arXiv subjects

Norihisa Ikoma

Publications and source records attributed to Norihisa Ikoma.

At least 19 recordsLinked to original sources

arXiv2026

Normalized solutions of $L^2$-supercritical NLS equations with periodic potentials

In this paper, we study normalized solutions to the following $L^2$-supercritical nonlinear Schrödinger equation with a periodic potential \begin{equation*} \begin{dcases} -Δu +V (x)u + λu=χ_{Ω}(x)f(u)\quad \text{in } \mathbb{R}^N, u>0 \quad \text{in} \ \mathbb{R}^N, \int_{\mathbb{R}^N}\abs{u}^2\, dx =μ. \end{dcases} \end{equation*} Here $N \geq 1$, $μ>0$ is prescribed, $λ\in \mathbb{R}$ is a Lagrange multiplier, $V\in C(\mathbb{R}^N)$ is $1$-periodic in $x_1,...,x_N$, $f \in C^1(\mathbb{R})$ is a nonlinearity having $L^2$-supercritical growth at infinity, $Ω\subset \mathbb{R}^N$ is either a (nonempty) bounded open set with smooth boundary $\partial Ω$ or the whole space $\mathbb{R}^N$, and $χ_{Ω}$ is the characteristic function of $Ω$. In both cases, we prove the existence of normalized solutions of mountain pass type when $μ>0$ is small and $f$ behaves like a power function $|t|^{p-2}t$ with $2+4/N 0$. On the other hand, when $Ω=\mathbb{R}^N$, we prove the existence of solutions corresponding to local minimizers when $μ>0$ is small. When $Ω$ is bounded, the results are obtained via a monotonicity trick for mountain pass values and blow-up analysis with the Morse index estimates. In the case where $Ω= \mathbb{R}^N$, we develop the concentration-compactness argument based on the Morse index for the existence of mountain pass type solutions.

math.AP↗

arXiv2025

Ground state solutions to the nonlinear Born-Infeld problem

In the paper we show the existence of ground state solutions to the nonlinear Born-Infeld problem \[ \mathrm{div}\, \left( \frac{\nabla u}{\sqrt{1-|\nabla u|^2}} \right) + f(u) = 0, \quad x \in \mathbb{R}^N \] in the zero and positive mass cases. Moreover, we find a new proof of the Sobolev-type inequality \[ \int_{\mathbb{R}^N} \left(1 - \sqrt{1-|\nabla u|^2}\right) \, dx \geq C_{N,p} \left( \int_{\mathbb{R}^N} |u|^p \, dx \right)^{\frac{N}{N+p}}, \] for $p > 2^*$ as well as the characterization of the optimal constant $C_{N,p}$ in terms of the ground state energy level. Previous approaches relied on approximation schemes and/or symmetry assumptions, which typically yield to compact embeddings and may lead to solutions that are not at the ground state energy level. In contrast, neither approximation arguments nor symmetry assumptions are employed in the paper to obtain a ground state solution. Instead, we develop a new direct variational approach based on minimization over a Pohožaev manifold combined with profile decomposition techniques. Finally, we show that nonradial solutions exist whenever $N \geq 4$; in particular, this settles a previously open problem in the case $N=5$.

math.AP↗

arXiv2025

Normalized Solutions for the $(2,q)$-Laplacian Operator Between Mass-Critical Exponents

This paper concerns the existence of normalized solutions to a class of $(2,q)$-Laplacian equations with a power type nonlinearity in the intermediate regime between the two mass critical exponents $2(1+2/N)$, $q(1+2/N)$. More precisely, we prove the existence of solutions with negative energy obtained through a global minimization procedure, and of solutions with positive energy established via a local minimization technique and a mountain-pass argument. Furthermore, we derive both existence and nonexistence results for the zero-mass case $λ= 0$, highlighting the role of the mixed diffusion in determining the qualitative behavior of solutions. Specifically, this paper's novelty lies in providing a comprehensive understanding of the intermediate cases that arise when the non-homogeneous $(2,q)$-Laplacian operator appears. Our analysis combines variational methods, compactness arguments, and delicate energy estimates adapted to the nonhomogeneous nature of the $(2,q)$-Laplacian operator.

