Nonresonance conditions on the potential with respect to the Fučik spectrum for the periodic boundary value problem
This article deals with the solvability of the periodic boundary value problem (1) \(u''+ g(u)= h(t)\), \(u(0)- u(T)= u'(0)- u'(T)= 0\), where \(g: \mathbb{R}\to \mathbb{R}\) is a continuous function. It is assumed that there are positive constants \(q_\pm\) and \(Q_\pm\) such that \[ q_\pm\leq \liminf_{s\to \pm \infty} {g(s)\over s}\leq \limsup_{s\to \pm\infty} {g(s)\over s}\leq Q_\pm\tag{2} \] and (3) \((q_-, q_+)\in {\mathcal C}_m\), \((Q_-, Q_+)\in {\mathcal C}_{m+ 1}\), where \({\mathcal C}_m\) and \({\mathcal C}_{m+ 1}\) are two consecutive branches of the Dancer-Fučik spectrum for the periodic problem (recall that \({\mathcal C}_k= \{(\mu, \nu): \mu^{-{1\over 2}}+ \nu^{-{1\over 2}}= 2/k\}\), for \(k= 1, 2,\dots)\). It is known that the hypotheses (2) and (3) alone are not sufficient to guarantee the existence of solutions to problem (1) for every \(h(t)\). Thus, in order to avoid resonance with the critical sets \({\mathcal C}_m\) and \({\mathcal C}_{m+ 1}\), the authors impose further conditions of the form \[ q_+< \limsup_{s\to + \infty} {2G(s)\over s^2}, \qquad \text{and} \qquad \liminf_{s\to+ \infty} {{2G(s)} \over s^2}< Q_+ \] (or similar ones at \(- \infty\), or, also, a choice of one condition at \(- \infty\) and one at \(+ \infty\), where \(G= \int g\)). Analogous results are obtained for the case ``below'' \({\mathcal C}_1\) and extended to the equation \(u''+ f(u) u'+ g(u)= h(t)\) as well, with no condition on \(f(x)\) (except continuity). The proofs are based on degree and variational arguments.
scientific article; zbMATH DE number 894653
scientific article