CPAA
A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation
Kazuhiro Kurata Tatsuya Watanabe
In this paper, we are concerned with radially symmetric solutions with a vortex to a nonlinear Schrödinger equation:

-ħ$^2 \Delta v+($ ħ$^2\omega^2/|x|^2+V(x))v=f(v)$ in $\mathbf R^2.$

We give precise asymptotic profiles of solutions as ħ$\rightarrow 0$ by variational methods and ODE arguments.

keywords: Variational methods asymptotic profiles vortex.
CPAA
Singular positive solutions for a fourth order elliptic problem in $R$
Tokushi Sato Tatsuya Watanabe
In this paper, we consider the following fourth order elliptic problem in $R^N$:

$\Delta^2 u-c_1\Delta u+c_2 u=u^p+\kappa \sum_{i=1}^m \alpha_i \delta_{a_i}$ in $\mathcal D'(R^N),$

$ u(x)>0, u(x) \rightarrow 0 $ as $ |x| \rightarrow \infty. $

We will prove if $0 < \kappa < \kappa^* $ for some $\kappa^*\in (0,\infty)$, then this problem has at least two singular positive solutions.

keywords: singular solutions minimal solutions Mountain Pass methods. Fourth order elliptic problem
CPAA
Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities
Shinji Adachi Masataka Shibata Tatsuya Watanabe
We study the asymptotic behavior of the ground state for a class of quasilinear Schrödinger equations with general nonlinearities. By the variational argument and dual approach, we show the asymptotic non-degeneracy and uniqueness of the ground state.
keywords: Quasilinear elliptic equation asymptotic behavior.

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