We prove a multiplicity result for non-constant weak solutions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u \in H^1(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>∈</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for the quasilinear elliptic equation <Equation ID="Equ25"> <EquationSource Format="TEX">$ {\left\{ \begin{array}{ll} -\textrm{div}(A(x,u)\nabla u) + \dfrac{1}{2} D_s A(x,u)\,\nabla u \cdot \nabla u = g(x,u) - \lambda u & \text {in } \Omega ,\\ A(x,u)\nabla u \cdot \eta = 0 & \text {on } \partial \Omega , \end{array}\right. } $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <mo>-</mo> <mtext>div</mtext> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> <msub> <mi>D</mi> <mi>s</mi> </msub> <mi>A</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo>·</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>λ</mi> <mi>u</mi> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>u</mi> <mo>·</mo> <mi>η</mi> <mo>=</mo> <mn>0</mn> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mtext>on</mtext> <mspace width="0.333333em" /> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda \in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a bounded Lipschitz domain, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation> is the outward normal to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, and g(x, u) is a Carathéodory function satisfying a general subcritical and superlinear growth condition. We also prove that any weak solution is bounded under a stronger growth assumption.

Quasilinear Equations with Neumann Boundary Conditions

期刊论文
发表日期：

2026年1月12日

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Annamaria Canino,
Simone Mauro

摘要

We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega )$ $u \in H^{1} (Ω)$ for the quasilinear elliptic equation $ {\left\{ \begin{array}{ll} -\textrm{div}(A(x,u)\nabla u) + \dfrac{1}{2} D_s A(x,u)\,\nabla u \cdot \nabla u = g(x,u) - \lambda u & \text {in } \Omega ,\\ A(x,u)\nabla u \cdot \eta = 0 & \text {on } \partial \Omega , \end{array}\right. } $ $\{\begin{matrix} - div (A (x, u) \nabla u) + \frac{1}{2} D_{s} A (x, u) \nabla u \cdot \nabla u = g (x, u) - λ u & in Ω, \\ A (x, u) \nabla u \cdot η = 0 & on \partial Ω, \end{matrix})$ where $\lambda \in \mathbb {R}$ $λ \in R$ , $\Omega $ $Ω$ is a bounded Lipschitz domain, $\eta $ $η$ is the outward normal to $\partial \Omega $ $\partial Ω$ , and g(x, u) is a Carathéodory function satisfying a general subcritical and superlinear growth condition. We also prove that any weak solution is bounded under a stronger growth assumption.