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Guosheng Jiang

Publications and source records attributed to Guosheng Jiang.

4 recordsLinked to original sources
arXiv2026

A Cheng-Yau type estimate for positive biharmonic functions

We establish a Cheng-Yau type estimate for positive biharmonic functions on complete Riemannian manifolds with Ricci curvature satisfies $Ric_g \ge -(n-1)K g$. If $u$ is a positive biharmonic function in $B_{2R}(p)$, then $$ -\frac{Δ_g u}{u} +\frac{1}{8n}\frac{|\nabla u|_g^2}{u^2} \le C_n\left(R^{-2}+K\right) \quad\text{on }B_R(p). $$ Further, if the Ricci curvature is nonnegative, every global positive biharmonic function satisfies $Δ_g u\equiv c$ and the sharp estimate $|\nabla u|_g^2\le2cu$ for some nonnegative constant $c$. We also show that every positive $k$-polyharmonic function has nonnegative constant $(k-1)$-st Laplacian and growth of order at most $2k-2$.

math.DG↗
arXiv2022

Stability of Bernstein type theorem for the minimal surface equation

Let $ Ω\subsetneq \mathbf{R}^n\,(n\geq 2)$ be an unbounded convex domain. We study the minimal surface equation in $Ω$ with boundary value given by the sum of a linear function and a bounded uniformly continuous function in $ \mathbf{R}^n$. If $ Ω$ is not a half space, we prove that the solution is unique. If $ Ω$ is a half space, we prove that graphs of all solutions form a foliation of $Ω\times\mathbf{R}$. This can be viewed as a stability type theorem for Edelen-Wang's Bernstein type theorem in \cite{EW2021}. We also establish a comparison principle for the minimal surface equation in $Ω$.

math.AP↗
arXiv2017

Remarks on the Clark theorem

The Clark theorem is important in critical point theory. For a class of even functionals it ensures the existence of infinitely many negative critical values converging to $0$ and it has important applications to sublinear elliptic problems. We study the convergence of the corresponding critical points and we give a characterization of accumulation points of critical points together with examples, in which critical points with negative critical values converges to non-zero critical point. Our results improve the abstract results in Kajikiya [Ka1] and Liu-Wang [LW].

math.AP↗