arXiv subjects

Hongjing Pan

Publications and source records attributed to Hongjing Pan.

3 recordsLinked to original sources

arXiv2024

Global bifurcation curves for fourth-order MEMS/NEMS models II

Global solution curve and exact multiplicity of positive solutions for a class of fourth-order beam equations with clamped boundary conditions are derived. The results extend atheorem of P. Korman (2004) by allowing the presence of a singularity in the nonlinearity. The paper also establishes an a priori estimate for C^3-norm of positive solutions, which is optimal in Holder regularity. Applications to MEMS/NEMS models are presented.

math.AP↗

arXiv2022

A Degenerate Hopf Bifurcation Theorem in Infinite Dimensions

A Hopf bifurcation theorem is established for the abstract evolution equation $\frac{\mathrm{d}x}{\mathrm{d}t}=F(x,\lambda)$ in infinite dimensions under the degeneracy condition $Re \mu ^{\prime}(\lambda_0)= 0$ and suitable assumptions. The stability properties of bifurcating periodic solutions are also derived. Interestingly, it is shown that a transcritical Hopf bifurcation still can occur at $\lambda_0$ although the stability property of the trivial solutions does not change near $\lambda_0$. Our results do not require the analyticity of $F$. The main tools are the Lyapunov--Schmidt reduction and a Morse lemma. Applications to a multi-parameter diffusive predator--prey system discover new branches of periodic solutions.

math.FA↗

arXiv2015

On the Existence of Positive Solutions for Some Nonlinear Boundary Value Problems II

We study a class of boundary value problems with $φ$-Laplacian (e.g., the prescribed mean curvature equation, in which $φ(s)=\frac{s}{\sqrt{1+s^2}}$) \begin{center} $-\left(φ(u')\right)'=λf(u)\; \text{ on }(-L, L),\quad u(-L)=u(L)=0,$ \end{center} where $λ$ and $L$ are positive parameters. For convex $f$ with $f(0)=0$, we establish various results on the exact number of positive solutions as well as global bifurcation diagrams. Some new bifurcation patterns are shown. This paper is a continuation of Pan and Xing [13], where the case $f(0)>0$ has been investigated.

math.CA↗