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

Abstract

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 a nontrivial branch of solutions emanating from the upper bound of every spectral gap of $-\Delta + V$. No convexity condition is assumed. The following result of independent interest is also proven: the direct sum $Y \oplus Z$ in $H^1(\IR^N)$ associated to a decomposition of the spectrum of $-\Delta+V$ 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).