We study the equation \[ -\Delta_{m,g}u=\lambda e^u|\nabla u|^q \] on complete, connected, noncompact Riemannian manifolds of dimension $n\ge2$ with $\Ric_g\ge0$. For every $m>1$, $q\in\mathbb R$ and $\lambda>0$, we prove that every positive $C^1$ weak solution is constant when $q>0$, while no such solution exists when $q\le0$. For $q>m$, we derive a local gradient estimate independent of the solution by Nash--Moser iteration, which leads to automatic boundedness and resolves the question raised by Lu and Zhu~\cite[Remark~1.6(B)]{LZ}. For $m-1<q<m$, we use an auxiliary function and a divergence inequality to extend the Liouville result beyond the range covered by existing gradient-inequality methods. The remaining cases are treated by suitable changes of variables and integral estimates. The conclusions also extend to real-valued solutions bounded below.
Let $1<m<n$, $q>0$, and $p\in\mathbb R$. We study positive $C^1_{\mathrm{loc}}$ weak solutions of \[-\Delta_m u=u^p|Du|^q\qquad\text{in }\mathbb R^n.\] If $0<q<m-1$, we prove that every such solution is constant provided \[p+q-m+1<\frac{(m-1)(m-q)^2}{(n-m)(m-1-q)}.\] If $q\geqslant m-1$, the same conclusion holds for e...
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 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...
We study stable solutions of the mean curvature equation $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) =|u|^{p-1}u $$ in domains of $\mathbb R^n$. We establish Liouville-type results in three settings. First, for globally stable solutions in $\mathbb R^n$, we prove that $u\equiv0$ if $1<p<p_...
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...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
Fan-Ze Kong, Xiao-Yu Zeng· 0 citations
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