We deal with the boundary value problem <Equation ID="Equ19"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_410_Article_Equ19.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="281" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u(x)= \lambda f(u(x)), & x\in \Omega ,\\ u(x)=0, & x\in \mathbb {R}^N \setminus \Omega , \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_410_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is an open and bounded subset of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_410_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation> with smooth boundary, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_410_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="47" /> </InlineMediaObject> <EquationSource Format="TEX">$(-\Delta )^s$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_410_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$s\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the fractional Laplacian, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13540_2025_410_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and f is locally Lipschitz and continuous. We provide necessary and sufficient conditions on f to ensure existence and multiplicity of bounded solutions between two zeros of f.

Existence, nonexistence and multiplicity of bounded solutions to a nonlinear BVP associated to the fractional Laplacian

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2025年5月5日

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José Carmona Tapia,
Rubén Fiñana Aránega

摘要

We deal with the boundary value problem $\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u(x)= \lambda f(u(x)), & x\in \Omega ,\\ u(x)=0, & x\in \mathbb {R}^N \setminus \Omega , \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} {(- Δ)}^{s} u (x) = λ f (u (x)), & x \in Ω, \\ u (x) = 0, & x \in R^{N} \ Ω, \end{matrix}) \end{matrix}$ where $\Omega $ $Ω$ is an open and bounded subset of $\mathbb {R}^N$ $R^{N}$ with smooth boundary, $(-\Delta )^s$ ${(- Δ)}^{s}$ , $s\in (0,1)$ $s \in (0, 1)$ denotes the fractional Laplacian, $\lambda \ge 0$ $λ \geq 0$ and f is locally Lipschitz and continuous. We provide necessary and sufficient conditions on f to ensure existence and multiplicity of bounded solutions between two zeros of f.