Oscillating solutions to second-order ODEs with indefinite superlinear nonlinearities

Abstract

We consider a class of ordinary differential equations of the following type:where α is bounded and changes sign. We study the effect of such a coefficient on the existence of oscillating solutions on bounded and unbounded domains.

Superlinear Indefinite Equations on the Real Line and Chaotic Dynamics

“…(1) We first observe that, in contrast with various situations which can be found in the literature, we do not assume any periodicity on q: A recent contribution where oscillatory functions (not necessarily periodic) are considered is due to Terracini and Verzini [19]; however, in [19] an appropriate boundedness condition on q is required.…”

“…We remark that, in [19] as well as in other papers where a variational approach is considered, the nonlinearity g must satisfy some homogeneity conditions. However, in [19] a non-zero right-hand side in (1.1) is allowed.…”

“…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]. We remark that, in [19] as well as in other papers where a variational approach is considered, the nonlinearity g must satisfy some homogeneity conditions. However, in [19] a non-zero right-hand side in (1.1) is allowed.…”

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).…”

Highly oscillatory solutions of a Neumann problem for ap-laplacian equation

“…The results obtained therein, which can be applied to the spatially inhomogeneous balanced Allen-Cahn equation 2 u + a(x)u(1 − u 2 ) = 0, (1.1) and to the equation for a pendulum of variable length 2 u + a(x) sin(πu) = 0, (1.2) can be roughly summarized as follows: the asymptotic behavior, for → 0 + , of solutions to (1.1) and (1.2) (with Neumann boundary conditions) can be characterized in term of a limit energy function and, conversely, highly oscillatory solutions corresponding to any admissible limit profile exist for small enough. More precisely, the admissible limit profiles are determined by an ordinary differential equation solved by the limit energy function and solutions to the boundary value problem are constructed using a variational approach, of broken-geodesic Nehari type (see also [21,23]). Notice that this in particular shows that the above equations possess an extremely rich set of (nodal) solutions.…”