We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a reaction which exhibits the competing effects of a parametric singular term <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1957_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$\lambda x^{-\eta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <msup> <mi>x</mi> <mrow> <mo>-</mo> <mi>η</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1957_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="72" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\eta <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>η</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) and of a Caratheodory perturbation f(z, x) which is “superlinear" in x but without satisfying the usual in such cases Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we prove an existence and multiplicity result which is global in the parameter <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1957_Article_IEq3.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>. In the process we show also the boundedness of the weak solutions.

Global Existence and Multiplicity of Positive Solutions for Singular Double Phase Equations

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发表日期：

2025年8月30日

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Jiangfeng Han,
Zhenhai Liu,
Nikolaos S. Papageorgiou

摘要

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a reaction which exhibits the competing effects of a parametric singular term $\lambda x^{-\eta }$ $λ x^{- η}$ ( $0<\eta <1$ $0 < η < 1$ ) and of a Caratheodory perturbation f(z, x) which is “superlinear" in x but without satisfying the usual in such cases Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we prove an existence and multiplicity result which is global in the parameter $\lambda >0$ $λ > 0$ . In the process we show also the boundedness of the weak solutions.