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On the existence and non-existence of bounded solutions for a fourth order ODE. (English) Zbl 1294.34041
Authors’ abstract: We prove that if \(F\in C^1(\mathbb R)\) is coercive and \(\{F'=0\}\) is discrete, then the EFK equation \[ u''''-c^2u''+F'(u)=0 \tag{1} \] possesses \(L^\infty (\mathbb R)\) solutions if and only if \(F'\) changes sign at least twice. As a corollary we prove that if \(u_n\) solves \[ u_n''''+c_n^2u_n''+F'(u_n)=0, \] then \(||u_n||_\infty \to +\infty \) if \(c_n\to 0\), provided \( F\) has a unique local minimum, its only minimum is nondegenerate and \(\mathrm{int}(\{F'=0\})=\emptyset \). Finally we give criteria ensuring existence and non-existence of \(T\)-periodic solutions to (1) when \( F\) has multiple wells.

MSC:
34C11 Growth and boundedness of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems, general theory
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