We are studying the stationary quantum Zakharov systems <Equation ID="Equ1"> <EquationNumber>1</EquationNumber> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2u-\Delta u+W_\lambda (x) u+\beta \phi u=g(x,u), & \quad x\in {\mathbb {R}}^3,\\ -\Delta \phi +\phi =u^2,& \quad x\in {\mathbb {R}}^3,\\ \end{array} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mi>u</mi> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <msub> <mi>W</mi> <mi>λ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>+</mo> <mi>β</mi> <mi>ϕ</mi> <mi>u</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>+</mo> <mi>ϕ</mi> <mo>=</mo> <msup> <mi>u</mi> <mn>2</mn> </msup> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Delta ^2:=\Delta (\Delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mo>:</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$W_\lambda (x)=\lambda W^+(x)-W^-(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>λ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>λ</mi> <msup> <mi>W</mi> <mo>+</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msup> <mi>W</mi> <mo>-</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$W^{\pm }(x)=\max \lbrace \pm W(x),0\rbrace $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mo>±</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true">max</mo> <mrow> <mo stretchy="false">{</mo> <mo>±</mo> <mi>W</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Under suitable assumptions, we prove the existence of nontrivial solution for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> large enough. Our main contribution is to consider stationary quantum Zakharov system taking into account the case where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta \in {\mathbb {R}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, particularly, if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\beta <0,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo><</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we are keen to determine whether Problem (<InternalRef RefID="Equ1">1</InternalRef>) has at least one solution considering the interaction between the terms g(x, u) and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\beta \phi u.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mi>ϕ</mi> <mi>u</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Our results extend the related results in the literature (e.g., Example Sun et al. in Appl Math Lett 95:172–178, 2019; Benhana in Complex Variables Elliptic Equ. <a href="https://doi.org/10.1080/17476933.2023.2229736">https://doi.org/10.1080/17476933.2023.2229736</a>. 2023).

Existence of a nontrivial solution for the stationary quantum Zakharov system

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发表日期：

2025年7月7日

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Abdelkader Hakim Benhana

摘要

We are studying the stationary quantum Zakharov systems 1 $\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2u-\Delta u+W_\lambda (x) u+\beta \phi u=g(x,u), & \quad x\in {\mathbb {R}}^3,\\ -\Delta \phi +\phi =u^2,& \quad x\in {\mathbb {R}}^3,\\ \end{array} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} Δ^{2} u - Δ u + W_{λ} (x) u + β ϕ u = g (x, u), & x \in R^{3}, \\ - Δ ϕ + ϕ = u^{2}, & x \in R^{3}, \end{matrix}) \end{matrix}$ where $\Delta ^2:=\Delta (\Delta )$ $Δ^{2} : = Δ (Δ)$ and $W_\lambda (x)=\lambda W^+(x)-W^-(x)$ $W_{λ} (x) = λ W^{+} (x) - W^{-} (x)$ with $W^{\pm }(x)=\max \lbrace \pm W(x),0\rbrace $ $W^{\pm} (x) = max {\pm W (x), 0}$ . Under suitable assumptions, we prove the existence of nontrivial solution for $\lambda $ $λ$ large enough. Our main contribution is to consider stationary quantum Zakharov system taking into account the case where $\beta \in {\mathbb {R}}$ $β \in R$ , particularly, if $\beta <0,$ $β < 0,$ we are keen to determine whether Problem (1) has at least one solution considering the interaction between the terms g(x, u) and $\beta \phi u.$ $β ϕ u .$ Our results extend the related results in the literature (e.g., Example Sun et al. in Appl Math Lett 95:172–178, 2019; Benhana in Complex Variables Elliptic Equ. https://doi.org/10.1080/17476933.2023.2229736. 2023).