A topological approach to superlinear indefinite boundary value problems

Abstract

We obtain the existence of infinitely many solutions with prescribed nodal properties for some boundary value problems associated to the second order scalar equationẍ + q(t)g(x) = 0, where g(x) has superlinear growth at infinity and q(t) changes sign.

Superlinear Indefinite Equations on the Real Line and Chaotic Dynamics

“…However, Lemma 2.4 and Lemma 2.5 hold true also when more general conditions are imposed on g: besides the sign condition (2.2), in [13] the authors assume a hypothesis related to the asymptotic behaviour of the time-maps associated to the autonomous equations .…”

“…The situation when the nonlinearity is, at the same time, both superlinear in the space-variable and sign-changing in the time-variable has been considered, with shooting methods, for boundary-value problems on a bounded interval by (among others) Butler [7], Papini [11,12], and Papini and Zanolin [13]; for related results, mainly in the framework of variational methods, we refer the reader to the papers by Alama and Tarantello [1], Badiale [2], Berestycki et al [4], and Ramos et al [15]. However, as for solutions defined on unbounded domains, much less is known; we refer the reader to the very recent result by Terracini and Verzini [19].…”

“…Indeed, the fact that, for every j 2 N; the set O þ j is non-empty and bounded directly follows from the study of the two-point boundary-value problem associated to (1.3) which was developed in [13]. In that paper, roughly speaking, a refined phase-plane analysis based on the study of the time-maps associated to the autonomous equations .…”

“…and O 2k 2kþ1 ; following the approach of [7], it is possible to prove (see [13]) that the intersections of these sets with the coordinate axes are bounded. Indeed, the following result holds true:…”

Second-order ordinary differential equations with indefinite weight: the Neumann boundary value problem

“…The investigation of the case in which the nonlinear term g(s) has superlinear growth at infinity (namely, g(s) ∼ |s| p−1 s, with p > 1) led to multiplicity results of oscillatory solutions for various boundary value problems associated to (1.1) (see [13,30,34] and the references therein). The search of positive solutions has been addressed both to the case of ODEs and to nonlinear elliptic PDEs of the form 4) under different conditions for u| ∂Ω (see [1,2,3,4,5,6] for some classical results in this direction).…”