<p>In this paper, we examine the following quasilinear Schrödinger equation <Equation ID="Equ42"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1857_Article_Equ42.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="343" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} -\Delta u-\Delta \left( u^{2}\right) u = \left( |x|^{\mu -n}*|u|^q\right) |u|^{q-2}u \text{ in } {\mathbb {R}}^{n}, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mfenced close=")" open="("> <msup> <mi>u</mi> <mn>2</mn> </msup> </mfenced> <mi>u</mi> <mo>=</mo> <mfenced close=")" open="("> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>μ</mi> <mo>-</mo> <mi>n</mi> </mrow> </msup> <mrow /> <mo>∗</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mi>q</mi> </msup> </mfenced> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>q</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1857_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$q>3+\sqrt{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>></mo> <mn>3</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </math></EquationSource> </InlineEquation>. Under certain appropriate assumptions on <i>n</i> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1857_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>, we establish a Liouville type theorem for the class of stable bounded sign-changing solutions to this problem.</p>

Stable Solutions to Quasilinear Schrödinger Equations of Choquard Type

期刊论文
发表日期：

2025年3月19日

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Belgacem Rahal

摘要

In this paper, we examine the following quasilinear Schrödinger equation $\begin{aligned} -\Delta u-\Delta \left( u^{2}\right) u = \left( |x|^{\mu -n}*|u|^q\right) |u|^{q-2}u \text{ in } {\mathbb {R}}^{n}, \end{aligned}$ $\begin{matrix} - Δ u - Δ (u^{2}) u = ({| x |}^{μ - n} * {| u |}^{q}) {| u |}^{q - 2} u in R^{n}, \end{matrix}$ where $q>3+\sqrt{2}$ $q > 3 + \sqrt{2}$ . Under certain appropriate assumptions on n and $\mu $ $μ$ , we establish a Liouville type theorem for the class of stable bounded sign-changing solutions to this problem.