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<p>For a generalized second-order differential equation of the Emden–Fowler type with an unbounded power-law potential, we examine the monotonicity and boundary behavior of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mu \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>-solutions defined on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((0;+\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>;</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the asymptotic properties of solutions defined on (0,&#xa0;<i>h</i>), where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(h&gt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>h</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, that tend to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(-\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> at zero, for various relationships between the parameters <i>k</i>, <i>m</i>, and&#xa0;<i>n</i>.</p>

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QUALITATIVE PROPERTIES OF SOLUTIONS OF SOME DIFFERENTIAL EQUATIONS WITH UNBOUNDED POWER-LAW POTENTIALS

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • O. D. Prokopenko

摘要

For a generalized second-order differential equation of the Emden–Fowler type with an unbounded power-law potential, we examine the monotonicity and boundary behavior of \(\mu \) μ -solutions defined on \((0;+\infty )\) ( 0 ; + ∞ ) and the asymptotic properties of solutions defined on (0, h), where \(h>0\) h > 0 , that tend to \(-\infty \) - ∞ at zero, for various relationships between the parameters k, m, and n.