Oscillating minimizers of a fourth order problem invariant under scaling

By variational methods, we prove the inequality: $${\int }_{\mathbb{R}}{u}^{\mathrm{\prime }\mathrm{\prime }}{}^{2}dx-{\int }_{\mathbb{R}}{u}^{\mathrm{\prime }\mathrm{\prime }}{u}^{2}dx\ge I{\int }_{\mathbb{R}}{u}^{4}dx\mathrm{\forall }u\in L^4(\mathbb{R})suchthat{u}^{\mathrm{\prime }\mathrm{\prime }}\in L^2(\mathbb{R})$$ for some constant $I\in (-9\mathrm{/}64,-1\mathrm{/}4)$. This inequality is connected to Lieb-Thirring type problems and has interesting scaling properties. The best constant is achieved by sign changing minimizers of a problem on periodic functions, but does not depend on the period. Moreover, we completely characterize the minimizers of the periodic problem.