{"paper":{"title":"On the asymptotics of ground states for a boundary value problem for the equation $-\\varepsilon \\Delta_p u = a|u|^{q-2}u - b|u|^{\\gamma-2}u$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Elvira I. Turianova, Yavdat Sh. Il'yasov","submitted_at":"2026-05-23T09:11:02Z","abstract_excerpt":"We study a singularly perturbed Dirichlet problem for the $p$-Laplacian with competing superlinear terms, \\[ -\\varepsilon \\Delta_p u = a(x)|u|^{q-2}u - b(x)|u|^{\\gamma-2}u, \\qquad u|_{\\partial\\Omega}=0, \\] where $1<p<q<\\gamma<p^*$, $a\\geq 0$, $b\\geq\\sigma_b>0$, and $\\varepsilon>0$ is small. By means of the nonlinear Rayleigh quotient method, we introduce two critical parameter values, $\\varepsilon^*$ and $\\varepsilon_e^*$, related respectively to the Nehari manifold and to the zero energy level. We prove the nonexistence of nontrivial weak solutions for $\\varepsilon>\\varepsilon^*$, and the exi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.24482","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.24482/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}