2019/05/12 by Boscaggin, Alberto, Feltrin, Guglielmo, Sovrano, Elisa
#34B08 #34B18 #34C25 #47H11 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.04671
We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation u'' + cu' + ( λa+(x) - μa-(x) ) g(u) = 0, where λ,μ>0 are parameters, c∈ℝ, a(x) is a locally integrable P-periodic sign-changing weight function, and g\colon\mathopen[0,1\mathclose]→ℝ is a continuous function such that g(0)=g(1)=0, g(u)>0 for all u∈\mathopen]0,1\mathclose[, with superlinear growth at zero. A typical example for g(u), that is of interest in population genetics, is the logistic-type nonlinearity g(u)=u2(1-u). Using a topological degree approach, we provide high multiplicity results by exploiting the nodal behaviour of a(x). More precisely, when m is the number of intervals of positivity of a(x) in a P-periodicity interval, we prove the existence of 3m-1 non-constant positive P-periodic solutions, whenever the parameters λ and μ are positive and large enough. Such a result extends to the case of subharmonic solutions. Moreover, by an approximation argument, we show the existence of a countable family of globally defined solutions with a complex behaviour, coded by (possibly non-periodic) bi-infinite sequences of 3 symbols.