Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system

In this paper, we are concerned with the physically engaging pseudo-relativistic Schrödinger system: \[ \begin{cases} \left(-\Delta+m^{2}\right)^{s}u(x)=f(x,u,v,\nabla u) & \hbox{in } \Omega,\\ \left(-\Delta+m^{2}\right)^{t}v(x)=g(x,u,v,\nabla v) & \hbox{in } \Omega,\\ u>0,v>0 & \hbox{in } \Omega, \\ u=v\equiv 0 & \hbox{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} \] where $s,t\in (0,1)$ and the mass $m>0.$ By using the direct method of moving plane, we prove the strict monotonicity, symmetry and uniqueness for positive solutions to the above system in a bounded domain, unbounded domain, $\mathbb {R}^{N}$ , $\mathbb {R}^{N}_{+}$ and a coercive epigraph domain $\Omega$ in $\mathbb {R}^{N}$ , respectively.

Liouville theorems and elliptic gradient estimates for a nonlinear parabolic equation involving the Witten Laplacian

In this paper, we establish local and global elliptic type gradient estimates for a nonlinear parabolic equation on a smooth metric measure space whose underlying metric and potential satisfy a ( k , m ) {(k,m)} -super Perelman–Ricci flow inequality. We discuss a number of applications and implications including curvature free global estimates and some constancy and Liouville type results.

Embeddings of weighted Sobolev spaces and a weighted fourth-order elliptic equation

New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth-order elliptic equation. We also obtain Liouville type results for the related equation. Some problems are still open.