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

P. AMSTER; M. P. KUNA

Córdoba

Congreso; IV CONGRESO LATINOAMERICANO DE MATEMÁTICOS; 2012

Let us consider the equation $$u''= g(x,u) + A(x)$$ with radiation boundary conditions $$u'(0)= a_0 u(0),\; u'(1)= a_1 u(1),\;\;\; \hbox{ with } a_0, a_1 >0$$ for $g: [0,1] \times \mathbb R \to \mathbb R$ superlinear, $$\lim_{ \left|u\right| \rightarrow + \infty} \frac{g(x,u)}{u}=+\infty.$$ We shall study existence, uniqueness and multiplicity of solutions using variational methods. A particular case of interest is $g(x,u)=\frac{1}{2}u^3 +(a+bx)u$ for $a,b, A, a_0$ and $a_1$ some specific constants. This Painelv\'e II model in two-ion electrodifussion was derived independently by Grafov and Chernenko in {GCh} and Bass in {B}. In {BBR}, Bracken et al associated flux quantization with the iteration of the B\"acklund transformations. Due to that connection, the Robin-type boundary value conditions were derived for the Painlev\'e II equation. Bibliography {B} L. Bass, Electric structures of interfaces in steady electrolysis, Transf. Faraday. Soc. {\bf 60}, 1656-1663 (1964). {BBR} L. Bass, A.J. Bracken and C. Rogers, B\"acklund flux-quantization in a model of electrodiffusion based on Painlev\'e II, J Phys. A Math. \& Theor. {\bf 45}, 105204 (2012). {GCh} B.M. Grafov and A.A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte, Dokl. Akad. Nauk SSR {\bf 146} 135- (1962). {MW} J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. New York: Springer-Verlag (1989).