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

In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem<disp-formula id="tmmp-2023-0003_eqn_001">
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2023-0003_eqn_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo>{</mo><mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"><mtr><mtd><mo>−</mo><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>=</mo><mfrac><mrow><mrow><mo stretchy="false">|</mo></mrow><mi>u</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mo>∗</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi></mrow><mrow><mrow><mo stretchy="false">|</mo></mrow><mi>y</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>t</mi></mrow></msup></mrow></mfrac><mo>+</mo><mi>μ</mi><mrow><mo stretchy="false">|</mo></mrow><mi>u</mi><msup><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi></mtd><mtd><mtext> in </mtext><mi mathvariant="normal">Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn></mtd><mtd><mtext> on </mtext><mi mathvariant="normal">∂</mi><mi mathvariant="normal">Ω</mi><mo>,</mo></mtd></mtr></mtable><mo fence="true" stretchy="true"></mo></mrow></math>
<tex-math>\begin{cases}-\Delta u=\frac{|u|^{2^*(t)-2} u}{|y|^t}+\mu|u|^{q-2} u &amp; \text { in } \Omega, \\ u=0 &amp; \text { on } \partial \Omega,\end{cases}</tex-math>
</alternatives>
</disp-formula> where Ω is an open bounded domain in ℝ<italic>N</italic> , which contains some points (0,<italic>z</italic>*), <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2023-0003_ieqn_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>μ</mi><mo>></mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>&lt;</mo><mi>q</mi><mo>&lt;</mo><mn>2</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>∗</mo></mrow></msup><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math><tex-math>\mu>0,1&amp;#x003C;q&amp;#x003C;2,2^*(t)=\frac{2(N-t)}{N-2}</tex-math></alternatives></inline-formula>, 0 ≤ <italic>t</italic> &lt; 2, <italic>x</italic> = (<italic>y</italic>, <italic>z</italic>) ∈ ℝ<italic>k</italic> × ℝ<italic>N−k</italic>, 2 ≤ <italic>k</italic> ≤ <italic>N</italic>. We prove that if <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/tmmp-2023-0003_ieqn_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>N</mi><mo>></mo><mn>2</mn><mfrac><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mi>t</mi></mrow></math><tex-math>$N > 2{{q + 1} \over {q - 1}} + t$</tex-math></alternatives></inline-formula>, then the above problem has two disjoint and infinite sets of solutions. Here, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in [1] for the case of the critical Hardy-Sobolev-Maz’ya problem.
</abstract>

In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem


{−Δu=|u|2∗(t)−2u|y|t+μ|u|q−2u in Ω,u=0 on ∂Ω,
\begin{cases}-\Delta u=\frac{|u|^{2^*(t)-2} u}{|y|^t}+\mu|u|^{q-2} u & \text { in } \Omega, \\ u=0 & \text { on } \partial \Omega,\end{cases}

 where Ω is an open bounded domain in ℝN , which contains some points (0,z*), μ>0,1<q<2,2∗(t)=2(N−t)N−2\mu>0,1&#x003C;q&#x003C;2,2^*(t)=\frac{2(N-t)}{N-2}, 0 ≤ t < 2, x = (y, z) ∈ ℝk × ℝN−k, 2 ≤ k ≤ N. We prove that if N>2q+1q−1+t$N > 2{{q + 1} \over {q - 1}} + t$, then the above problem has two disjoint and infinite sets of solutions. Here, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in [1] for the case of the critical Hardy-Sobolev-Maz’ya problem.

Two Disjoint and Infinite Sets of Solutions for An Elliptic Equation with Critical Hardy-Sobolev-Maz’ya Term and Concave-Convex Nonlinearities

Tatra Mountains Mathematical Publications

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

In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem


{−Δu=|u|2∗(t)−2u|y|t+μ|u|q−2u in Ω,u=0 on ∂Ω,
\begin{cases}-\Delta u=\frac{|u|^{2^*(t)-2} u}{|y|^t}+\mu|u|^{q-2} u & \text { in } \Omega, \\ u=0 & \text { on } \partial \Omega,\end{cases}

 where Ω is an open bounded domain in ℝN , which contains some points (0,z*), μ>0,10,1&#x003C;q&#x003C;2,2^*(t)=\frac{2(N-t)}{N-2}, 0 ≤ t 2q+1q−1+t$N > 2{{q + 1} \over {q - 1}} + t$, then the above problem has two disjoint and infinite sets of solutions. Here, we give a positive answer to one open problem proposed by Ambrosetti, Brezis and Cerami in [1] for the case of the critical Hardy-Sobolev-Maz’ya problem.

In this paper, we consider the following critical Hardy-Sobolev-Maz’ya problem


{−Δu=|u|2∗(t)−2u|y|t+μ|u|q−2u in Ω,u=0 on ∂Ω,
\begin{cases}-\Delta...

Two Disjoint and Infinite Sets of Solutions for An Elliptic Equation with Critical Hardy-Sobolev-Maz’ya Term and Concave-Convex Nonlinearities

Published Online: Mar 07, 2023

Page range: 25 - 42

Received: Nov 01, 2022

DOI: https://doi.org/10.2478/tmmp-2023-0003

Keywords
critical Hardy-Sobolev-Maz’ya term, infnitely many solutions, concave-convex, Fountain theorem

© 2023 Mathematical Institute, Slovak Academy of Sciences

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

Two Disjoint and Infinite Sets of Solutions for An Elliptic Equation with Critical Hardy-Sobolev-Maz’ya Term and Concave-Convex Nonlinearities

Published Online: Mar 07, 2023

Page range: 25 - 42

Received: Nov 01, 2022

DOI: https://doi.org/10.2478/tmmp-2023-0003

Keywordscritical Hardy-Sobolev-Maz’ya term, infnitely many solutions, concave-convex, Fountain theorem

© 2023 Mathematical Institute, Slovak Academy of Sciences

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

Keywords
critical Hardy-Sobolev-Maz’ya term, infnitely many solutions, concave-convex, Fountain theorem