Details of a Researcher - Ikoma, Norihisa

Papers - Ikoma, Norihisa

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Existence and Regularity for Prescribed Lorentzian Mean Curvature Hypersurfaces, and the Born–Infeld Model

Jaeyoung Byeon, Norihisa Ikoma, Andrea Malchiodi, Luciano Mari

Annals of PDE (Springer Science and Business Media LLC) 10 ( 1 ) 2024.06

Accepted, ISSN 25245317

Given a measure ρ on a domain Ω ⊂ Rm , we study spacelike graphs over Ω in Minkowski space with Lorentzian mean curvature ρ and Dirichlet boundary condition on ∂Ω , which solve [Figure not available: see fulltext.] The graph function also represents the electric potential generated by a charge ρ in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer uρ of the associated action Iρ(ψ)≐∫Ω(1-1-|Dψ|2)dx-⟨ρ,ψ⟩ among functions ψ satisfying | Dψ| ≤ 1 , by the lack of smoothness of the Lagrangian density for | Dψ| = 1 one cannot guarantee that uρ satisfies the Euler-Lagrange equation (BI). A chief difficulty comes from the possible presence of light segments in the graph of uρ . In this paper, we investigate the existence of a solution for general ρ . In particular, we give sufficient conditions to guarantee that uρ solves (BI) and enjoys log -improved energy and Wloc2,2 estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of ρ to ensure the solvability of (BI).

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The existence and multiplicity of L^2-normalized solutions to nonlinear Schrödinger equations with variable coefficients

Norihisa Ikoma, Mizuki Yamanobe

Advanced Nonlinear Studies (Walter de Gruyter GmbH) 24 ( 2 ) 477 - 509 2024.04

Accepted, ISSN 1536-1365

Abstract

The existence of L 2–normalized solutions is studied for the equation − Δ u + μ u = f ( x , u ) in R N , ∫ R N u 2 d x = m . $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R } }^{N},\quad {\int }_{ { \mathbf{R } }^{N } }{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L 2-subcritical growth and (μ, u) ∈ R × H 1(R N ) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.

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On weak solutions to a fractional Hardy–Hénon equation, Part II: Existence

Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami

Nonlinear Analysis (Elsevier BV) 227 113165 - 113165 2023.02

Accepted, ISSN 0362-546X

This paper and Hasegawa et al. (2021) treat the existence and nonexistence of stable weak solutions to a fractional Hardy–Hénon equation (−Δ)su=|x|ℓ|u|p−1u in RN, where 0<s<1, ℓ>−2s, p>1, N≥1 and N>2s. In this paper, when p is critical or supercritical in the sense of the Joseph–Lundgren, we prove the existence of a family of positive radial stable solutions, which satisfies the separation property. We also show the multiple existence of the Joseph–Lundgren critical exponent for some ℓ∈(0,∞) and s∈(0,1), and this property does not hold in the case s=1.

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Nonlinear elliptic equations of sublinearity: qualitative behavior of solutions

Norihisa Ikoma, Kazunaga Tanaka, Zhi-Qiang Wang, Chengxiang Zhang

Indiana University Mathematics Journal (Indiana University Mathematics Journal) 71 ( 5 ) 2001 - 2043 2022

Accepted, ISSN 0022-2518

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Existence and asymptotic behavior of positive solutions for a class of locally superlinear Schrödinger equation

Adachi S., Ikoma N., Watanabe T.

Manuscripta Mathematica (Manuscripta Mathematica) 172 ( 3-4 ) 933 - 970 2022

ISSN 00252611

This paper treats the existence of positive solutions of - Δ u+ V(x) u= λ f(u) in RN. Here N≥ 1 , λ > 0 is a parameter and f(u) satisfies conditions only in a neighborhood of u= 0. We shall show the existence of positive solutions with potential of trapping type or G-symmetric potential where G⊂ O(N). Our results extend previous results (Adachi and Watanabe in J Math Anal Appl 507:125765, 2022; Costa and Wang in Proc Am Math Soc 133(3):787–794, 2005; do Ó et al. in J Math Anal Appl 342:432–445, 2008) as well as we also study the asymptotic behavior of a family (uλ)λ≥λ0 of positive solutions as λ → ∞.

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On weak solutions to a fractional hardy-henon equation: Part i: nonexistence

Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami

Communications on Pure and Applied Analysis (American Institute of Mathematical Sciences (AIMS)) 20 ( 4 ) 1559 - 1600 2021.04

Accepted, ISSN 15340392

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On the existence of positive solutions to a certain class of semilinear elliptic equations

Norihisa Ikoma

Partial Differential Equations and Applications (Partial Differential Equations and Applications) 2 ( 2 ) 2021.04

Accepted, ISSN 26622963

In this paper, we study the following semilinear elliptic equation Δu=φ(V(x)u-f(x,u(x)))inRN,u∈H1(RN)where N≥ 1 and φ(s) , V(x), f(x, s) are given functions. Under some conditions on φ(s) , V(x) , f(x, s) , we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.

