Exsitence and multiplicity results for a second order equation with radiation boundary conditions

MARIEL PAULA KUNA; PABLO AMSTER

Barcelona

Workshop; Twelfth Workshop on Interactions between Dynamical Systems and Partial Differential Equations (JISD2014); 2014

Universidad Politécnica de Catalunya

Let us consider the equation for a function u : [0, 1] → Ru (x) = g(x, u) + A(x),where A ∈ L2 (0, 1) and g : [0, 1] × R → R continuous and superlinear, thatis:g(x, u)= +∞l ́ımu|u|→+∞uniformly in x.Subject to radiation-type boundary conditions (see [BBR]),u (0) = a0 u(0), u (1) = a1 u(1),with a0 , a1 > 0.(1)We shall study existence, multiplicity and uniqueness of solution viavariational methods. Moreover, we will present some qualitative propertiesof the solutions.Note that the boundary conditions (1) are different from the standardRobin-type conditions in the fact that both a0 and a1 are positive, thus,the associated functional has the form J(u) := J0 (u) − a1 J1 (u), where J0 iscoercive and J1 takes arbitrarily large positive values over a one-dimensionalsubspace of H 1 (0, 1). In this setting, the proof will consist of showing thatthe functional is still coercive, and deducing from this fact the existence ofa global minimum.A particular case of interest is g(x, u) = 1 u3 +(a+bx)u for a, b, A, a0 and2a1 some specific constants. This Painlev ́ II model in two-ion electrodifus-esion was derived independently by Grafov and Chernenko in [GCh] and Bassin [B]. In [BBR], Bracken et al associated flux quantization with the itera-tion of the B ̈cklund transformations. Due to that connection, Robin-typeaboundary conditions such as (1) were derived for the Painlev ́ II equation.e[A] P. Amster, Topological Methods in the Study of Boundary Value Prob-lems. Springer, 2014.[AKR] P. Amster, M. K. Kwong, and C. Rogers, A Painlev II model in two-ion electrodiffusion with radiation boundary conditions. Nonlinear Analysis:Real World Applications, 16, 120-131. 2013.