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This paper shows that the nonlinear periodic eigenvalue problem $${cases}-\Delta u + V(x) u - f(x,u) = \lambda u, u \in H^1(\IR^N), {cases}$$ has anontrivial branch of solutions emanating from the upper bound of every spectralgap of $-\Delta + V$. No convexity condition is assumed. The following resultof independent interest is also proven: the direct sum $Y \oplus Z$ in$H^1(\IR^N)$ associated to a decomposition of the spectrum of $-\Delta+V$remains "topologically direct" in the $L^p$'s (in the sense that theprojections from $Y+Z$ onto $Y$ and $Z$ are $L^p$-continuous).

Bifurcation into spectral gaps for a noncompact semilinear Schr\"odinger equation with nonconvex potential