변재형 교수

This paper is concerned with the qualitative property of the ground state solutions for the Hénon equation. By studying a limiting equation on the upper half space \mathbf{R}_{ + }^{N} , we investigate the asymptotic energy and the asymptotic profile of the ground states for the Hénon equation. The limiting problem is related to a weighted Sobolev type inequality which we establish in this paper. Résumé Nous nous intéresserons, dans cet article, aux propriétés qualitatives des fonctions minimisa

We consider a singularly perturbed elliptic equation \varepsilon^2\Delta u - V(x) u + f(u)=0, \ u(x) > 0 \text{ on } \mathbb R^N, \, \lim_{|x| \to \infty}u(x) = 0, where V(x) > 0 for any x \in \mathbb R^N. The singularly perturbed problem has corresponding limiting problems \Delta U - c U + f(U)=0, \ U(x) > 0 \text{ on } \ \mathbb R^N, \, \lim_{|x| \to \infty}U(x) = 0, \ c > 0. Berestycki–Lions found almost necessary and sufficient conditions on nonlinearity f for existence of a solu

For N = 1,2, we consider singularly perturbed elliptic equations ϵ2Δ u − V(x) u + f(u)= 0, u(x)> 0 on R N , lim|x|→∞ u(x)= 0. For small ϵ > 0, we show the existence of a localized bound state solution concentrating at an isolated component of positive local minimum of V under conditions on f we believe to be almost optimal; when N ≥ 3, it was shown in Byeon and Jeanjean (2007 Byeon , J. , Oshita , Y. ( 2004 ). Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödi

For singularly perturbed Schrödinger equations with decaying potentials at infinity we construct semiclassical states of a critical frequency concentrating on spheres near zeroes of the potentials. The results generalize some recent work of Ambrosetti–Malchiod–Ni [3] which gives solutions concentrating on spheres where the potential is positive. The solutions we obtain exhibit different behaviors from the ones given in [3].

We consider singularly perturbed elliptic equations $\varepsilon^2\Delta u- V(x) u + f(u)=0, x\in R^N, N \ge 3.$ For small $\varepsilon> 0,$ we glue together localized bound state solutionsconcentrating at isolated components of positive local minimum of$V$ under conditions on $f$ we believe to be almost optimal.

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