<p>The existence of regular and singular bound state solutions to <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2092_Article_Equa.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="223" /> </MediaObject> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">△</mi> <mi>p</mi> </msub> <mi>u</mi> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.3em" /> <mspace width="0.3em" /> <mspace width="0.3em" /> <mi>r</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> <mi mathvariant="normal">∖</mi> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </math></EquationSource> <EquationSource Format="TEX">\( \triangle _{p}u+f(u)=0,~~~r\in \mathbb{R}^{n}\backslash \{0\} \)</EquationSource> </Equation> is considered. Our result concerns the solution according to its behavior as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2092_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>r</mi> <mo stretchy="false">→</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$r\rightarrow 0$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2092_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>r</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></EquationSource> <EquationSource Format="TEX">$r\rightarrow \infty $</EquationSource> </InlineEquation>. Under the assumption that <i>f</i> is supercritical for small <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2092_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>u</mi> <mo>&gt;</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$u&gt;0$</EquationSource> </InlineEquation> and is subcritical for large <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2025_2092_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>u</mi> <mo>&gt;</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$u&gt;0$</EquationSource> </InlineEquation>, we show the existence of various types of solutions. The Pohozaev identity plays a crucial role in our investigation.</p>

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Existence of regular and singular bound state solutions to a quasilinear equation

  • Wei-Chuan Wang

摘要

The existence of regular and singular bound state solutions to p u + f ( u ) = 0 , r R n { 0 } \( \triangle _{p}u+f(u)=0,~~~r\in \mathbb{R}^{n}\backslash \{0\} \) is considered. Our result concerns the solution according to its behavior as r 0 $r\rightarrow 0$ and r $r\rightarrow \infty $ . Under the assumption that f is supercritical for small u > 0 $u>0$ and is subcritical for large u > 0 $u>0$ , we show the existence of various types of solutions. The Pohozaev identity plays a crucial role in our investigation.