De Gruyter Open

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

Existence and uniqueness of solution for a class of nonlinear degenerate elliptic equation in weighted Sobolev spaces

Published Online: Aug 05, 2017

Page range: 26 - 44

Received: May 17, 2016

DOI: https://doi.org/10.1515/ausm-2017-0003

Keywordsdegenerate nonlinear elliptic equation, weighted Sobolev spaces

© 2017 Albo Carlos Cavalheiro, published by De Gruyter Open

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

Keywords
degenerate nonlinear elliptic equation, weighted Sobolev spaces