Positive solutions for super-sublinear indefinite problems: High multiplicity results via coincidence degree

Abstract

We study the periodic boundary value problem associated with the second order nonlinear equationwhere g(u) has superlinear growth at zero and sublinear growth at infinity. For λ, µ positive and large, we prove the existence of 3 m − 1 positive T -periodic solutions when the weight function a(t) has m positive humps separated by m negative ones (in a T -periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.

Three positive solutions to an indefinite Neumann problem: A shooting method

“…Without loss of generality, we can suppose that A − (σ, t) > 0 for all t ∈ ]σ, τ ]. Indeed, it is always possible to choose a suitable σ as in (a * ) that satisfies this additional hypothesis, as pointed out in [3,7,8,21].…”

“…Moreover, we say that a solution is positive if u(t) > 0 for all t ∈ [0, T ]. Our study is motivated by the results achieved in [1,2,3,6,7,8,10,11,14] in which, dealing with different boundary value problems compared to the one treated here, the authors established multiplicity results of positive solutions in relation to the features of the graph of the weight a(t). This way, we would like to pursue further the investigation of the dynamical effects produced by the weight term associated with nonlinearities satisfying conditions (g * ) and (g 0 ).…”

Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model

“…Remark 4.1. In order to clarify the derivation of formula (4.8), we now give the direct computation of the degree in the case N = 2, without using the combinatorial argument developed in [9]. As an example we compute the degree in A ∅,{1} .…”

“…. For a general integer N one can prove formula (4.8) using the combinatorial argument in [9] or by induction using the excision property of the degree. ⊳…”

Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities

“…Moreover, following a standard terminology (cf. [6]), we can look at (1.1) as an indefinite equation, meaning that the sign of the weight is non-constant. Our main goal is to provide multiplicity results of positive solutions to equation (1.1) together with the Sturm-Liouville boundary conditions, namely conditions of the form αu(0) − βu (0) = 0 γu(L) + δu (L) = 0, (1.2) where α, β, γ, δ ≥ 0 with γβ + αγ + αδ > 0.…”