Existence and multiplicity results for a Ermakov-Painlevè II-type equation with radiation boundary condition

MARIEL PAULA KUNA; PABLO AMSTER

Buenos Aires

Congreso; X Americas Conference on Differential Equations and Nonlinear Analysis; 2015

Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires

Let us consider the next equation for a function $u: [0,1] o mathbb R$, subject to radiation boundary conditions$$ left{egin{array}{ll}           u´´=g(x,u) + h(u) \           u´(0)= a_0 u(0),; u´(1)= a_1 u(1),          end{array} ight.$$where $h: (0, +infty) o mathbb R$ has a singularity in $u=0$, $g: [0,1] imes mathbb R o mathbb R$ is continuous and superlinear, that is: $$lim_{ left|u ight| ightarrow + infty} rac{g(x,u)}{u}= +infty$$uniformly in $x$, and $a_0, a_1 >0$.We shall study existence and multiplicity of positive solution.A particular case of interest is $g(x,u)=au^3 +bxu$ and $h(u)= rac{c}{u^3}$ for $a,b, c, a_0$ and $a_1$ some specific constants. This Ermakov-Painlev´e II equation arises out of a reduction of a Nernst-Planck system for three-ion electrodiffusion.