{"paper":{"title":"A note on semilinear elliptic equation with biharmonic operator and multiple critical nonlinearities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mousomi Bhakta","submitted_at":"2014-05-01T14:06:37Z","abstract_excerpt":"We study the existence and non-existence of nontrivial weak solution of $$ {\\Delta^2u-\\mu\\frac{u}{|x|^{4}} = \\frac{|u|^{q_{\\beta}-2}u}{|x|^{\\beta}}+|u|^{q-2}u\\quad\\textrm{in ${\\mathbb R}^N$,}} $$ where $N\\geq 5$, $q_{\\beta}=\\frac{2(N-\\beta)}{N-4}$, $0<\\beta<4$, $1<q\\leq 2^{**}$ and $\\mu<\\mu_1:=\\big(\\frac{N(N-4)}{4}\\big)^2$. Using Pohozaev type of identity, we prove the non-existence result when $1<q< 2^{**}$. On the other hand when the equation has multiple critical nonlinearities i.e. $q=2^{**}$ and $-(N-2)^2\\leq\\mu<\\mu_1$, we establish the existence of nontrivial solution using the Mountain-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0162","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"}