Normalized ground-states for the Sobolev critical Kirchhoff equation with at least mass critical growth
Abstract In this article, we investigate the following nonlinear Kirchhoff equation with Sobolev critical growth: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:mfenced open="(" close=")"> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mi>b</m:mi> <m:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle="true"> <m:mo>∫</m:mo> </m:mstyle> </m:mrow> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>μ</m:mi> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>4</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mspace width="0.1em"/> <m:mtext>in</m:mtext> <m:mspace width="0.1em"/> <m:mspace width="0.33em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:mi>u</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle="true"> <m:mo>∫</m:mo> </m:mstyle> </m:mrow> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mspace width="0.1em"/> <m:mtext>in</m:mtext> <m:mspace width="0.1em"/> <m:mspace width="0.33em"/> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> <m:mspace width="2.0em"/> <m:mspace width="2.0em"/> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u+\lambda u=\mu f\left(u)+{| u| }^{4}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ u\gt 0,\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1
Authors
- Qiongfen Zhang (ORCID: https://orcid.org/0000-0002-7037-1961)
- Shiyong Zhang (ORCID: https://orcid.org/0000-0001-7633-358X)
Institutions
- Guangxi University (CN)
- Guilin University of Aerospace Technology (CN)
- Guilin University of Technology (CN)
- Guilin University of Electronic Technology (CN)
Publication Details
- Journal
- Open Mathematics
- Published
- 2025-01-01
- DOI
- https://doi.org/10.1515/math-2025-0182
- Primary Topic
- Nonlinear Partial Differential Equations
- Type
- article
- Field-Weighted Citation Impact
- 0.00