{"paper":{"title":"Nonlinear Schr\\\"odinger problems: symmetries of some variational solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Christopher Grumiau","submitted_at":"2012-12-20T16:02:27Z","abstract_excerpt":"In this paper, we are interested in the nonlinear Schr\\\"odinger problem $-\\Delta u + Vu = \\abs{u}^{p-2}u$ submitted to the\n  Dirichlet boundary conditions. We consider $p>2$ and we are working with an open bounded domain $\\Omega\\subset\\IR^N$ ($N\\geq 2$). Potential $V$ satisfies $\\max(V,0)\\in L^{N/2}(\\Omega)$ and $\\min(V,0)\\in L^{+\\infty}(\\Omega)$. Moreover, $-\\Delta + V$ is positive definite and has one and only one principal eigenvalue. When $p\\simeq 2$, we prove the uniqueness of the solution once we fix the projection on an eigenspace of $-\\Delta + V$. It implies partial symmetries (or symm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.5111","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"}