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Naoki Shioji

Publications and source records attributed to Naoki Shioji.

6 recordsLinked to original sources
arXiv2025

Uniqueness and multiplicity of positive radial solutions to the super-critical Brezis-Nirenberg problem in an annulus

The super-critical Brezis-Nirenberg problem in an annulus is considered. The new uniqueness result of positive radial solutions is established for the three-dimensional case. It is also proved that the problem has at least three positive radial solutions when the inner radius of the annulus is sufficiently small and the outer radius of the annulus is in a certain range. Moreover, for each positive integer $k$, the problem has at least $k$ positive radial solutions when the exponent of the equation is greater than the critical Sobolev exponent and is less than the Joseph-Lundgren exponent.

math.AP↗
arXiv2024

Uniqueness and multiple existence of positive radial solutions of the Brezis-Nirenberg Problem on annular domains in ${\Bbb S}^{3}$

The uniqueness and multiple existence of positive radial solutions to the Brezis-Nirenberg problem on a domain in the 3-dimensional unit sphere ${\mathbb S}^3$ \begin{equation*} \left\{ \begin{aligned} \Delta_{{\mathbb S}^3}U -\lambda U + U^p&=0,\, U>0 && \text{in $\Omega_{\theta_1,\theta_2}$,}\\ U &= 0&&\text{on $\partial \Omega_{\theta_1,\theta_2}$,} \end{aligned} \right. \end{equation*} for $-\lambda_{1}<\lambda\leq 1$ are shown, where $\Delta_{{\mathbb S}^3}$ is the Laplace-Beltrami operator, $\lambda_{1}$ is the first eigenvalue of $-\Delta_{{\mathbb S}^3}$ and $\Omega_{\theta_1,\theta_2}$ is an annular domain in ${\mathbb S}^3$: whose great circle distance (geodesic distance) from $(0,0,0,1)$ is greater than $\theta_1$ and less than $\theta_2$. A solution is said to be radial if it depends only on this geodesic distance. It is proved that the number of positive radial solutions of the problem changes with respect to the exponent $p$ and parameter $\lambda$ when $\theta_1=\varepsilon$, $\theta_2=\pi-\varepsilon$ and $0<\varepsilon$ is sufficiently small.

math.AP↗
arXiv2017

Rabinowitz alternative for non-cooperative elliptic systems on geodesic balls

The purpose of this paper is to study properties of continua (closed connected sets) of nontrivial solutions of non-cooperative elliptic systems considered on geodesic balls in $S^n$. In particular, we have shown that if the geodesic ball is a hemisphere, then these continua are unbounded. It is also shown that the phenomenon of global symmetry-breaking bifurcation of such solutions occurs. Since the problem is variational and SO(n)-symmetric, we apply the techniques of equivariant bifurcation theory to prove the main results of this article. As the topological tool we use the degree theory for SO(n)-invariant strongly indefinite functionals defined in A. Gołȩbiewska, S. Rybicki, \emph{Global bifurcations of critical orbits of $G$-invariant strongly indefinite functionals}, Nonl. Anal. {74} (2011), 1823-1834..

math.AP↗
arXiv2014

Coron problem for fractional equations

We prove that the critical problem for the fractional Laplacian in an annular type domain admits a nontrivial solution provided that the inner hole is sufficiently small.

math.AP↗