<p>In this paper, we investigate the existence of sign-changing solutions for the following problem: where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1170_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="132" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\alpha <1,\ m>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>α</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> <mspace width="4pt" /> <mi>m</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1170_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$N\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1170_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="153" /> </InlineMediaObject> <EquationSource Format="TEX">$ 0<\mu < \min \{N,4\alpha \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>μ</mi> <mo><</mo> <mo movablelimits="true">min</mo> <mo stretchy="false">{</mo> <mi>N</mi> <mo>,</mo> <mn>4</mn> <mi>α</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1170_Article_IEq6.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="99" /> </InlineMediaObject> <EquationSource Format="TEX">$2< p<\frac{2N-\mu }{N-2\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mfrac> <mrow> <mn>2</mn> <mi>N</mi> <mo>-</mo> <mi>μ</mi> </mrow> <mrow> <mi>N</mi> <mo>-</mo> <mn>2</mn> <mi>α</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. We show that problem <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2025_1170_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$(P_0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> possesses at least a radial sign-changing solution.</p>

Least energy radial sign-changing solutions for pseudo-relativistic Hartree equations in $\mathbb {R}^N$

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发表日期：

2025年3月10日

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Zhihua Huang,
Jianfu Yang,
Weilin Yu

摘要

In this paper, we investigate the existence of sign-changing solutions for the following problem: where $0<\alpha <1,\ m>0$ $0 < α < 1, m > 0$ , $N\ge 3$ $N \geq 3$ , $ 0<\mu < \min \{N,4\alpha \}$ $0 < μ < min {N, 4 α}$ and $2< p<\frac{2N-\mu }{N-2\alpha }$ $2 < p < \frac{2 N - μ}{N - 2 α}$ . We show that problem $(P_0)$ $(P_{0})$ possesses at least a radial sign-changing solution.

Least energy radial sign-changing solutions for pseudo-relativistic Hartree equations in \(\mathbb {R}^N\)

摘要