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<p>We consider a nonlinear Dirichlet equation driven by the weighted <i>p</i>-Laplacian with weight <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2162_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="28" /> </InlineMediaObject> <EquationSource Format="TEX">\(a(\cdot )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> which is not bounded away from zero (degenerate problem). The reaction (right hand side) is parametric and exhibits the combined effects of a singular term and of indefinite (sign-changing), (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2162_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(p-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>)-superlinear perturbation. Using variational tools, truncations, comparisons and critical groups, we show that for all small values of the parameter, the problem has at least two bounded positive solutions.</p>

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A degenerate singular equation with a superlinear indefinite perturbation

  • 期刊论文

  • 发表日期：

    2025年8月26日

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  • Jiangfeng Han,
  • Zhenhai Liu,
  • Nikolaos S. Papageorgiou

摘要

We consider a nonlinear Dirichlet equation driven by the weighted p-Laplacian with weight \(a(\cdot )\) a ( · ) which is not bounded away from zero (degenerate problem). The reaction (right hand side) is parametric and exhibits the combined effects of a singular term and of indefinite (sign-changing), ( \(p-1\) p - 1 )-superlinear perturbation. Using variational tools, truncations, comparisons and critical groups, we show that for all small values of the parameter, the problem has at least two bounded positive solutions.