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Semi-classical states for logarithmic Schrödinger equations

Norihisa Ikoma, Kazunaga Tanaka, Zhi Qiang Wang, Chengxiang Zhang

Nonlinearity (Nonlinearity) 34 ( 4 ) 1900 - 1942 2021.04

Accepted, ISSN 09517715

In this paper, we investigate semi-classical bound states for logarithmic Schrödinger type equations with a potential function which has a finite number of singularities of at most logarithmic strength. We construct localized solutions concentrating at a logarithmic type singular point of the potential, and we also characterize the asymptotic limiting profile of the localized solutions. To accomplish these, we develop new penalization techniques for treating the difficulties associated with the non-smoothness of the variational formulation and the singularity of the potentials.

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The compactness of minimizing sequences for a nonlinear Schrödinger system with potentials

Norihisa Ikoma, Yasuhito Miyamoto

Communications in Contemporary Mathematics (Communications in Contemporary Mathematics) 25 ( 2 ) 2021

Accepted, ISSN 02191997

In this paper, we consider the following minimizing problem with two constraints: inf{E(u)|u = (u1,u2),u1L22 = α 1,u2L22 = α 2}, where α1,α2 > 0 and E(u) is defined by E(u):= RN 1 2i=12(|u i|2 + V i(x)|ui|2) - i=12 μi 2pi + 2|ui|2pi+2 - β p3 + 1|u1|p3+1|u 2|p3+1 dx. Here N ≥ 1, μ1,μ2,β > 0 and Vi(x) (i = 1, 2) are given functions. For Vi(x), we consider two cases: (i) both of V1 and V2 are bounded, (ii) one of V1 and V2 is bounded. Under some assumptions on Vi and pj, we discuss the compactness of any minimizing sequence.

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Foliation by area-constrained willmore spheres near a nondegenerate critical point of the scalar curvature

Norihisa Ikoma, Andrea Malchiodi, Andrea Mondino

International Mathematics Research Notices (International Mathematics Research Notices) 2020 ( 19 ) 6539 - 6568 2021

Accepted, ISSN 10737928

Let (M, g) be a three-dimensional Riemannian manifold. The goal of the paper is to show that if P0 ∈ M is a nondegenerate critical point of the scalar curvature, then a neighborhood of P0 is foliated by area-constrained Willmore spheres. Such a foliation is unique among foliations by area-constrained Willmore spheres having Willmore energy less than 32π; moreover, it is regular in the sense that a suitable rescaling smoothly converges to a round sphere in the Euclidean three-dimensional space. We also establish generic multiplicity of foliations and the 1st multiplicity result for area-constrained Willmore spheres with prescribed (small) area in a closed Riemannian manifold. The topic has strict links with the Hawking mass.

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Multiplicity of radial and nonradial solutions to equations with fractional operators

Norihisa Ikoma

Communications on Pure and Applied Analysis (Communications on Pure and Applied Analysis) 19 ( 7 ) 3501 - 3530 2020.07

Accepted, ISSN 15340392

In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under N ≥ 2. We also show the existence of least energy solution (with the Pohozaev identity) and its mountain pass characterization. For nonradial solutions, we prove the existence of at least one nonradial solution under N ≥ 4 and infinitely many nonradial solutions under either N = 4 or N ≥ 6. We treat both of the zero mass and the positive mass cases.

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Stable standing waves of nonlinear Schrodinger equations with potentials and general nonlinearities

Norihisa Ikoma, Yasuhito Miyamoto

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (SPRINGER HEIDELBERG) 59 ( 2 ) 2020.04

Accepted, ISSN 09442669

The existence and nonexistence of the minimizer of the L-2-constraint minimization problem e(alpha) := inf{E(u)vertical bar u is an element of H1(RN), parallel to u parallel to(2)(L2(RN)) = alpha} are studied. Here,E(u) := 1/2 integral(RN) vertical bar del u vertical bar(2) + V(x)vertical bar u vertical bar(2)dx - integral(RN) F(vertical bar u vertical bar)dx,V(x) is an element of C(R-N), 0 not equivalent to V(x) <= 0, V(x) -> 0 (vertical bar x vertical bar -> infinity) and F(s) = integral(s)(0) f (t)dt is a rather general nonlinearity. We show that there exists alpha(0) >= 0 such that e(a) is attained for alpha > alpha(0) and e(alpha) is not attained for 0 < alpha < alpha(0). We study differences between the cases V(x) not equivalent to 0 and V(x) = 0, and obtain sufficient conditions for alpha(0) = 0. In particular, if N = 1, 2, then alpha(0) = 0, and hence e(alpha) is attained for all alpha > 0.

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A Note On Deformation Argument For L2 Normalized Solutions Of Nonlinear Schrödinger Equations And Systems

Norihisa Ikoma, Kazunaga Tanaka

Advances in Differential Equations (Advances in Differential Equations) 24 ( 11-12 ) 609 - 646 2019.11

Accepted, ISSN 10799389

Abstract. We study the existence of L2 normalized solutions for nonlinear Schrödinger equations and systems. Under new Palais-Smale type conditions, we develop new deformation arguments for the constrained functional on (Formula Presented). As applications, we give other proofs to the results of [5, 8, 22]. As to the results of [5, 22], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [8], via our deformation result, we can show the existence of vector solution without using constraint related to the Pohozaev identity.

Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

Takafumi Akahori, Slim Ibrahim, Norihisa Ikoma, Hiroaki Kikuchi, Hayato Nawa

Calculus of Variations and Partial Differential Equations (Calculus of Variations and Partial Differential Equations) 58 ( 4 ) 2019.08

Accepted, ISSN 09442669

The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In Akahori et al. (Global dynamics above the ground state energy for the combined power-type nonlinear Schrödinger equation with energy-critical growth at low frequencies, preprint), semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schrödinger and Klein–Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.

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Existence and non-existence of maximizers for the Moser–Trudinger type inequalities under inhomogeneous constraints

Norihisa Ikoma, Michinori Ishiwata, Hidemitsu Wadade

Mathematische Annalen (Mathematische Annalen) 373 ( 1-2 ) 831 - 851 2019.02

Accepted, ISSN 00255831

In this paper, we study the existence and non-existence of maximizers for the Moser–Trudinger type inequalities in R N of the form DN,α(a,b):=supu∈W1,N(RN),‖∇u‖LN(RN)a+‖u‖LN(RN)b=1∫RNΦN(α|u|N′)dx.Here N≥2,N′=NN-1,a,b>0,α∈(0,αN] and ΦN(t):=et-∑j=0N-2tjj! where αN:=NωN-11/(N-1) and ω N-1 denotes the surface area of the unit ball in R N . We show the existence of the threshold α ∗ = α ∗ (a, b, N) ∈ [0 , α N ] such that D N,α (a, b) is not attained if α∈ (0 , α ∗ ) and is attained if α∈ (α ∗ , α N ). We also provide the conditions on (a, b) in order that the inequality α ∗ < α N holds.

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A NOTE ON DEFORMATION ARGUMENT FOR L-2 NORMALIZED SOLUTIONS OF NONLINEAR SCHRODINGER EQUATIONS AND SYSTEMS

Ikoma Norihisa, Tanaka Kazunaga

ADVANCES IN DIFFERENTIAL EQUATIONS 24 ( 11-12 ) 609 - 646 2019

ISSN 1079-9389

Existence and nonexistence of positive solutions to some fully nonlinear equation in one dimension

Patricio Felmer, Norihisa Ikoma

Journal of Functional Analysis (Journal of Functional Analysis) 275 ( 8 ) 2162 - 2196 2018.10

Accepted, ISSN 00221236

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EMBEDDED AREA-CONSTRAINED WILLMORE TORI OF SMALL AREA IN RIEMANNIAN THREE-MANIFOLDS, II: MORSE THEORY

Norihisa Ikoma, Andrea Malchiodi, Andrea Mondino

AMERICAN JOURNAL OF MATHEMATICS (JOHNS HOPKINS UNIV PRESS) 139 ( 5 ) 1315 - 1378 2017.10

Accepted, ISSN 0002-9327

This is the second part of a series of two papers where we construct embedded Willmore tori with small area constraint in Riemannian three -manifolds. In both papers the construction relies on a Lyapunov-Schmidt reduction, the difficulty being the Mobius degeneration of the tori. In the first paper the construction was performed via minimization, here by Morse Theory. To this aim we establish new geometric expansions of the derivative of the Willmore functional on small Clifford tori (in geodesic normal coordinates) which degenerate to small geodesic spheres with a small handle under the action of the Mobius group. By using these sharp asymptotics we give sufficient conditions, in terms of the ambient curvature tensors and Morse inequalities, for having existence/multiplicity of embedded tori which are stationary for the Willmore functional under the constraint of prescribed (sufficiently small) area.

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Embedded area-constrained Willmore tori of small area in Riemannian three-manifolds I: minimization

Norihisa Ikoma, Andrea Malchiodi, Andrea Mondino

PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY (WILEY) 115 ( 3 ) 502 - 544 2017.09

Accepted, ISSN 0024-6115

We construct embedded Willmore tori with small area constraint in Riemannian three-manifolds under some curvature condition used to prevent Mobius degeneration. The construction relies on a Lyapunov-Schmidt reduction; to this aim we establish new geometric expansions of exponentiated small symmetric Clifford tori and analyze the sharp asymptotic behaviour of degenerating tori under the action of the Mobius group. In this first work we prove two existence results by minimizing or maximizing a suitable reduced functional, in particular we obtain embedded area-constrained Willmore tori (or, equivalently, toroidal critical points of the Hawking mass under area-constraint) in compact 3-manifolds with constant scalar curvature and in the double Schwarzschild space. In a forthcoming paper new existence theorems will be achieved via Morse theory.

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Erratum to: Existence of solutions of scalar field equations with fractional operator (Journal of Fixed Point Theory and Applications, (2017), 19, 1, (649-690), 10.1007/s11784-016-0369-x)

Norihisa Ikoma

Journal of Fixed Point Theory and Applications 19 ( 2 ) 1649 - 1652 2017.06

Accepted, ISSN 1661-7738

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