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

<p>We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model problems is
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2022-0022_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><mo>-</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow><mo>(</mo><mrow><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mfrac><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></mrow><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow><mo>)</mo></mrow><mi>γ</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mrow><mo>∇</mo><mi>u</mi></mrow> <mo>|</mo></mrow></mrow></mrow><mi>p</mi></msup></mrow><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mi>θ</mi></msup></mrow></mfrac><mo>=</mo><mi>f</mi><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mi>i</mi><mi>n</mi><mi> </mi><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mi>o</mi><mi>n</mi><mi> </mi><mo>∂</mo><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{   { - div\left( {b\left( x \right){{{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \over {\left( {1 + \left| u \right|} \right)\gamma }}} \right) + {{{{\left| {\nabla u} \right|}^p}} \over {{{\left| u \right|}^\theta }}} = f,} \hfill &amp; {in\,\Omega ,} \hfill  \cr    {u = 0,} \hfill &amp; {on\,\partial \Omega ,} \hfill  \cr  } } \right.</tex-math>
</alternatives>
</disp-formula> swhere Ω is a bounded open subset in ℝ<sup>N</sup>, 1 ≤ <italic>θ</italic> &lt; 2, <italic>p</italic> > 2 and <italic>γ</italic> > 0. We will show that, even if the lower order term is singular, we obtain existence and regularity of positive solution, under various assumptions on the summability of the source <italic>f</italic>.</p>
</abstract>

We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model problems is



{ -div(b(x)| ∇u |p-2∇u(1+| u |)γ)+| ∇u |p| u |θ=f,in Ω,u=0,on ∂Ω, 
\left\{ {\matrix{   { - div\left( {b\left( x \right){{{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \over {\left( {1 + \left| u \right|} \right)\gamma }}} \right) + {{{{\left| {\nabla u} \right|}^p}} \over {{{\left| u \right|}^\theta }}} = f,} \hfill & {in\,\Omega ,} \hfill  \cr    {u = 0,} \hfill & {on\,\partial \Omega ,} \hfill  \cr  } } \right.

 swhere Ω is a bounded open subset in ℝN, 1 ≤ θ < 2, p > 2 and γ > 0. We will show that, even if the lower order term is singular, we obtain existence and regularity of positive solution, under various assumptions on the summability of the source f.

Existence and Regularity for Solution to a Degenerate Problem with Singular Gradient Lower Order Term

Moroccan Journal of Pure and Applied Analysis

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

We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model problems is



{ -div(b(x)| ∇u |p-2∇u(1+| u |)γ)+| ∇u |p| u |θ=f,in Ω,u=0,on ∂Ω, 
\left\{ {\matrix{   { - div\left( {b\left( x \right){{{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \over {\left( {1 + \left| u \right|} \right)\gamma }}} \right) + {{{{\left| {\nabla u} \right|}^p}} \over {{{\left| u \right|}^\theta }}} = f,} \hfill & {in\,\Omega ,} \hfill  \cr    {u = 0,} \hfill & {on\,\partial \Omega ,} \hfill  \cr  } } \right.

 swhere Ω is a bounded open subset in ℝN, 1 ≤ θ  2 and γ > 0. We will show that, even if the lower order term is singular, we obtain existence and regularity of positive solution, under various assumptions on the summability of the source f.

We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model...

Existence and Regularity for Solution to a Degenerate Problem with Singular Gradient Lower Order Term

Publié en ligne: 11 oct. 2022

Pages: 310 - 327

Reçu: 28 août 2021

Accepté: 24 sept. 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0022

Mots clés
Nonlinear Elliptic Equations, Degenerate Problem, Singular Natural growth gradient terms

© 2022 Hichem Khelifi, published by Sciendo

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

Existence and Regularity for Solution to a Degenerate Problem with Singular Gradient Lower Order Term

Publié en ligne: 11 oct. 2022

Pages: 310 - 327

Reçu: 28 août 2021

Accepté: 24 sept. 2022

DOI: https://doi.org/10.2478/mjpaa-2022-0022

Mots clésNonlinear Elliptic Equations, Degenerate Problem, Singular Natural growth gradient terms

© 2022 Hichem Khelifi, published by Sciendo

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

Mots clés
Nonlinear Elliptic Equations, Degenerate Problem, Singular Natural growth gradient terms