In this paper, we show the following equation \begin{document}$\begin{cases} Δ u+u^{p}+λ u = 0&\text{ in }Ω,\\ on }\partialΩ, \end{cases}$ \end{document} has at most one positive radial solution for a certain range of $λ>0$. Here $p>1$ and $Ω$ is annulus $\{x∈{{\mathbb{R}}^{n}}:a<|x|<b\}$, $0<a<b$. We also radially non-degenerate via bifurcation methods.

This paper is focused on the gradient blowup rate for following general semilinear parabolic equationwith being a smoothly bounded domain in and smooth function. Under some suitable assumptions , we establish lower bound by rescaling method any . While upper which established comparison can be obtained only At last, provide specific examples to verify our results.

This paper investigates semilinear elliptic problems in planar triangles with Dirichlet conditions specified on one side of the boundary and Neumann imposed remaining two sides. By employing moving plane method, we establish that positive solution is monotone normal direction when vertex non-obtuse. In case where obtuse, longer provided some technical conditions. Furthermore, this monotonicity property extends to first mixed eigenfunction through continuity method via domain deformation. It...

We investigate the location of maximal gradient torsion function on some non-symmetric planar domains. First, for triangles, by reflection method, we show that always occurs longest sides, lying between foot altitude and middle point. Moreover, via nodal line analysis continuity demonstrate restricted each side, critical point is unique nondegenerate. Second, establishing uniform estimates narrow domains, prove as a domain bounded two graphs becomes increasingly narrow, its tends toward...

.It has been a widely-accepted belief that for planar convex domain with two coordinate axes of symmetry, the location maximal norm gradient torsion function is either linked to contact points largest inscribed circle or connected on boundary minimal curvature. However, we show this not quite true in general. Actually, derive precise formula nearly ball domains \(\mathbb{R}^n\) , which displays nonlocal nature and thus does inherently establish connection aforementioned types points....

We study the bifurcation and exact multiplicity of solutions for a class Neumann boundary value problem with indefinite weight. prove that all obtained form smooth reversed S-shaped curve by topological degree theory, Crandall-Rabinowitz theorem, uniform antimaximum principle in terms eigenvalues. Moreover, we obtain equation has exactly either one, two, or three depending on real parameter. The stability is eigenvalue comparison principle.

This paper is concerned with the positive solution of a scalar problem We show that maximum and minimum w are monotone respect to diffusion rate d . also prove there exists threshold value 0 such admits unique when > not other way around.

It has been a widely belief that for planar convex domain with two coordinate axes of symmetry, the location maximal norm gradient torsion function is either linked to contact points largest inscribed circle or connected on boundary minimal curvature. However, we show this not quite true in general. Actually, derive precise formula nearly ball domains $\mathbb{R}^n$, which displays nonlocal nature and thus does inherently establish connection aforementioned types points. Consequently,...

This paper deals with the second Neumann eigenfunction ${u}$ of any planar triangle ${T}$. In a recent work by C. Judge and S. Mondal [Ann. Math., 2022], it was established that does not have critical point within interior this paper, we show uniqueness non-vertex monotonicity property eigenfunction. To be more precise, when ${T}$ is an equilateral triangle, exists if only acute super-equilateral global extrema are achieved at endpoints longest side. establishes origin theorem conjecture...

Let $\Omega$ be a bounded Lipshcitz domain in $\mathbb{R}^n$ and we study boundary behaviors of solutions to the Laplacian eigenvalue equation with constant Neumann data. \begin{align} \label{cequation0} \begin{cases} -\Delta u=cu\quad &\mbox{in $\Omega$}\\ \frac{\partial u}{\partial \nu}=-1\quad &\mbox{on $\partial \Omega$}. \end{cases} \end{align}First, by using properties Bessel functions proving new inequalities on elementary symmetric polynomials, obtain following inequality for...

Abstract The paper is devoted to the qualitative properties of positive solutions a semilinear elliptic equation in planar sub-spherical sector. Under certain range amplitudes, we prove some monotonicity via method moving planes. symmetry follow from uniqueness corresponding over-determined problem by Farina and Valdinoci (2013 Am. J. Math. ).