{"paper":{"title":"Multiple solutions for Grushin operator without odd nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mohamed Karim Hamdani","submitted_at":"2019-09-08T09:50:18Z","abstract_excerpt":"We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively: \\begin{eqnarray*} (P_g)\\quad - \\Delta_{\\lambda} u + V(x) u = f(x,u)+g(x),\\;\\mbox{ in } \\R^N,\\; \\end{eqnarray*} and \\begin{eqnarray*} (P_0)\\quad - \\Delta_{\\lambda} u + V(x) u = K(x)f(x,u),\\;\\mbox{ in } \\R^N,\\; \\end{eqnarray*} where $\\Delta_{\\lambda}$ is the strongly degenerate operator, $V(x)$ is allowed to be sign-changing, $K\\in C(\\R^N,\\R)$, $g:\\R^N\\to\\R$ is a perturbation and the nonlinearity $f(x,u)$ is a continuous function does not satisfy the Ambrosetti-Rabinowitz sup"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.03417","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/1909.03417/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"}