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

<p>This article shows the existence and multiplicity of weak solutions for the singular subelliptic system on the Heisenberg group
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2022-0006_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><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><msup><mrow><mi>ℍ</mi></mrow><mi>n</mi></msup></mrow></msub><mi>u</mi><mo>+</mo><mi>a</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mfrac><mi>u</mi><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>z</mi> <mo>|</mo></mrow></mrow></mrow><mn>4</mn></msup><mo>+</mo><msup><mrow><mi>t</mi></mrow><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfrac><mo>=</mo><mi>λ</mi><msub><mrow><mi>F</mi></mrow><mi>u</mi></msub><mrow><mo>(</mo><mrow><mi>ξ</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>i</mi><mi>n</mi><mi> </mi><mi> </mi><mi> </mi><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><msup><mrow><mi>ℍ</mi></mrow><mi>n</mi></msup></mrow></msub><mi>v</mi><mo>+</mo><mi>b</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mfrac><mi>v</mi><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>z</mi> <mo>|</mo></mrow></mrow></mrow><mn>4</mn></msup><mo>+</mo><msup><mrow><mi>t</mi></mrow><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfrac><mo>=</mo><mi>λ</mi><msub><mrow><mi>F</mi></mrow><mi>v</mi></msub><mrow><mo>(</mo><mrow><mi>ξ</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>v</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow><mi>i</mi><mi>n</mi><mi> </mi><mi> </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><mi>v</mi><mo>=</mo><mn>0</mn></mrow></mtd><mtd columnalign="left"><mrow><mi>o</mi><mi>n</mi><mi> </mi><mi> </mi><mo>∂</mo><mi mathvariant="normal">Ω</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{   { - {\Delta _{{\mathbb{H}^n}}}u + a\left( \xi  \right){u \over {{{\left( {{{\left| z \right|}^4} + {t^2}} \right)}^{{1 \over 2}}}}} = \lambda {F_u}\left( {\xi ,u,v} \right)} \hfill &amp; {in\,\,\,\Omega ,} \hfill  \cr    { - {\Delta _{{\mathbb{H}^n}}}v + b\left( \xi  \right){v \over {{{\left( {{{\left| z \right|}^4} + {t^2}} \right)}^{{1 \over 2}}}}} = \lambda {F_v}\left( {\xi ,u,v} \right)} \hfill &amp; {in\,\,\,\Omega ,} \hfill  \cr    {u = v = 0} \hfill &amp; {on\,\,\partial \Omega .} \hfill  \cr  } } \right.</tex-math>
</alternatives>
</disp-formula>
The approach is based on variational methods.</p>
</abstract>

This article shows the existence and multiplicity of weak solutions for the singular subelliptic system on the Heisenberg group



{ -Δℍnu+a(ξ)u(| z |4+t2)12=λFu(ξ,u,v)in   Ω,-Δℍnv+b(ξ)v(| z |4+t2)12=λFv(ξ,u,v)in   Ω,u=v=0on  ∂Ω. 
\left\{ {\matrix{   { - {\Delta _{{\mathbb{H}^n}}}u + a\left( \xi  \right){u \over {{{\left( {{{\left| z \right|}^4} + {t^2}} \right)}^{{1 \over 2}}}}} = \lambda {F_u}\left( {\xi ,u,v} \right)} \hfill & {in\,\,\,\Omega ,} \hfill  \cr    { - {\Delta _{{\mathbb{H}^n}}}v + b\left( \xi  \right){v \over {{{\left( {{{\left| z \right|}^4} + {t^2}} \right)}^{{1 \over 2}}}}} = \lambda {F_v}\left( {\xi ,u,v} \right)} \hfill & {in\,\,\,\Omega ,} \hfill  \cr    {u = v = 0} \hfill & {on\,\,\partial \Omega .} \hfill  \cr  } } \right.


The approach is based on variational methods.

Existence results of infinitely many weak solutions of a singular subelliptic system on the Heisenberg group

Department of Pure Mathematics, Faculty of Science

Acta Universitatis Sapientiae, Mathematica

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

This article shows the existence and multiplicity of weak solutions for the singular subelliptic system on the Heisenberg group



{ -Δℍnu+a(ξ)u(| z...

Existence results of infinitely many weak solutions of a singular subelliptic system on the Heisenberg group

Pubblicato online: 18 nov 2022

Pagine: 90 - 103

Ricevuto: 14 feb 2021

DOI: https://doi.org/10.2478/ausm-2022-0006

Parole chiave
singular potential, variational methods, infinitely many solutions, Heisenberg group

© 2022 S. Heidari et al., published by Sciendo

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

Existence results of infinitely many weak solutions of a singular subelliptic system on the Heisenberg group

Pubblicato online: 18 nov 2022

Pagine: 90 - 103

Ricevuto: 14 feb 2021

DOI: https://doi.org/10.2478/ausm-2022-0006

Parole chiavesingular potential, variational methods, infinitely many solutions, Heisenberg group

© 2022 S. Heidari et al., published by Sciendo

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

Parole chiave
singular potential, variational methods, infinitely many solutions, Heisenberg group