{"paper":{"title":"Curves of equiharmonic solutions, and problems at resonance","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Philip Korman","submitted_at":"2016-09-19T16:24:59Z","abstract_excerpt":"We consider the semilinear Dirichlet problem \\[ \\Delta u+kg(u)=\\mu _1 \\varphi _1+\\cdots +\\mu _n \\varphi _n+e(x) \\;\\; \\mbox{for $x \\in \\Omega$}, \\;\\; u=0 \\;\\; \\mbox{on $\\partial \\Omega$}, \\] where $\\varphi _k$ is the $k$-th eigenfunction of the Laplacian on $\\Omega$ and $e(x) \\perp \\varphi _k$, $k=1, \\ldots, n$. Write the solution in the form $u(x)= \\Sigma _{i=1}^n \\xi _i \\varphi _i+U(x)$, with $ U \\perp \\varphi _k$, $k=1, \\ldots, n$. Starting with $k=0$, when the problem is linear, we continue the solution in $k$ by keeping $\\xi =(\\xi _1, \\ldots,\\xi _n)$ fixed, but allowing for $\\mu =(\\mu _1, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05817","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"}