In this paper, we investigate a class of nonlinear Kirchhoff-type equation <Equation ID="Equa"> <EquationSource Format="MATHML"><math> <mo>−</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msub> <mo>∫</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> </msub> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mn>2</mn> </msup> <mi>d</mi> <mi>x</mi> <mo>)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>u</mi> <mo>+</mo> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mrow> <mi>p</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>,</mo> <mi>x</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> <mo>,</mo> </math></EquationSource> <EquationSource Format="TEX">$ -\left (a+b\int _{\mathbb{R}^{N}}|\nabla {u}|^{2}dx\right )\Delta {u}-f(u)= \lambda {u}+|u|^{p-2}{u},x\in {\mathbb{R}^{N}}, $</EquationSource> </Equation> where <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>></mo> <mn>0</mn> </math></EquationSource> <EquationSource Format="TEX">$a,b,c>0$</EquationSource> </InlineEquation> are prescribed, <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </math></EquationSource> <EquationSource Format="TEX">$\lambda \in \mathbb{R}$</EquationSource> </InlineEquation> arises as a Lagrange multiplier and the normalized constrain <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msub> <mo>∫</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> </msub> <mo stretchy="false">|</mo> <mi>u</mi> <msup> <mo stretchy="false">|</mo> <mn>2</mn> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msup> <mi>c</mi> <mn>2</mn> </msup> </math></EquationSource> <EquationSource Format="TEX">$\int _{\mathbb{R}^{N}}|u|^{2}dx=c^{2}$</EquationSource> </InlineEquation> is satisfied in the case <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mn>1</mn> <mo>≤</mo> <mi>N</mi> <mo>≤</mo> <mn>3</mn> </math></EquationSource> <EquationSource Format="TEX">$1\leq {N}\leq 3$</EquationSource> </InlineEquation>. The nonlinearity f is mass subcritical or supercritical and <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn> <mo>+</mo> <mfrac> <mn>8</mn> <mi>N</mi> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$2< p<2+\frac{8}{N}$</EquationSource> </InlineEquation> or <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mn>2</mn> <mo>+</mo> <mfrac> <mn>8</mn> <mi>N</mi> </mfrac> <mo><</mo> <mi>p</mi> <mo><</mo> <msup> <mn>2</mn> <mo>∗</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$2+\frac{8}{N}< p<2^{*}$</EquationSource> </InlineEquation>. By making a series of assumptions about f, we obtain the existence or the nonexistence of ground state solutions via minimizing methods.

Normalized solutions to a class of Kirchhoff equation with general nonlinearity: mass subcritical or supercritical

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2026年4月20日

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Ruqun Yi,
Aixia Qian

摘要

In this paper, we investigate a class of nonlinear Kirchhoff-type equation $- (a + b \int_{R^{N}} | \nabla u |^{2} d x) Δ u - f (u) = λ u + | u |^{p - 2} u, x \in R^{N},$ $ -\left (a+b\int _{\mathbb{R}^{N}}|\nabla {u}|^{2}dx\right )\Delta {u}-f(u)= \lambda {u}+|u|^{p-2}{u},x\in {\mathbb{R}^{N}}, $ where $a, b, c > 0$ $a,b,c>0$ are prescribed, $λ \in R$ $\lambda \in \mathbb{R}$ arises as a Lagrange multiplier and the normalized constrain $\int_{R^{N}} | u |^{2} d x = c^{2}$ $\int _{\mathbb{R}^{N}}|u|^{2}dx=c^{2}$ is satisfied in the case $1 \leq N \leq 3$ $1\leq {N}\leq 3$ . The nonlinearity f is mass subcritical or supercritical and $2 < p < 2 + \frac{8}{N}$ $2< p<2+\frac{8}{N}$ or $2 + \frac{8}{N} < p < 2^{*}$ $2+\frac{8}{N}< p<2^{*}$ . By making a series of assumptions about f, we obtain the existence or the nonexistence of ground state solutions via minimizing methods.