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 $1<p<p_{JL}$, where $p_{JL}$ is the Joseph--Lundgren exponent. Second, for solutions in $\mathbb R^n$ that are stable outside a compact set, we show that if $n>2$ and $p=p_S$, where $p_S$ is the Sobolev critical exponent, then $u\equiv0$. Third, for solutions of the Dirichlet problem on connected unbounded domains with smooth boundary satisfying an epigraph-type geometric condition, we prove that if $n>2$, $1<p\le p_S$, and the solution is stable outside a compact set, then $u\equiv0$. The proofs are based on a gradient estimate for stable solutions, an integrability result, conservation laws and Pohozaev-type identities, and a unique continuation argument. These results extend Liouville-type theorems for stable solutions from the semilinear Lane--Emden equation to the mean curvature equation.
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 the following prescribed scalar curvature problem $$ -\Delta u=K(y)u^{2^*-1},\ \ u>0 \,\ \ \text{in}\ \ \mathbb{R}^N, \tag*{$(*)$} $$ where $2^*=\frac{2N}{N-2}$. For $N\geq4$, we construct a positive $K(y)$ with maximum set $\{|y|=1\}$ and critical order of flatness $m=N-2$ at $|y|=1$, such that problem $(*)$...
We consider entire solutions $u$: $\mathbb{R}^n \rightarrow \mathbb{R}$, $n \ge 2$, of the equation $\operatorname{div}(\nabla f(\nabla u)) = 0$ with convex density $f$: $\mathbb{R}^n \rightarrow \mathbb{R}$ given for example by $f(\nabla u) = \sum_{i=1}^n |\partial_i u|^{p_i}$ with exponents $p_i \ge 2$ including Giaq...
We study Liouville-type theorems for stationary solutions of the incompressible magnetohydrodynamics equations in $\mathbb{R}^n$, with directional fractional dissipation $(-\partial_i^2)^{\alpha}u$ and $(-\partial_j^2)^{\beta}b$, where $n\ge 2$ and $0<\alpha,\beta\le 1$. We prove that $u\equiv b\equiv 0$ by assuming th...
Let $\mathbb H^n$ be the $n$-dimensional hyperbolic space, and $\Delta_{\mathbb H^n}$ be the Laplace--Beltrami operator on $\mathbb H^n$. Let $u=u(t,\cdot)$ be the solution to the wave equation $\partial^2_{t}u-\big(\Delta_{\mathbb H^n}+\frac{1}{4}(n-1)^2\operatorname{Id}\big)u=0$ with initial conditions $u(0,x)=f(x)$...
Peng Chen, X. Duong, Yun-Xiang Wang et al.· 0 citations
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...
Yu-Hua Sun, Zhen Wang, Meng-Yu Zhang· 0 citations
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