<p>In this work, we consider the Hénon- Kirchhoff–Boussinesq type problem <Equation ID="Equ18"> <EquationSource Format="TEX">$\begin{aligned} \Delta ^2 u \pm \operatorname {div} \big (|x|^{\kappa } | \nabla u |^{p-2} \nabla u \big ) = |x|^\ell f(u) {\text { in }} B,\qquad u = \frac{\partial u}{\partial \nu } = 0 {\text { on }} \partial B, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msup> <mi>u</mi> <mo>±</mo> <msup> <mrow> <mo>div</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mi>κ</mi> </msup> <msup> <mrow> <mo stretchy="false">|</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mi>ℓ</mi> </msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">in</mi> <mi>B</mi> <mo>,</mo> <mspace width="2em" /> <mi>u</mi> <mo>=</mo> <mfrac> <mrow> <mi>∂</mi> <mi>u</mi> </mrow> <mrow> <mi>∂</mi> <mi>ν</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mi mathvariant="normal">on</mi> <mi>∂</mi> <mi>B</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$B\subset \mathbb {R}^{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N \ge 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, is the unit ball, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\ell >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ℓ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$2<p<2(N+\kappa )/(N-2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mi>κ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$f(u) \sim |u|^{q-2}u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>∼</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>q</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$2<q<2(N+\ell )/(N-4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>N</mi> <mo>+</mo> <mi>ℓ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. By using Variational Methods, we obtain the existence and multiplicity of weak solution when <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\kappa >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is small. Note that <i>f</i> may grow faster than the critical exponent <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$2N/(N-4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>N</mi> <mo stretchy="false">/</mo> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the Sobolev embedding of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$H^2(B)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

On a Kirchhoff–Boussinesq equation with Hénon-type nonlinearity

期刊论文
发表日期：

2025年11月27日

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R. D. Carlos,
M. F. Furtado,
T. G. Melo

摘要

In this work, we consider the Hénon- Kirchhoff–Boussinesq type problem $\begin{aligned} \Delta ^2 u \pm \operatorname {div} \big (|x|^{\kappa } | \nabla u |^{p-2} \nabla u \big ) = |x|^\ell f(u) {\text { in }} B,\qquad u = \frac{\partial u}{\partial \nu } = 0 {\text { on }} \partial B, \end{aligned}$ $\begin{matrix} Δ^{2} u \pm {div (| x |}^{κ} {| \nabla u |}^{p - 2} \nabla u) = {| x |}^{ℓ} f (u) in B, u = \frac{\partial u}{\partial ν} = 0 on \partial B, \end{matrix}$ where $B\subset \mathbb {R}^{N}$ $B \subset R^{N}$ , $N \ge 5$ $N \geq 5$ , is the unit ball, $\ell >0$ $ℓ > 0$ , $2<p<2(N+\kappa )/(N-2)$ $2 < p < 2 (N + κ) / (N - 2)$ and $f(u) \sim |u|^{q-2}u$ $f (u) \sim {| u |}^{q - 2} u$ , with $2<q<2(N+\ell )/(N-4)$ $2 < q < 2 (N + ℓ) / (N - 4)$ . By using Variational Methods, we obtain the existence and multiplicity of weak solution when $\kappa >0$ $κ > 0$ is small. Note that f may grow faster than the critical exponent $2N/(N-4)$ $2 N / (N - 4)$ for the Sobolev embedding of $H^2(B)$ $H^{2} (B)$ .