We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, is a bounded domain with a low-regularity boundary <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>. The coefficients <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$c, h \in L^p(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>,</mo> <mi>h</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$p > n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>></mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$c^\pm \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>c</mi> <mo>±</mo> </msup> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="173" /> </InlineMediaObject> <EquationSource Format="TEX">$c_\lambda (x):= \lambda c^+(x) - c^-(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mi>λ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mi>λ</mi> <msup> <mi>c</mi> <mo>+</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msup> <mi>c</mi> <mo>-</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for a real parameter <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. The matrix A(x) is uniformly positive definite and bounded, while M(x) is positive definite and bounded. Under suitable assumptions, we characterize the solution continuum of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$(P_\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mi>λ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, including its bifurcation points. We establish existence and uniqueness results in the coercive case (<InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda \le 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≤</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) and prove multiplicity results in the non-coercive case (<InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_459_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>).

Multiplicity and Bifurcation Results for a Class of Quasilinear Elliptic Problems with Quadratic Growth on the Gradient

期刊论文
发表日期：

2025年6月13日

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Fiorella Rendón,
Mayra Soares

摘要

We investigate the existence, non-existence, and multiplicity of solutions to the following class of quasilinear elliptic equations where $\Omega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ , $n\ge 3$ $n \geq 3$ , is a bounded domain with a low-regularity boundary $\partial \Omega $ $\partial Ω$ . The coefficients $c, h \in L^p(\Omega )$ $c, h \in L^{p} (Ω)$ for some $p > n$ $p > n$ , with $c^\pm \ge 0$ $c^{\pm} \geq 0$ and $c_\lambda (x):= \lambda c^+(x) - c^-(x)$ $c_{λ} (x) : = λ c^{+} (x) - c^{-} (x)$ for a real parameter $\lambda $ $λ$ . The matrix A(x) is uniformly positive definite and bounded, while M(x) is positive definite and bounded. Under suitable assumptions, we characterize the solution continuum of $(P_\lambda )$ $(P_{λ})$ , including its bifurcation points. We establish existence and uniqueness results in the coercive case ( $\lambda \le 0$ $λ \leq 0$ ) and prove multiplicity results in the non-coercive case ( $\lambda > 0$ $λ > 0$ ).