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Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line. (English) Zbl 1325.34056

The authors study the nonlinear indefinite equation
\[ \ddot{u}+a_{\mu}(t)u^{3}=0,\;t\in\mathbb{R} (1) \]
where \(\mu>0\) is a large parameter and \(a_{\mu}(t)=a^{+}(t)-\mu a^{-}(t)\) for every \(t\in\mathbb{R}\), with \(a^{+}(t)\) and \(a^{-}(t)\) denoting the positive and negative part of a sign-changing, \(T\)-periodic function \(a\in L^{\infty}(\mathbb{R})\). They prove that for \(\mu\) large enough, the nonlinear scalar ODE (1) possesses infinitely many positive solutions, defined on the real line and which are characterized by the fact of being either small or large in each interval of positivity of \(a\). Moreover, they find periodic solutions having minimal period arbitrary large, and bounded non-periodic solutions, exhibing a complex behavior. The proof is based on an approximation procedure and on variational methods exploiting suitable natural contraints of Nehari type.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
34D10 Perturbations of ordinary differential equations
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References:

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