Bifurcation into spectral gaps for a noncompact semilinear Schrödinger equation with nonconvex potential

2012/07/04 by Christophe Troestler, Troestler, Christophe
Mathematics · #35J20 #35P30 (Primary) 35J10 #47A30 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35J10 #msc:35J20 #msc:35P30 #msc:47A30

paper · pdf · doi:10.48550/arxiv.1207.1052

arxiv created 2012/07/04 · arxiv updated 2012/07/05

Abstract

This paper shows that the nonlinear periodic eigenvalue problem cases -Δu + V(x) u - f(x,u) = λu, u ∈ H¹(\IR^N), cases has a nontrivial branch of solutions emanating from the upper bound of every spectral gap of -Δ+ V. No convexity condition is assumed. The following result of independent interest is also proven: the direct sum Y ⊕ Z in H¹(\IR^N) associated to a decomposition of the spectrum of -Δ+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).

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