{"paper":{"title":"Explicit solutions and multiplicity results for some equations with the $p$-Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"Philip Korman","submitted_at":"2016-06-24T15:53:46Z","abstract_excerpt":"We derive explicit ground state solutions for several equations with the $p$-Laplacian in $R^n$, including (here $\\varphi (z)=z|z|^{p-2}$, with $p>1$) \\[ \\varphi \\left(u'(r)\\right)' +\\frac{n-1}{r} \\varphi \\left(u'(r)\\right)+u^M+u^Q=0 \\,. \\] The constant $M>0$ is assumed to be below the critical power, while $Q=\\frac{M p-p+1}{p-1}$ is above the critical power. This explicit solution is used to give a multiplicity result, similarly to C.S. Lin and W.-M. Ni [11]. We also give the $p$-Laplace version of G. Bratu's solution [3].\n  In another direction, we present a change of variables which removes"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07734","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"}