arXiv subjects

Masataka Shibata

Publications and source records attributed to Masataka Shibata.

7 recordsLinked to original sources

arXiv2025

Classification and qualitative properties of positive solutions to double-power nonlinear stationary Schr\"{o}dinger equations

In this paper, we investigate positive radial solutions to double-power nonlinear stationary Schrodinger equations in three space dimensions. It is now known that the non-uniqueness of H^{1}-positive solutions can occur in three dimensions when the frequency is sufficiently small. Under suitable conditions, in addition to the ground state solution (whose L^{\infty} norm vanishes as the frequency tends to zero), there exists another positive solution that minimizes a different constrained variational problem, with an L^{\infty} norm diverging as the frequency tends to zero (see Theorem 1.4). We classify all positive solutions with small frequency into two categories: the ground state and the Aubin-Talenti type solution. As a consequence, we establish the multiplicity of positive solutions. Finally, we also examine the non-degeneracy and Morse index of each positive solution.

math.AP↗

arXiv2024

Minimal volume product of three dimensional convex bodies invariant under certain groups of order four

We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under two kinds of discrete subgroups of $O(3)$ of order four. We also characterize the convex bodies with the minimal volume product in each case. This provides new partial results of the non-symmetric version of Mahler's conjecture in the three dimensional case.

math.MG↗

arXiv2022

Minimal volume product of convex bodies with certain discrete symmetries and its applications

We give the sharp lower bound of the volume product of $n$-dimensional convex bodies which are invariant under a discrete subgroup $SO(K)=\{ g \in SO(n); g(K)=K \}$, where $K$ is an $n$-cube or $n$-simplex. This provides new partial results of Mahler's conjecture and its non-symmetric version. In addition, we give partial answers for Viterbo's isoperimetric type conjecture in symplectic geometry from the view point of Mahler's conjecture.

math.MG↗

arXiv2020

Minimal volume product of three dimensional convex bodies with various discrete symmetries

We give the sharp lower bound of the volume product of three dimensional convex bodies which are invariant under a discrete subgroup of $O(3)$ in several cases. We also characterize the convex bodies with the minimal volume product in each case. In particular, this provides a new partial result of the non-symmetric version of Mahler's conjecture in the three dimensional case.

math.MG↗

arXiv2019

Symmetric Mahler's conjecture for the volume product in the three dimensional case

In this paper, we prove Mahler's conjecture concerning the volume product of centrally symmetric convex bodies in $\mathbb{R}^n$ in the case where $n=3$. Furthermore, we determine the equality condition.

math.MG↗

arXiv2016

A note on the uniqueness and the non-degeneracy of positive radial solutions for semilinear elliptic problems and its application

In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE analysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified Schr\"odinger equations.

math.AP↗

arXiv2013

A new rearrangement inequality and its application for L^2-constraint minimizing problems

We study $L^2$-constraint minimizing problems related to elliptic systems. We introduce a new rearrangement to show $H^1$-precompactness of any minimizing sequences. Some strict inequality for the new rearrangement is obtained. It is a key to exclude the dichotomy of minimizing sequences.

math.AP↗