arXiv subjects

Wenguo Liang

Publications and source records attributed to Wenguo Liang.

3 recordsLinked to original sources

arXiv2026

Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient

In this paper, we study local and global properties of positive solutions to the equation $-Δu=u^p|\nabla u|^q$ in a domain $Ω$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouville-type theorems. As an application, we derive universal estimates for local solutions. Some of our results are new, as we extend the condition $p+q<(N+3)/(N-1)$ considered by He, Hu and Wang [Math. Z. 313 (2026), No. 6] to a wider range of parameters.

math.AP↗

arXiv2026

Gradient estimates and Liouville-type theorems for the semilinear elliptic equation involving the nonlinear gradient source

We study local and global properties of positive solutions to the equation $-Δu=u^p+M|\nabla u|^q$ in a domain $Ω$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate auxiliary functions over various regions and employing the maximum principle, we derive the local gradient estimates for all $(p,q)\in \mathbb R^2$, and further establish Liouville-type theorems. As an application, we acquire universal estimates for local solutions of elliptic equations with general nonlinearities. Our results extend partial conclusions established in Bidaut-Véron, Garcia-Huidobro and Véron [Math. Ann. 378 (1-2) (2020) 13-56].

math.AP↗

arXiv2024

A priori estimates and Liouville-type theorems for the semilinear parabolic equations involving the nonlinear gradient source

This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation $u_t-Δu=u^p+M|\nabla u|^q$ in $Ω\times I\subset \R^N\times \R$, where $M>0$, and $p,q>1$. We first establish the local pointwise gradient estimates when $q$ is subcritical, critical and supercritical with respect to $p$. With these estimates, we can prove the parabolic Liouville-type theorems for time-decreasing ancient solutions. Next, we use Gidas-Spruck type integral methods to prove the Liouville-type theorem for the entire solutions when $q$ is critical. Finally, as an application of the Liouville-type theorem, we use the doubling lemma to derive universal priori estimates for local solutions of parabolic equations with general nonlinearities. Our approach relies on a parabolic differential inequality containing a suitable auxiliary function rather than Keller-Osserman type inequality, which allows us to generalize and extend the partial results of the elliptic equation (Bidaut-Véron, Garcia-Huidobro and Véron (2020) \cite{veron-sum}) to the parabolic case.

math.AP↗