{"paper":{"title":"A Landesman-Lazer type result for periodic parabolic problems on $\\mathbb{R}^N$ at resonance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aleksander Cwiszewski, Renata Lukasiak","submitted_at":"2014-10-13T17:28:09Z","abstract_excerpt":"We are concerned with $T$-periodic solutions of nonautonomous parabolic problem of the form $u_t = \\Delta u + V(x) u + f(t,x,u)$, $t >0$, $x \\in \\mathbb{R}^N$, with $V \\in L^\\infty (\\mathbb{R}^N)+L^p(\\mathbb{R}^N)$, $p \\geq N$ and $T$-periodic continuous perturbation $f:\\mathbb{R}^N\\times \\mathbb{R} \\to \\mathbb{R}$. The so-called resonant case is considered, i.e. when ${\\cal N}:=\\mathrm{Ker} (\\Delta + V) \\neq \\{0\\}$ and $f$ is bounded by a square-integrable function. We derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the Brouwer "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.3400","kind":"arxiv","version":2},"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"}