Scholarlink

<p>We consider a discrete analog of the differential equation of the Emden–Fowler type <Equation ID="Equ39"> <EquationSource Format="TEX">\( \Delta ^2v(k)=-k^s (\Delta v(k))^3, \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>-</mo> <msup> <mi>k</mi> <mi>s</mi> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(s \ne 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≠</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(s\in \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\Delta v(k)=v(k+1)-v(k)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>-</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. It is a discrete analog of the second-order nonlinear equation <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(y''(x)=y^s(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>y</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>y</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We prove the existence of an approximate solution of the form <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(V(k)=\pm \dfrac{\sqrt{2s+2}}{s-1} k^{(1-s)/2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo>±</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <msqrt> <mrow> <mn>2</mn> <mi>s</mi> <mo>+</mo> <mn>2</mn> </mrow> </msqrt> <mrow> <mi>s</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> <msup> <mi>k</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and a nontrivial solution tending to 0 as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(k \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

ANALOG OF BLOW-UP SOLUTIONS OF A DISCRETE SECOND-ORDER EQUATION OF THE EMDEN–FOWLER TYPE

  • 期刊论文

  • 发表日期：

    2026年2月1日

查看全文
  • E. V. Korobko

摘要

We consider a discrete analog of the differential equation of the Emden–Fowler type \( \Delta ^2v(k)=-k^s (\Delta v(k))^3, \) Δ 2 v ( k ) = - k s ( Δ v ( k ) ) 3 , where \(k \rightarrow \infty \) k → ∞ , \(s \ne 1\) s ≠ 1 , \(s\in \mathbb {R}\) s ∈ R , and \(\Delta v(k)=v(k+1)-v(k)\) Δ v ( k ) = v ( k + 1 ) - v ( k ) . It is a discrete analog of the second-order nonlinear equation \(y''(x)=y^s(x)\) y ′ ′ ( x ) = y s ( x ) . We prove the existence of an approximate solution of the form \(V(k)=\pm \dfrac{\sqrt{2s+2}}{s-1} k^{(1-s)/2}\) V ( k ) = ± 2 s + 2 s - 1 k ( 1 - s ) / 2 and a nontrivial solution tending to 0 as \(k \rightarrow \infty \) k → ∞ .