math.AP↗

arXiv2025

Normalized ground states for NLS equations with mass critical nonlinearities

We study normalized solutions $(μ,u)\in \mathbb{R} \times H^1(\mathbb{R}^N)$ to nonlinear Schrödinger equations $$ -Δu + μu = g(u)\quad \hbox{in}\ \mathbb{R}^N, \qquad \frac{1}{2}\int_{\mathbb{R}^N} u^2 dx = m, $$ where $N\geq 2$ and the mass $m>0$ is given. Here $g$ has an $L^2$-critical growth, both at the origin and at infinity, that is $g(s)\sim |s|^{p-1}s$ as $s\sim 0$ and $s\sim\infty$, where $p=1+\frac{4}{N}$. We continue the analysis started in [Cingolani-Gallo-Ikoma-Tanaka, 2024], where we found two (possibly distinct) minimax values $\underline{b} \leq 0 \leq \overline{b}$ of the Lagrangian functional. In this paper we furnish explicit examples of $g$ satisfying $\underline{b}<0<\overline{b}$, $\underline{b}=0<\overline{b}$ and $\underline{b}<0=\overline{b}$; notice that $\underline{b}=0=\overline{b}$ in the power case $g(t)=|t|^{p-1}t$. Moreover, we deal with the existence and non-existence of a solution with minimal energy. Finally, we discuss the assumptions required on $g$ to obtain the existence of a positive solution for perturbations of $g$.

math.AP↗

arXiv2025

Existence and order of the self--binding transition in non--local non--linear Schrödinger equations

We consider a class of non--linear and non--local functionals giving rise to the Choquard equation with a suitably regular interaction potential, modelling, i.e., gases with impurities and axion stars. We study how existence of minimizers depends on the coupling constant, and find that there is a critical interaction strength needed for the minimizers to exist, both in dimensions two and three. In $d=3$, a minimizer exists also at the critical coupling but none do in $d=2$ under suitable assumptions on the potential. We also establish that in $d=3$ there exist other critical points beyond the global minimizer.

math.AP↗

arXiv2025

Normalized solutions for nonlinear Schrödinger equations with $L^2$-critical nonlinearity

We study the following nonlinear Schrödinger equation and we look for normalized solutions $(μ,u)\in {\bf R}\times H^1({\bf R}^N)$ for a given $m>0$ and $N\geq 2$ \[ -Δu + μu = g(u)\quad \text{in}\ {\bf R}^N, \qquad \frac{1}{2}\int_{{\bf R}^N} u^2 dx = m. \] We assume that $g$ has an $L^2$-critical growth, both at the origin and at infinity. That is, for $p=1+\frac{4}{N}$, $g(s)=|s|^{p-1}s +h(s)$, $h(s)=o(|s|^p)$ as $s\sim 0$ and $s\sim\infty$. The $L^2$-critical exponent $p$ is very special for this problem; in the power case $g(s) = |s|^{p-1}s$ a solution exists only for the specific mass $m=m_1$, where $m_1=\frac{1}{2}\int_{{\bf R}^N}ω_1^2\, dx$ is the mass of a least energy solution $ω_1$ of $-Δω+ω=ω^p$ in ${\bf R}^N$. We prove the existence of a positive solution for $m=m_1$ when $h$ has a sublinear growth at infinity, i.e., $h(s)=o(s)$ as $s\sim\infty$. In contrast, we show non-existence results for $h(s)\not=o(s)$ ($s\sim 0$) under a suitable monotonicity condition.

math.AP↗

arXiv2024

Compactness via monotonicity in nonsmooth critical point theory, with application to Born-Infeld type equations

