Behaviors of solutions to a scalar-field equation involving the critical Sobolev exponent with the Robin condition
Y. Kabeya
Discrete & Continuous Dynamical Systems - A 2006, 14(1): 117-134 doi: 10.3934/dcds.2006.14.117
The blowup behaviors of solutions to a scalar-field equation with the Robin condition are discussed. For some range of the parameter, there exist at least two positive solutions to the equation. Here, the blowup rate of the large solution and the scaling properties are discussed.
keywords: bifurcation. asymptotic behaviors Semilinear elliptic equations
Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space
Yen-Lin Wu Zhi-You Chen Jann-Long Chern Y. Kabeya
Communications on Pure & Applied Analysis 2014, 13(2): 949-960 doi: 10.3934/cpaa.2014.13.949
In this article, we consider the following semilinear elliptic equation on the hyperbolic space \begin{eqnarray} \Delta_{H^n} u-\lambda u+|u|^{p-1}u=0\quad on\quad H^n\setminus \{Q\} \end{eqnarray} where $\Delta_{H^n}$ is the Laplace-Beltrami operator on the hyperbolic space \begin{eqnarray} H^n=\{(x_1,\cdots, x_n,x_{n+1})|x_1^2+\cdots+x_n^2-x_{n+1}^2=-1\}, \end{eqnarray} $n>10,\ p>1, \lambda>0, $ and $Q=(0,\cdots,0,1)$. We provide the existence and uniqueness of a singular positive ``radial'' solution of the above equation for big $p$ (greater than the Joseph-Lundgren exponent, which appears if $n > 10$) as well as its asymptotic behavior.
keywords: hyperbolic space. singular solutions Nonlinear elliptic equations
Hot spots for the two dimensional heat equation with a rapidly decaying negative potential
Kazuhiro Ishige Y. Kabeya
Discrete & Continuous Dynamical Systems - S 2011, 4(4): 833-849 doi: 10.3934/dcdss.2011.4.833
We consider the Cauchy problem of the two dimensional heat equation with a radially symmetric, negative potential $-V$ which behaves like $V(r)=O(r^{-\kappa})$ as $r\to\infty$, for some $\kappa > 2$. We study the rate and the direction for hot spots to tend to the spatial infinity. Furthermore we give a sufficient condition for hot spots to consist of only one point for any sufficiently large $t>0$.
keywords: heat equation large time behavior. Hot spots
Canonical forms and structure theorems for radial solutions to semi-linear elliptic problems
Y. Kabeya Eiji Yanagida Shoji Yotsutani
Communications on Pure & Applied Analysis 2002, 1(1): 85-102 doi: 10.3934/cpaa.2002.1.85
We propose a method to investigate the structure of positive radial solutions to semilinear elliptic problems with various boundary conditions. It is already shown that the boundary value problems can be reduced to a canonical form by a suitable change of variables. We show structure theorems to canonical forms to equations with power nonlinearities and various boundary conditions. By using these theorems, it is possible to study the properties of radial solutions of semilinear elliptic equations in a systematic way, and make clear unknown structure of various equations.
keywords: Semilinear elliptic equations radial solutions canonical forms.
Imperfect bifurcations in nonlinear elliptic equations on spherical caps
C. Bandle Y. Kabeya Hirokazu Ninomiya
Communications on Pure & Applied Analysis 2010, 9(5): 1189-1208 doi: 10.3934/cpaa.2010.9.1189
We consider elliptic boundary value problems on large spherical caps with parameter dependent power nonlinearities. In this paper we show that imperfect bifurcation occurs as in the work [13]. When the domain is the whole sphere, there is a constant solution. In the case where the domain is a spherical cap, however, the constant solution disappears due to the boundary condition. For large spherical caps we construct solutions which are close to the constant solution in the whole n-dimensional sphere, using the eigenvalues of the linearized problem in the whole sphere and fixed point arguments based on a Lyapunov-Schmidt type reduction. Numerically there is a surprising similarity between the diagrams of this problem and the ones obtained in [18], also [5], for a Brezis-Nirenberg type problem on spherical caps.
keywords: Nonlinear elliptic equations spherical coordinates fixed point theorems. bifurcation

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