<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

Pubblicato online: 11 ott 2022

Pagine: 310 - 327

Ricevuto: 28 ago 2021

Accettato: 24 set 2022

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

Parole chiave
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

Pubblicato online: 11 ott 2022

Pagine: 310 - 327

Ricevuto: 28 ago 2021

Accettato: 24 set 2022

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

Parole chiaveNonlinear 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.

Parole chiave
Nonlinear Elliptic Equations, Degenerate Problem, Singular Natural growth gradient terms