In this paper, we prove new existence and multiplicity results for critical points of lower semicontinuous functionals in Banach spaces, complementing the nonsmooth critical point theory set forth by Szulkin and avoiding the need of the Palais-Smale condition. We apply our abstract results to get entire solutions with finite energy to Born-Infeld type autonomous equations. More precisely, under almost optimal conditions on the nonlinearity, we construct a positive solution and infinitely many solutions both in the classes of radially symmetric functions and nonradiallly symmetric ones.

math.AP↗

arXiv2023

Existence and regularity for prescribed Lorentzian mean curvature hypersurfaces, and the Born-Infeld model

Given a measure $ρ$ on a domain $Ω\subset \mathbb{R}^m$, we study spacelike graphs over $Ω$ in Minkowski space with Lorentzian mean curvature $ρ$ and Dirichlet boundary condition on $\partial Ω$. The graph function $u_ρ: Ω\rightarrow \mathbb{R}$ also represents the electric potential generated by a charge $ρ$ in electrostatic Born-Infeld theory. While $u_ρ$ minimizes the action $$ I_ρ(ψ) = \int_Ω \Big( 1 - \sqrt{1-|Dψ|^2} \Big) \mathrm{d} x - \langle ρ, ψ\rangle $$ among competitors with $|Dψ| \le 1$, because of a lack of smoothness of the Lagrangian density when $|Dψ| = 1$ a direct approach via minimization may not produce a solution to the Euler-Lagrange equation (BI). In this paper, we study existence and regularity of $u_ρ$ for general $ρ$, in a bounded domain and in the entire $\mathbb{R}^m$. In particular, we find sufficient conditions to guarantee that $u_ρ$ solves (BI) and enjoys log-improved $W^{2,2}_{\mathrm{loc}}$ estimates, and we construct examples helping to identify sharp thresholds for the regularity of $ρ$ to ensure the validity of (BI). One of the main difficulties is the possible presence of light segments in the graph of $u_ρ$, which will be discussed in detail.

math.AP↗

arXiv2022

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

This paper treats the existence of positive solutions of $-Δu + V(x) u = λf(u)$ in $\mathbb{R}^N$. Here $N \geq 1$, $λ> 0$ 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 $\mathcal{G}$-symmetric potential where $\mathcal{G} \subset O(N)$. Our results extend previous results as well as we also study the asymptotic behavior of a family $(u_λ)_{λ\geq λ_0}$ of positive solutions as $λ\to \infty$.

math.AP↗

arXiv2021

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

This paper and [29] treat the existence and nonexistence of stable weak solutions to a fractional Hardy--Hénon equation $(-Δ)^s u = |x|^\ell |u|^{p-1} u$ in $\mathbb{R}^N$, where $0 < s < 1$, $\ell > -2s$, $p>1$, $N \geq 1$ 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 $\ell \in (0,\infty)$ and $s \in (0,1)$, and this property does not hold in the case $s=1$.

math.AP↗

arXiv2021

On weak solutions to a fractional Hardy--Hénon equation: Part I: Nonexistence

This paper and [17] treat the existence and nonexistence of stable (resp. outside stable) weak solutions to a fractional Hardy--Hénon equation $(-Δ)^s u = |x|^\ell |u|^{p-1} u$ in $\mathbb{R}^N$ where $0 < s < 1$, $\ell > -2s$, $p>1$, $N \geq 1$ and $N > 2s$. In this paper, the nonexistence part is proved for the Joseph--Lundgren subcritical case.

math.AP↗

arXiv2020

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

In this paper, we consider the following minimizing problem with two constraints: \[ \inf \left\{ E(u) | u=(u_1,u_2), \ \| u_1 \|_{L^2}^2 = α_1, \ \| u_2 \|_{L^2}^2 = α_2 \right\}, \] where $α_1,α_2 > 0$ and $E(u)$ is defined by \[ E(u) := \int_{\mathbf{R}^N} \left\{\frac{1}{2} \sum_{i=1}^2 \left( |\nabla u_1|^2 + V_i (x) |u_i|^2 \right) - \sum_{i=1}^2 \frac{μ_i}{2p_i+2} |u_i|^{2p_i+2} - \fracβ{p_3+1} |u_1|^{p_3+1} |u_2|^{p_3+1} \right\} \mathrm{d} x. \] Here $N \geq 1$, $ μ_1,μ_2,β> 0$ and $V_i(x)$ $(i=1,2)$ are given functions. For $V_i(x)$, we consider two cases: (i) both of $V_1$ and $V_2$ are bounded, (ii) one of $V_1$ and $V_2$ is bounded. Under some assumptions on $V_i$ and $p_j$, we discuss the compactness of any minimizing sequence.

