<p>We prove existence of solutions for the following nonlinear Dirichlet system: <Equation ID="Equ71"> <EquationSource Format="TEX">$ \left\{ \begin{array}{cl} -\mathop {{\textrm{div}}}(A(x)\nabla u) + u = - \mathop {{\textrm{div}}}(u\,M(x)\nabla \psi ) + f(x) & \text{ in } \Omega \text{, } \\ -\mathop {{\textrm{div}}}(M(x)\nabla \psi ) = u^{\theta } & \text{ in } \Omega \text{, }\\ u = 0 = \psi & \text{ on } \partial \Omega \text{. } \end{array} \right. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd> <mrow> <mo>-</mo> <mtext>div</mtext> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>u</mi> <mo>=</mo> <mo>-</mo> <mtext>div</mtext> <mo stretchy="false">(</mo> <mi>u</mi> <mspace width="0.166667em" /> <mi>M</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mtext>,</mtext> <mspace width="0.333333em" /> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mo>-</mo> <mtext>div</mtext> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>u</mi> <mi>θ</mi> </msup> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mtext>,</mtext> <mspace width="0.333333em" /> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mi>ψ</mi> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="0.333333em" /> <mtext>on</mtext> <mspace width="0.333333em" /> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mtext>.</mtext> <mspace width="0.333333em" /> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>under various assumptions on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\theta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>θ</mi> </math></EquationSource> </InlineEquation>: either <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0< \theta < \frac{1}{2} + \frac{1}{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>θ</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\theta = \frac{2}{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>2</mn> <mi>N</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation>.</p>

Some existence results for nonlinear elliptic systems with a convection term

期刊论文
发表日期：

2026年2月28日

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Lucio Boccardo,
Luigi Orsina

摘要

We prove existence of solutions for the following nonlinear Dirichlet system: $ \left\{ \begin{array}{cl} -\mathop {{\textrm{div}}}(A(x)\nabla u) + u = - \mathop {{\textrm{div}}}(u\,M(x)\nabla \psi ) + f(x) & \text{ in } \Omega \text{, } \\ -\mathop {{\textrm{div}}}(M(x)\nabla \psi ) = u^{\theta } & \text{ in } \Omega \text{, }\\ u = 0 = \psi & \text{ on } \partial \Omega \text{. } \end{array} \right. $ $\{\begin{matrix} - div (A (x) \nabla u) + u = - div (u M (x) \nabla ψ) + f (x) & in Ω, \\ - div (M (x) \nabla ψ) = u^{θ} & in Ω, \\ u = 0 = ψ & on \partial Ω . \end{matrix})$ under various assumptions on $\theta $ $θ$ : either $0< \theta < \frac{1}{2} + \frac{1}{N}$ $0 < θ < \frac{1}{2} + \frac{1}{N}$ , or $\theta = \frac{2}{N}$ $θ = \frac{2}{N}$ .