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

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