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<p>In this note, by establishing a priori estimate, we reprove the optimal Liouville theorem for a large class of subcritical semilinear elliptic equations on Riemannian manifolds with nonnegative Ricci curvature. Based on the classical Bernstein method, our proofs rely on the new auxiliary functions constructed in Lu (Logarithmic gradient estimate and universal bounds for semilinear elliptic equations revisited) and the lower bounds of nonnegative superharmonic functions inspired by Catino and Monticelli (J Eur Math Soc, 2024. <a href="https://doi.org/10.4171/jems/1484">https://doi.org/10.4171/jems/1484</a>), Serrin and Zou (Acta Math 189(1):79–142, 2002) and Wu (Liouville theorem for one kind of elliptic equations on complete Riemannian manifold). Besides recovering the classical results, our theorems have three advantages as follows: first, we do not need the nonlinear terms to be superlinear; second, we find a new Liouville property, which claims the non-existence of bounded solutions for a class of critical equations; third, using the concept of Bakry-Émery Ricci curvature, we relax the conditions on the coefficient functions to consider Lane-Emden equations with gradient term. These conditions are optimal for the validity of the Liouville property considered in this work.</p>

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Some remarks for semilinear elliptic equations on Riemannian manifolds with nonnegative curvature

  • 期刊论文

  • 发表日期：

    2025年4月25日

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  • Zhihao Lu

摘要

In this note, by establishing a priori estimate, we reprove the optimal Liouville theorem for a large class of subcritical semilinear elliptic equations on Riemannian manifolds with nonnegative Ricci curvature. Based on the classical Bernstein method, our proofs rely on the new auxiliary functions constructed in Lu (Logarithmic gradient estimate and universal bounds for semilinear elliptic equations revisited) and the lower bounds of nonnegative superharmonic functions inspired by Catino and Monticelli (J Eur Math Soc, 2024. https://doi.org/10.4171/jems/1484), Serrin and Zou (Acta Math 189(1):79–142, 2002) and Wu (Liouville theorem for one kind of elliptic equations on complete Riemannian manifold). Besides recovering the classical results, our theorems have three advantages as follows: first, we do not need the nonlinear terms to be superlinear; second, we find a new Liouville property, which claims the non-existence of bounded solutions for a class of critical equations; third, using the concept of Bakry-Émery Ricci curvature, we relax the conditions on the coefficient functions to consider Lane-Emden equations with gradient term. These conditions are optimal for the validity of the Liouville property considered in this work.