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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.

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
Full Text: DOI
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