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<p>For all non-positive indices and partial positive indices, we derive logarithmic gradient estimate and boundedness estimate for positive solutions of Lane–Emden system on Riemannian manifolds with Ricci curvature bounded below. Especially, logarithmic gradient estimate implies Harnack inequality and our boundedness estimate is stronger than <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1967_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> estimates of solutions for suplinear case. To our knowledge, these estimates are also new for Lane–Emden system on Euclidean spaces. As its application, we obtain a Liouville theorem for Lane–Emden system on Riemannian manifolds with non-negative Ricci curvature.</p>

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Logarithmic Gradient Estimate and Boundedness Estimate for Lane–Emden System on Riemannian Manifolds

  • 期刊论文

  • 发表日期：

    2025年3月18日

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  • Zhihao Lu

摘要

For all non-positive indices and partial positive indices, we derive logarithmic gradient estimate and boundedness estimate for positive solutions of Lane–Emden system on Riemannian manifolds with Ricci curvature bounded below. Especially, logarithmic gradient estimate implies Harnack inequality and our boundedness estimate is stronger than \(L^{\infty }\) L ∞ estimates of solutions for suplinear case. To our knowledge, these estimates are also new for Lane–Emden system on Euclidean spaces. As its application, we obtain a Liouville theorem for Lane–Emden system on Riemannian manifolds with non-negative Ricci curvature.