The aim of this paper is to establish the existence of the first (smallest) eigenvalue <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3767_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> for a nonlinear elliptic problem driven by the nonhomogeneous (p, q)-Laplace operator <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3767_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$-\Delta _p -\Delta _q $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </msub> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>q</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> in a bounded domain with a source term involving the exponent <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3767_Article_IEq3.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3767_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$q<\gamma \le p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo><</mo> <mi>γ</mi> <mo>≤</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation>. We show that <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3767_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>λ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is simple and associated to a unique and bounded eigenfunction <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3767_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$u_1>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. In the second part, using variational arguments, we study two types of nonlinear problems involving the nonhomogeneous (p, q)-Laplace operator, in particular we study two classes of sublinear and superlinear (p, q)-Laplacian problems with parameters.

On the first eigenvalue of the (p, q)-Laplacian and some related problems

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发表日期：

2025年5月6日

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Said El Manouni,
Kanishka Perera,
Patrick Winkert

摘要

The aim of this paper is to establish the existence of the first (smallest) eigenvalue $\lambda _1$ $λ_{1}$ for a nonlinear elliptic problem driven by the nonhomogeneous (p, q)-Laplace operator $-\Delta _p -\Delta _q $ $- Δ_{p} - Δ_{q}$ in a bounded domain with a source term involving the exponent $\gamma $ $γ$ with $q<\gamma \le p$ $q < γ \leq p$ . We show that $\lambda _1$ $λ_{1}$ is simple and associated to a unique and bounded eigenfunction $u_1>0$ $u_{1} > 0$ . In the second part, using variational arguments, we study two types of nonlinear problems involving the nonhomogeneous (p, q)-Laplace operator, in particular we study two classes of sublinear and superlinear (p, q)-Laplacian problems with parameters.