{"paper":{"title":"A concentration phenomenon for semilinear elliptic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrzej Szulkin, Nils Ackermann","submitted_at":"2012-06-14T17:50:09Z","abstract_excerpt":"For a domain $\\Omega\\subset\\dR^N$ we consider the equation $ -\\Delta u + V(x)u = Q_n(x)\\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\\in(2,2^*)$. Here $V\\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $\\Omega$ and negative outside, and such that the sets $\\{Q_n>0\\}$ shrink to a point $x_0\\in\\Omega$ as $n\\to\\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\\{Q_n>0\\}$ shrink to two points "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3196","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"}