{"paper":{"title":"Blow-up for biharmonic Schrodinger equation with critical nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Thanh Viet Phan","submitted_at":"2018-07-24T10:05:36Z","abstract_excerpt":"We consider the minimizers for the biharmonic nonlinear Schr\\\"odinger functional $$ \\mathcal{E}_a(u)=\\int_{\\mathbb{R}^d} |\\Delta u(x)|^2 d x + \\int_{\\mathbb{R}^d} V(x) |u(x)|^2 d x - a \\int_{\\mathbb{R}^d} |u(x)|^{q} d x $$ with the mass constraint $\\int |u|^2=1$. We focus on the special power $q=2(1+4/d)$, which makes the nonlinear term $\\int |u|^q$ scales similarly to the biharmonic term $\\int |\\Delta u|^2$. Our main results are the existence and blow-up behavior of the minimizers when $a$ tends to a critical value $a^*$, which is the optimal constant in a Gagliardo--Nirenberg interpolation i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.09002","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"}