{"paper":{"title":"A problem involving the $p$-Laplacian operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"D. Choudhuri, Ratan K. Giri","submitted_at":"2016-01-15T19:27:04Z","abstract_excerpt":"Using a variational technique we guarantee the existence of a solution to the \\emph{resonant Lane-Emden} problem $-\\Delta_p u=\\lambda |u|^{q-2}u$, $u|_{\\partial\\Omega}=0$ if and only if a solution to $-\\Delta_p u=\\lambda |u|^{q-2}u+f$, $u|_{\\partial\\Omega}=0$, $f\\in L^{p'}(\\Omega)$ ($p'$ being the conjugate of $p$), exists for $q\\in (1,p)\\bigcup (p,p^{*})$ under a certain condition for both the cases, i.e., $1<q<p<p^{*}$ and $1< p < q < p^{*}$ - the sub-linear and the super-linear cases."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04039","kind":"arxiv","version":3},"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"}