Limit Property of an <i>L</i><sup>2</sup>-Normalized Solution for an <i>L</i><sup>2</sup>-Subcritical Kirchhoff-Type Equation with a Variable Exponent

Limit Property of an <i>L</i><sup>2</sup>-Normalized Solution for an <i>L</i><sup>2</sup>-Subcritical Kirchhoff-Type Equation with a Variable Exponent

This paper is concerned with the following <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-su...

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Bibliographic Details
Main Authors:	Xincai Zhu, Hanxiao Wu
Format:	Article
Language:	English
Published:	MDPI AG 2024-08-01
Series:	Axioms
Subjects:	L2-subcritical Kirchhoff-type equation variable exponent L2-normalized solution limit property
Online Access:	https://www.mdpi.com/2075-1680/13/9/571
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Summary:	This paper is concerned with the following <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-subcritical Kirchhoff-type equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mfenced separators="" open="(" close=")"><mi>a</mi><mo>+</mo><mi>b</mi><msup><mfenced separators="" open="(" close=")"><msub><mo>∫</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mn>2</mn></msup></msub><msup><mrow><mo>\|</mo><mo>∇</mo><mi>u</mi><mo>\|</mo></mrow><mn>2</mn></msup><mi>d</mi><mi>x</mi></mfenced><mi>s</mi></msup></mfenced><mo>Δ</mo><mi>u</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>u</mi><mo>=</mo><mi>μ</mi><mi>u</mi><mo>+</mo><mi>β</mi><msup><mrow><mo>\|</mo><mi>u</mi><mo>\|</mo></mrow><mn>2</mn></msup><mi>u</mi><mo>,</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mi>x</mi><mo>∈</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mn>2</mn></msup></mrow></semantics></math></inline-formula>, with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo>∫</mo><msup><mrow><mi mathvariant="double-struck">R</mi></mrow><mn>2</mn></msup></msub><msup><mrow><mo>\|</mo><mi>u</mi><mo>\|</mo></mrow><mn>2</mn></msup><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>. We give a detailed analysis of the limit property of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-normalized solution when exponent <i>s</i> tends toward 0 from the right (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>↘</mo><mn>0</mn></mrow></semantics></math></inline-formula>). Our research extends previous works, in which the authors have displayed the limit behavior of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-normalized solutions when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>↘</mo><mn>0</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>b</mi><mo>↘</mo><mn>0</mn></mrow></semantics></math></inline-formula>.
ISSN:	2075-1680