Infinitely Many Solutions for a Floquet-Type BVP with Superlinearity Indefinite in Sign

Oscillating solutions to second-order ODEs with indefinite superlinear nonlinearities

“…The periodic problem was already considered by Butler [12], who proved the existence of infinitely many oscillating periodic solutions to equation (ODE), without giving any information on the locations of their zeros. This result has been extended very recently by Papini in [22], for a wide class of nonlinearities, also giving a nodal characterization, though weaker than ours. In the periodic case our result allows us to prove the existence of infinitely many subharmonic solutions, that is solutions having minimal period nT , for every integer n. We stress that the minimal number of zeros ki on each interval of positivity of α only depends on the local behaviour of α itself.…”

Superlinear Indefinite Equations on the Real Line and Chaotic Dynamics

“…(1.1) in the general case c=0 can be reduced, by means of a change of variable (see [6,11]), to the study of the case c ¼ 0: Therefore, from now on, we will always refer to the undamped equation…”

“…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].…”

A topological approach to superlinear indefinite boundary value problems

“…It is possible to obtain extensions of Theorems 1, 3, 4 for the equationẍ + cẋ + q(t)g(x) = 0, as well (see also [41]).…”

“…We also point out that, using our approach, it is easy to deal with the case in which there are some intervals where q ≡ 0. More details about the meaning of the assumptions for q are given in Section 2 (see also [41], where a similar kind of weights is considered with respect to a Floquet type BVP). Concerning g(x), we note that, by the use of mollifiers and proving the fact that the solutions of (1.3) with fixed nodal properties will be subjected to a priori bounds which are uniform with respect to perturbations of g which are small in the compact-open topology, it is possible to check that the condition of local lipschitzianity for g can be dropped and the continuity of g (paired by an upper bound for g(x)/x in a neighbourhood of zero) is enough to prove all our results.…”