Oct 2026· Publicationes mathematicae (Debrecen)· Vol 109, pp. 365-388· 0 citations
Abstract
In this paper, through the combination of the Nash--Moser iteration and the Saloff-Coste-type Sobolev inequality, we establish Cheng--Yau logarithmic gradient estimates for solutions to the following nonlinear elliptic equation
$$
\Delta_f u+p\frac{|\nabla u|^2}{u}+\mu(x)u^{r}=0,
$$
which is closely related to the gradient generalized $\tau$-quasi-Einstein metrics. Especially, we obtain the Liouville property and Harnack inequality for this equation when $\mu(x)$ is constant.
In this paper, we consider the inequality $\Delta_pu+\Delta_qu+u^s|\nabla u|^t\leq 0$ on geodesically complete noncompact Riemannian manifolds. By a test function argument, we establish Liouville-type theorems under the upper bound of volume of geodesic ball. In the Euclidean space $\mathbb{R}^n$, we obtain new nonexis...
In this paper, we apply Nash-Moser iteration and Saloff-Coste's Sobolev inequalities to derive gradient estimates for positive solutions to a class of nonlinear parabolic equations of the form \begin{equation*} u_{t}=\Delta u^{m}+a(x, t)u\log u+b(x, t)u, \quad 1-\frac{2}{n}<m<1 \quad \text{or}\quad 1<m<1+\frac{1}{1+\sq...
We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ 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 aux...
We prove the local $L^4$ Strichartz estimate without derivative loss for the Zakharov--Kuznetsov flow on $\mathbb{R}\times\mathbb{T}$. This reaches the sharp spatial regularity endpoint and implies the embedding $X^{0,b}\hookrightarrow L^4$ for every $b>1/2$. The proof reduces the estimate to a semi-algebraic mixed-mea...
E. Başakoğlu, Wen-Ting Hu, Yu-Zhao Wang et al.· 0 citations
We study pointwise estimates for positive, $C^4$-solution $u$ to the fourth-order H\'enon equation \[ \Delta^2 u = |x|^\sigma u^p \quad \text{in } \mathbf R^n \setminus \{0\} \] with $n \geq 5$, $p>1$, and $\sigma>-4 $. In an earlier work published in J. Anal. Math. 140 (2020) 371-394, Z. Guo, X. Huang, L. Wang, and J....
We study the equations $\Delta_p^H u=u^{-q}$, $u>0$, and $-\Delta_p^H u=e^u$ in $\mathbb{R}^N$, where $\Delta_p^H u=\operatorname{div}(H(\nabla u)^{p-1}\nabla H(\nabla u))$ is the Finsler $p$-Laplacian associated with a uniformly convex norm $H$, and $p\ge2$. We prove nonexistence of solutions stable outside a compact...
Nhat Vy Huynh, Phuong Le· 0 citations
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