In this paper, we investigate the minimization problem<Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_516_Article_Equ1.gif" Format="GIF" Height="35" Rendition="HTML" Resolution="72" Type="Linedraw" Width="234" /> </MediaObject> <EquationSource Format="TEX">$e_{s}(\rho)=\inf_{u\in W^{1,N}_{V}(\mathbb{R}^{N}),\|u\|^{N}_{N}=\rho>0}E(u),$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>e</mi> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo form="prefix" movablelimits="true">inf</mo> <mrow> <mi>u</mi> <mo>∈</mo> <msubsup> <mi>W</mi> <mrow> <mi>V</mi> </mrow> <mrow> <mn>1</mn> <mo>,</mo> <mi>N</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> <mo>∥</mo> <mi>u</mi> <msubsup> <mo>∥</mo> <mrow> <mi>N</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msubsup> <mo>=</mo> <mi>ρ</mi> <mo>></mo> <mn>0</mn> </mrow> </munder> <mi>E</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>,</mo> </math></EquationSource> </Equation> where <Equation ID="Equ2"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_516_Article_Equ2.gif" Format="GIF" Height="41" Rendition="HTML" Resolution="72" Type="Linedraw" Width="440" /> </MediaObject> <EquationSource Format="TEX">$E(u)={{1}\over {N}}\int_{\mathbb{R}^{N}}|\nabla u|^{N}{\rm d}x+{{1}\over {N}}\int_{\mathbb{R}^{N}}V(x)|u|^{N}{\rm d}x-{{1}\over {s}}\int_{\mathbb{R}^{N}}|u|^{s}{\rm d}x.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>N</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msup> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>N</mi> </mrow> </msup> <mrow> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>+</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>N</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msup> </mrow> </msub> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>u</mi> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>N</mi> </mrow> </msup> <mrow> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>−</mo> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>s</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msup> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>u</mi> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>s</mi> </mrow> </msup> <mrow> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>.</mo> </math></EquationSource> </Equation>Here s > N, V is a spherically symmetric increasing function satisfying<Equation ID="Equ3"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_516_Article_Equ3.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="210" /> </MediaObject> <EquationSource Format="TEX">$V(0)=0,~\lim_{|x|\rightarrow\infty}V(x)=+\infty.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>V</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <munder> <mo form="prefix" movablelimits="true">lim</mo> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>V</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </math></EquationSource> </Equation>We discuss the problem in three cases. First, for the case s > 2N, es(ρ) = −∞ for any ρ > 0. Secondly, for the case N < s < 2N, for any ρ > 0, we prove that it admits a minimizer which is nonnegative, spherically symmetric and decreasing via the N-Laplacian Gagliardo-Nirenberg inequality. When s = 2N, the existence and nonexistence of minimizers of es(ρ) will also be given. During the arguments, we provide the detailed proof of the N-Laplacian Gagliardo-Nirenberg inequality and N-Laplacian Pohozaev identity.

Minimizers for the N-Laplacian

期刊论文
发表日期：

2025年8月4日

查看全文

Wenbo Wang,
Quanqing Li,
Wei Zhang,
Chunlei Tang

摘要

In this paper, we investigate the minimization problem $e_{s}(\rho)=\inf_{u\in W^{1,N}_{V}(\mathbb{R}^{N}),\|u\|^{N}_{N}=\rho>0}E(u),$ $e_{s} (ρ) = inf_{u \in W_{V}^{1, N} (R^{N}), ∥ u ∥_{N}^{N} = ρ > 0} E (u),$ where $E(u)={{1}\over {N}}\int_{\mathbb{R}^{N}}|\nabla u|^{N}{\rm d}x+{{1}\over {N}}\int_{\mathbb{R}^{N}}V(x)|u|^{N}{\rm d}x-{{1}\over {s}}\int_{\mathbb{R}^{N}}|u|^{s}{\rm d}x.$ $E (u) = \frac{1}{N} \int_{R^{N}} | \nabla u |^{N} d x + \frac{1}{N} \int_{R^{N}} V (x) | u |^{N} d x - \frac{1}{s} \int_{R^{N}} | u |^{s} d x .$

Here s > N, V is a spherically symmetric increasing function satisfying $V(0)=0,~\lim_{|x|\rightarrow\infty}V(x)=+\infty.$ $V (0) = 0, lim_{| x | \to \infty} V (x) = + \infty .$

We discuss the problem in three cases. First, for the case s > 2N, e_s(ρ) = −∞ for any ρ > 0. Secondly, for the case N < s < 2N, for any ρ > 0, we prove that it admits a minimizer which is nonnegative, spherically symmetric and decreasing via the N-Laplacian Gagliardo-Nirenberg inequality. When s = 2N, the existence and nonexistence of minimizers of e_s(ρ) will also be given. During the arguments, we provide the detailed proof of the N-Laplacian Gagliardo-Nirenberg inequality and N-Laplacian Pohozaev identity.