In this work, we are concerned with the class of systems <Equation ID="Equ90"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda a(x)u^q+\tau c(x)u^{\alpha -1} v^\beta & \text{ in }\Omega \\ -\Delta v=\mu b(x)v^p+\delta c(x)u^\alpha v^{\beta -1} & \text{ in }\Omega \\ 0\not \equiv u\ge 0,\,\,0\not \equiv v\ge 0\,& \text{ in }\Omega \\ u=v=0& \text{ on }\partial \Omega \end{array} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>=</mo> <mi>λ</mi> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>u</mi> <mi>q</mi> </msup> <mo>+</mo> <mi>τ</mi> <mi>c</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>u</mi> <mrow> <mi>α</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>v</mi> <mi>β</mi> </msup> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi mathvariant="normal">Ω</mi> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>=</mo> <mi>μ</mi> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>v</mi> <mi>p</mi> </msup> <mo>+</mo> <mi>δ</mi> <mi>c</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>u</mi> <mi>α</mi> </msup> <msup> <mi>v</mi> <mrow> <mi>β</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi mathvariant="normal">Ω</mi> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mn>0</mn> <mo>≢</mo> <mi>u</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <mn>0</mn> <mo>≢</mo> <mi>v</mi> <mo>≥</mo> <mn>0</mn> <mspace width="0.166667em" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi mathvariant="normal">Ω</mi> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>u</mi> <mo>=</mo> <mi>v</mi> <mo>=</mo> <mn>0</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mspace width="3.33333pt" /> <mspace width="3.33333pt" /> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a smooth bounded domain in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {R}}^N$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>, a, b, c belong to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^\infty (\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lambda ,\,\mu ,\,\tau ,\,\delta >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>μ</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>τ</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>δ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p,q\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha ,\,\beta \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mspace width="0.166667em" /> <mi>β</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha +\beta >2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.

Some results on a gradient system with concave-convex and supercritical nonlinearities

期刊论文
发表日期：

2025年12月13日

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João Pablo Pinheiro Da Silva

摘要

In this work, we are concerned with the class of systems $\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda a(x)u^q+\tau c(x)u^{\alpha -1} v^\beta & \text{ in }~~\Omega \\ -\Delta v=\mu b(x)v^p+\delta c(x)u^\alpha v^{\beta -1} & \text{ in }~~\Omega \\ 0\not \equiv u\ge 0,\,\,0\not \equiv v\ge 0\,& \text{ in }~~\Omega \\ u=v=0& \text{ on }~~\partial \Omega \end{array} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} - Δ u = λ a (x) u^{q} + τ c (x) u^{α - 1} v^{β} & in Ω \\ - Δ v = μ b (x) v^{p} + δ c (x) u^{α} v^{β - 1} & in Ω \\ 0 ≢ u \geq 0, 0 ≢ v \geq 0 & in Ω \\ u = v = 0 & on \partial Ω \end{matrix}) \end{matrix}$ where $\Omega $ $Ω$ is a smooth bounded domain in ${\mathbb {R}}^N$ $R^{N}$ , a, b, c belong to $L^\infty (\Omega )$ $L^{\infty} (Ω)$ , $\lambda ,\,\mu ,\,\tau ,\,\delta >0$ $λ, μ, τ, δ > 0$ , $p,q\in (0,1)$ $p, q \in (0, 1)$ , $\alpha ,\,\beta \ge 1$ $α, β \geq 1$ and $\alpha +\beta >2$ $α + β > 2$ .