Papers - Ikoma, Norihisa

Area-constrained Willmore surfaces of small area in Riemannian three-manifolds: an approach via Lyapunov-Schmidt reduction

Norihisa Ikoma, Andrea Malchiodi, Andrea Mondino

RIMS Kokyuroku Bessatsu B63   31 - 50 2017.05

Accepted

Existence of solutions of scalar field equations with fractional operator

Norihisa Ikoma

JOURNAL OF FIXED POINT THEORY AND APPLICATIONS (SPRINGER BASEL AG)  19 ( 1 ) 649 - 690 2017.03

Accepted,  ISSN  1661-7738

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In this paper, the existence of least energy solution and infinitely many solutions is proved for the equation in where , and f(s) is a Berestycki-Lions type nonlinearity. The characterization of the least energy by the mountain pass value is also considered and the existence of optimal path is shown. Finally, exploiting these results, the existence of positive solution for the equation in is established under suitable conditions on f(x, s).

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Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

Norihisa Ikoma, Hitoshi Ishii

BULLETIN OF MATHEMATICAL SCIENCES (SPRINGER BASEL AG)  5 ( 3 ) 451 - 510 2015.10

Accepted,  ISSN  1664-3607

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This is a continuation of Ikoma and Ishii (Ann Inst H Poincar, Anal Non Lin,aire 29:783-812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous of degree one, on finite intervals or balls. In the multi-dimensional case, we consider only radial eigenpairs. Our eigenvalue problem has a general first-order boundary condition which includes, as a special case, the Dirichlet, Neumann and Robin boundary conditions. Given a nonnegative integer n, we prove the existence and uniqueness, modulo multiplication of the eigenfunction by a positive constant, of an eigenpair whose eigenfunction, as a radial function in the multi-dimensional case, has exactly n zeroes. When an eigenfunction has n zeroes, we call the corresponding eigenvalue of nth order. Furthermore, we establish results concerning comparison of two eigenvalues, characterizations of nth order eigenvalues via differential inequalities, the maximum principle for the boundary value problem in connection with the principal eigenvalue, and existence of a solution having n zeroes, as a radial function in the multi-dimensional case, of the boundary value problem with an inhomogeneous term.

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EXISTENCE OF GROUND STATE SOLUTIONS TO THE NONLINEAR KIRCHHOFF TYPE EQUATIONS WITH POTENTIALS

Norihisa Ikoma

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS (AMER INST MATHEMATICAL SCIENCES-AIMS)  35 ( 3 ) 943 - 966 2015.03

Accepted,  ISSN  1078-0947

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In this paper, we study the existence of ground state solutions to the nonlinear Kirchhoff type equations
-m (parallel to del u parallel to(2)(L2(RN))) Delta u + V(x)u = vertical bar u vertical bar(p-1)u in R-N; u is an element of H-1(R-N); N >= 1
where 1 < p < infinity when N = 1, 2, 1 < p < ( N + 2)/(N - 2) when N >= 3, m : [0, infinity) -> (0, infinity) is a continuous function and V : R-N -> R a smooth function. Under suitable conditions on m(s) and V, it is shown that a ground state solution to the above equation exists.

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Existence and Concentration Result for the Kirchhoff Type Equations with General Nonlinearities

Giovany M. Figueiredo, Norihisa Ikoma, Joao R. Santos Junior

ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS (SPRINGER)  213 ( 3 ) 931 - 979 2014.09

ISSN  0003-9527

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In this paper we study the existence and concentration behaviors of positive solutions to the Kirchhoff type equations
-epsilon M-2(epsilon(2-N) integral(RN) vertical bar del u vertical bar(2) dx) Delta u + V(x)u = f(u) in R-N, u is an element of H-1(R-N), N >= 1,
where M and V are continuous functions. Under suitable conditions on M and general conditions on f, we construct a family of positive solutions (u epsilon)epsilon is an element of(0,(epsilon) over tilde] which concentrates at a local minimum of V after extracting a subsequence (epsilon(k)).

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On compactness of minimizing sequences to nonlinear Schrödinger systems under multiconstraint condition

Norihisa Ikoma

Advanced Nonlinear Studies (ADVANCED NONLINEAR STUDIES, INC)  14 ( 1 ) 115 - 136 2014.02

ISSN  1536-1365

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In this paper, the precompactness of minimizing sequences under multiconstraint conditions are discussed. This minimizing problem is related to a coupled nonlinear Schrodinger system which appears in the field of nonlinear optics. As a consequence of the compactness of each minimizing sequence, the orbital stability of the set of all minimizers is obtained.

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Existence of minimizers for some coupled nonlinear Schrödinger equations

Norihisa Ikoma

Springer INdAM Series (Springer International Publishing)  2   143 - 164 2013

Accepted,  ISSN  2281-5198

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The existence and nonexistence of minimizers for two minimizing problems are considered. These problems are related to three coupled nonlinear Schrödinger equations which appear in the field of nonlinear optics. We observe the relationship between two minimizing problems and also study the asymptotic behavior when the coefficient standing for the nonlocal nonlinearity tend to 0.

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Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls

Norihisa Ikoma, Hitoshi Ishii

ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE (GAUTHIER-VILLARS/EDITIONS ELSEVIER)  29 ( 5 ) 783 - 812 2012.09

Accepted,  ISSN  0294-1449

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We study the eigenvalue problem for positively homogeneous, of degree one, elliptic ODE on finite intervals and PDE on balls. We establish the existence and completeness results for principal and higher eigenpairs, i.e., pairs of an eigenvalue and its corresponding eigenfunction. (c) 2012 Elsevier Masson SAS. All rights reserved.