math.AP↗

arXiv2019

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

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 [AIKN3], 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.

math.AP↗

arXiv2019

Multiplicity of radial and nonradial solutions to equations with fractional operators

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 $N \geq 2$. 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 $N \geq 4$ and infinitely many nonradial solutions under either $N =4$ or $N \geq 6$. We treat both of the zero mass and the positive mass cases.

math.AP↗

arXiv2019

A note on deformation argument for $L^2$ constraint problem

We study the existence of $L^2$ normalized solutions for nonlinear Schrödinger equations and systems. Under new Palais-Smale type conditions we develop new deformation arguments for the constraint functional on $S_m=\{ u; \, \int_{\mathbf{R}^N} | u |^2=m\}$ or $S_{m_1} \times S_{m_2}$. As applications, we give other proofs to the results of [\cite[J:20], \cite[BdV:6], \cite[BS1:7]]. As to the results of [\cite[J:20], \cite[BdV:6]], our deformation result enables us to apply the genus theory directly to the corresponding functional to obtain infinitely many solutions. As to the result [\cite[BS1:7]], via our deformation result we can show the existence of vector solution without using constraint related to the Pohozaev identity.

math.AP↗

arXiv2018

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

In this paper, we study the existence and non-existence of maximizers for the Moser-Trudinger type inequalities in $\Bbb R^N$ of the form \[ D_{N,α}(a,b):= \sup_{u\in W^{1,N}(\Bbb R^N),\,\|\nabla u\|_{L^N(\Bbb R^N)}^a+\|u\|_{L^N(\Bbb R^N)}^b=1} \int_{\Bbb R^N}Φ_N\left(α|u|^{N'}\right)dx. \] Here $N\geq 2$, $N'=\frac{N}{N-1}$, $a,b>0$, $α\in (0,α_N]$ and $Φ_N(t):=e^t-\sum_{j=0}^{N-2}\frac{t^j}{j!}$ where $α_N:= N ω_{N-1}^{1/(N-1)}$ and $ω_{N-1}$ denotes the surface area of the unit ball in $\Bbb R^N$. We show the existence of the threshold $α_\ast = α_\ast(a,b,N) \in [0,α_N]$ such that $D_{N,α}(a,b)$ is not attained if $α\in (0,α_\ast)$ and is attained if $ α\in (α_\ast , α_N)$. We also provide the conditions on $(a,b)$ in order that the inequality $α_\ast < α_N$ holds.

math.AP↗

arXiv2018

Foliation by area-constrained Willmore spheres near a non-degenerate critical point of the scalar curvature

Let $(M,g)$ be a 3-dimensional Riemannian manifold. The goal of the paper it to show that if $P_{0}\in M$ is a non-degenerate critical point of the scalar curvature, then a neighborhood of $P_{0}$ 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 first 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.

math.DG↗

arXiv2017

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

In this paper, we consider the existence (and nonexistence) of solutions to \[ -\mathcal{M}_{λ,Λ}^\pm (u'') + V(x) u = f(u) \quad {\rm in} \ \mathbf{R} \] where $\mathcal{M}_{λ,Λ}^+$ and $\mathcal{M}_{λ,Λ}^-$ denote the Pucci operators with $0< λ\leq Λ< \infty$, $V(x)$ is a bounded function, $f(s)$ is a continuous function and its typical example is a power-type nonlinearity $f(s) =|s|^{p-1}s$ $(p>1)$. In particular, we are interested in positive solutions which decay at infinity, and the existence (and nonexistence) of such solutions is proved.

math.AP↗