Keio University · Mathematics
Professor Norihisa Ikoma's research lab specializes in nonlinear partial differential equations, with a focus on variational methods, constrained minimization problems, and the existence and stability of solutions to Schrödinger-type and Kirchhoff-type equations. The lab investigates ground states, normalized solutions, and semi-classical bound states under various constraints and potential conditions, including singular or logarithmic-type potentials. Key themes include compactness of minimizing sequences, orbital stability, and the development of novel analytical techniques such as deformation arguments and penalization methods for non-smooth variational problems.
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Abstract In this paper, the precompactness of minimizing sequences under multiconstraint conditions are discussed. This minimizing problem is related to a coupled nonlinear Schrödinger 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.
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 constrained functional on $S_m=\{ u; \, \int_{\mathbb R^N} |u^2 | =m\}$ or $S_{m_1}\times S_{m_2}$. 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 infin
In this paper, we studythe existence of ground statesolutions to the nonlinear Kirchhoff type equations\[- m \left( \| \nabla u \|_{L^2(\mathbf{R}^N)}^{2}\right) \Delta u + V(x) u = |u|^{p-1} u\quad {\rm in}\ \mathbf{R}^N,\u \in H^1(\mathbf{R}^N), \N \geq 1\]where $ 1 < p < \infty$ when$N=1,2$, $1 < p < (N+2)/(N-2)$when $N \geq 3$,$m: [0,\infty) \to (0,\infty)$ is a continuous function and$V:\mathbf{R}^N \to \mathbf{R}$ a smooth function.Under suitable conditions on $m(s)$ and $V$,it is shown th
In this paper, we consider the following minimizing problem with two constraints: [Formula: see text] where [Formula: see text] and [Formula: see text] is defined by [Formula: see text] [Formula: see text] Here [Formula: see text], [Formula: see text] and [Formula: see text] [Formula: see text] are given functions. For [Formula: see text], we consider two cases: (i) both of [Formula: see text] and [Formula: see text] are bounded, (ii) one of [Formula: see text] and [Formula: see text] is bounded
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.
Abstract 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
This is a continuation of Ikoma and Ishii (Ann Inst H Poincar Anal Non Linaire 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 p
In this paper we study the existence of standing waves for coupled nonlinear Schrödinger 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
Abstract The existence of L 2 –normalized solutions is studied for the equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>μ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mtext> </m:mtext> <m:mtext> </m:mtext> <m:mtext>in</m:mtext> <m:mspace widt
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