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On radial solutions of inhomogeneous nonlinear scalar field equations

Norihisa Ikoma

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS (ACADEMIC PRESS INC ELSEVIER SCIENCE)  386 ( 2 ) 744 - 762 2012.02

Accepted,  ISSN  0022-247X

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We study the existence of radially symmetric solutions u is an element of H(1)(Omega) of the following nonlinear scalar field equation -Delta u = g(vertical bar x vertical bar, u) in Omega. Here Omega = R(N) or {x is an element of R(N) vertical bar vertical bar x vertical bar > R}, N >= 2. We generalize the results of Li and Li (1993) [13] and Li (1990) [14] in which they studied the problem in R(N) and {vertical bar x vertical bar > R} with the Dirichlet boundary condition. Furthermore, we extend it to the Neumann boundary problem and we also consider the nonlinear Schrodinger equation that is the case g(r, s) = -V(r)s + (g) over tilde (s). (C) 2011 Elsevier Inc. All rights reserved.

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A local mountain pass type result for a system of nonlinear Schrodinger equations

Norihisa Ikoma, Kazunaga Tanaka

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (SPRINGER)  40 ( 3-4 ) 449 - 480 2011.03

Accepted,  ISSN  0944-2669

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We consider a singular perturbation problem for a system of nonlinear Schrodinger equations:
-epsilon(2)Delta v(1) + V(1)(x)v(1) = mu(1)v(1)(3) + beta v(1)v(2)(2) in R(N),
-epsilon(2)Delta v(2) + V(2)(x)v(2) = mu(2)v(2)(3) + beta v(1)(2)v(2) in R(N),
v(1)(x), v(2)(x) > 0 in R(N), (*)
v(1)(x), v(2)(x) is an element of H(1)(R(N)).
where N = 2, 3, mu(1), mu(2), beta > 0 and V(1)(x), V(2)(x) : R(N) -> (0,infinity) are positive continuous functions. We consider the case where the interaction beta > 0 is relatively small and we define for P is an element of R(N) the least energy levelm(P) for non- trivial vector solutions of the rescaled " limit" problem:
-Delta v(1) + V(1)(P)v(1) = mu(1)v(1)(3) + beta v(1)v(2)(2) in R(N),
-Delta v(2) + V(2)(P)v(2) = mu(2)v(2)(3) + beta v(1)(2)v(2) in R(N),
v(1)(x), v(2)(x) > 0 in R(N), (**)
v(1)(x), v(2)(x) is an element of H(1)(R(N)).
We assume that there exists an open bounded set Lambda subset of R(N) satisfying
inf(P is an element of Lambda) m(P) < inf(P is an element of partial derivative Lambda) m(P).
We show that (*) possesses a family of non-trivial vector positive solutions {(v(1)epsilon(x), v(2 epsilon) (x))}(epsilon.(0, epsilon 0]) which concentrates-after extracting a subsequence epsilon(n) -> 0-to a point P(0) is an element of Lambda with m(P(0)) = inf(P is an element of Lambda)m(P). Moreover (v(1 epsilon)(x), v(2 epsilon)(x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.

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Existence of Standing Waves for Coupled Nonlinear Schrodinger Equations

Norihisa Ikoma

TOKYO JOURNAL OF MATHEMATICS (TOKYO JOURNAL MATHEMATICS EDITORIAL OFFICE ACAD CENTER)  33 ( 1 ) 89 - 116 2010.06

Accepted,  ISSN  0387-3870

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In this paper we study the existence of standing waves for coupled nonlinear Schrodinger equations. The interaction between equations plays an important role in our study. When the interaction is strong, the least energy solution is a solution whose both components are positive. When the interaction is weak, the least energy solution is a semitrivial solution, namely a solution of a form (u(1), 0) or (0, u(2)). Moreover, minimizing method on the Nehari type manifold with codimension 2 gives us a positive solution when the interaction is weak.

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NONLINEAR SCALAR FIELD EQUATIONS IN R-N: MOUNTAIN PASS AND SYMMETRIC MOUNTAIN PASS APPROACHES

Jun Hirata, Norihisa Ikoma, Kazunaga Tanaka

TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS (JULIUSZ SCHAUDER CTR NONLINEAR STUDIES)  35 ( 2 ) 253 - 276 2010.06

Accepted,  ISSN  1230-3429

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We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in R-N:
-Delta u = g(u) in R-N,
u is an element of H-1(R-N).
We give an extension of the existence results due to H. Berestycki, T. Gallouet and O. Kavian [2].
We take a mountain pass approach in H-1(R-N) and introduce a new method generating a Palais-Smale sequence with an additional property related to Pohozaev identity.

Uniqueness of Positive Solutions for a Nonlinear Elliptic System

Norihisa Ikoma

NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS (BIRKHAUSER VERLAG AG)  16 ( 5 ) 555 - 567 2009.10

Accepted,  ISSN  1021-9722

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In this paper we study the uniqueness of nontrivial positive solutions for the following second order nonlinear elliptic system:
[GRAPHICS]
We prove that for sufficiently small beta > 0, the above system has a unique nontrivial positive solution.

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