Existence and multiplicity of solutions for a superlinear second order equation arising in a two-ion electrodiffusion model

PABLO AMSTER; MARIEL PAULA KUNA

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

Let us consider the equationegin{equation} label{sis1}u´´= g(x,u) + A(x),end{equation}with radiation boundary conditions egin{equation} label{borde}u´(0)= a_0 u(0),; u´(1)= a_1 u(1),;;; hbox{ with } a_0, a_1 >0,end{equation}for $A in L^2(0,1)$ and $g: [0,1] imes mathbb R o mathbb R$  superlinear, $$lim_{ left|u ight| ightarrow + infty} rac{g(x,u)}{u}= +infty.$$  We shall study existence, uniqueness and multiplicity of solutions using variational methods.A particular case of interest is $g(x,u)=rac{1}{2}u^3 +(a+bx)u$ for $a,b, A, a_0$ and $a_1$ some specific constants. This Painlev´e II model in two-ion electrodifussion was derived independently by Grafov and Chernenko in [GCh]and Bass in [B]. In [BBR], Bracken {em et al} associated flux quantization with the iteration of the B"acklund transformations. Due to that connection, Robin-type boundary conditions such as ( ef{borde})were derived for the Painlev´e II equation.