{"paper":{"title":"Bifurcation into spectral gaps for a noncompact semilinear Schr\\\"odinger equation with nonconvex potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Christophe Troestler","submitted_at":"2012-07-04T16:28:13Z","abstract_excerpt":"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)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.1052","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}