In this paper, we construct multi-bump solutions for the following coupled system of Schrödinger equations with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2529_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">$\chi ^{(2)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>χ</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> nonlinearity <Equation ID="Equ41"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2529_Article_Equ41.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="347" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_{1}+(1+\varepsilon P(x)) u_{1} = \alpha u_{1} u_{2}, & x \in \mathbb {R}^{N},\\ -\Delta u_{2}+(1+\varepsilon Q(x)) u_{2} = \frac{\alpha }{2}u_{1}^{2}+\beta u_{2}^{2},& x \in \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>α</mi> <msub> <mi>u</mi> <mn>1</mn> </msub> <msub> <mi>u</mi> <mn>2</mn> </msub> <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>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mi>α</mi> <mn>2</mn> </mfrac> <msubsup> <mi>u</mi> <mrow> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mi>β</mi> <msubsup> <mi>u</mi> <mrow> <mn>2</mn> </mrow> <mn>2</mn> </msubsup> <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>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2529_Article_IEq4.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2529_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha >\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mi>β</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2529_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$2\le N <6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>≤</mo> <mi>N</mi> <mo><</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2529_Article_IEq7.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a small parameter, the positive potentials P(x), Q(x) are continuous and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2529_Article_IEq8.gif" Format="GIF" Height="29" Rendition="HTML" Resolution="72" Type="Linedraw" Width="220" /> </InlineMediaObject> <EquationSource Format="TEX">$\lim \limits _{|x|\rightarrow \infty }P(x)=0, \lim \limits _{|x|\rightarrow \infty }Q(x)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder> <mo movablelimits="false">lim</mo> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <munder> <mo movablelimits="false">lim</mo> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

Existence of multi-bump solutions for a nonlinearly coupled Schrödinger equations with $\chi ^{(2)}$ nonlinearity

期刊论文
发表日期：

2025年6月23日

查看全文

Xiong Gu,
Weiming Liu

摘要

In this paper, we construct multi-bump solutions for the following coupled system of Schrödinger equations with $\chi ^{(2)}$ $χ^{(2)}$ nonlinearity $\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_{1}+(1+\varepsilon P(x)) u_{1} = \alpha u_{1} u_{2}, & x \in \mathbb {R}^{N},\\ -\Delta u_{2}+(1+\varepsilon Q(x)) u_{2} = \frac{\alpha }{2}u_{1}^{2}+\beta u_{2}^{2},& x \in \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} - Δ u_{1} + (1 + ε P (x)) u_{1} = α u_{1} u_{2}, & x \in R^{N}, \\ - Δ u_{2} + (1 + ε Q (x)) u_{2} = \frac{α}{2} u_{1}^{2} + β u_{2}^{2}, & x \in R^{N}, \end{matrix}) \end{matrix}$ where $\alpha >0$ $α > 0$ , $\alpha >\beta $ $α > β$ , $2\le N <6$ $2 \leq N < 6$ and $\varepsilon >0$ $ε > 0$ is a small parameter, the positive potentials P(x), Q(x) are continuous and $\lim \limits _{|x|\rightarrow \infty }P(x)=0, \lim \limits _{|x|\rightarrow \infty }Q(x)=0$ $lim_{| x | \to \infty} P (x) = 0, lim_{| x | \to \infty} Q (x) = 0$ .

Existence of multi-bump solutions for a nonlinearly coupled Schrödinger equations with \(\chi ^{(2)}\) nonlinearity

摘要