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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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##### References:
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