<p>In this paper, we are concerned with the following nonlocal problem: <Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2024_1986_Article_Equa.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="623" /> </MediaObject> <EquationSource Format="MATHML"><math> <mo>−</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>−</mo> <mi>ϵ</mi> <msub> <mo>∫</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> </msub> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mn>2</mn> </msup> <mi>d</mi> <mi>x</mi> <mo>)</mo> </mrow> <mtext>div</mtext> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>q</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>+</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mn>4</mn> </msup> <mi>u</mi> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mn>3</mn> </msup> <mo>,</mo> </math></EquationSource> <EquationSource Format="TEX">$ -\left (a-\epsilon \displaystyle \int _{\mathbb{R}^{3}} K(x)| \nabla u|^{2}dx\right )\text{div}(K(x)\nabla u)=\lambda K(x)f(x)|u|^{q-2}u+K(x)|u|^{4}u, \quad x\in \mathbb{R}^{3}, $</EquationSource> </Equation> where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2024_1986_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>a</mi> <mo>,</mo> <mi>λ</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$a, \lambda >0$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2024_1986_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>1</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$1< q<2$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2024_1986_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="141" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>K</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>exp</mo> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>x</mi> <msup> <mo stretchy="false">|</mo> <mi>α</mi> </msup> <mo stretchy="false">/</mo> <mn>4</mn> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$K(x)=\exp ({|x|^{\alpha}/4})$</EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2024_1986_Article_IEq4.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>α</mi> <mo>≥</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$\alpha \geq 2$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2024_1986_Article_IEq5.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$\epsilon >0$</EquationSource> </InlineEquation> is small enough, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13661_2024_1986_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$f(x)\ge 0$</EquationSource> </InlineEquation> satisfies some integrability condition. By using the Ekeland variational principle and the concentration compactness principle, we establish the existence of two positive solutions for the problem and prove that at least one of them is a positive ground state solution.</p>

Multiple positive solutions for a nonlocal problem with fast increasing weight and critical exponent

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2025年1月6日

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Xiaotao Qian,
Zhigao Shi

摘要

In this paper, we are concerned with the following nonlocal problem: $- (a - ϵ \int_{R^{3}} K (x) | \nabla u |^{2} d x) div (K (x) \nabla u) = λ K (x) f (x) | u |^{q - 2} u + K (x) | u |^{4} u, x \in R^{3},$ $ -\left (a-\epsilon \displaystyle \int _{\mathbb{R}^{3}} K(x)| \nabla u|^{2}dx\right )\text{div}(K(x)\nabla u)=\lambda K(x)f(x)|u|^{q-2}u+K(x)|u|^{4}u, \quad x\in \mathbb{R}^{3}, $ where $a, λ > 0$ $a, \lambda >0$ , $1 < q < 2$ $1< q<2$ , $K (x) = exp (| x |^{α} / 4)$ $K(x)=\exp ({|x|^{\alpha}/4})$ with $α \geq 2$ $\alpha \geq 2$ , $ϵ > 0$ $\epsilon >0$ is small enough, and $f (x) \geq 0$ $f(x)\ge 0$ satisfies some integrability condition. By using the Ekeland variational principle and the concentration compactness principle, we establish the existence of two positive solutions for the problem and prove that at least one of them is a positive ground state solution.