{"paper":{"title":"Mixed local-nonlocal equations with critical nonlinearity on $\\mathbb{R}^N$: Non-existence, Existence, and Multiplicity of positive solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nirjan Biswas, Paramananda Das, Souptik Chakraborty","submitted_at":"2025-10-17T10:21:13Z","abstract_excerpt":"We consider the following quasilinear critical problem involving the mixed local-nonlocal operator: \\begin{equation}\\label{main_prob_abstract_1}\\tag{$\\mathcal{P}_p$}\n  -\\Delta_p u+(-\\Delta_p)^s u=|u|^{p^*-2}u+f(x)\\text{ in }\\mathbb{R}^N, \\end{equation} where $s \\in (0,1), p \\in (1, \\infty), N>p$, $p^*=\\frac{Np}{N-p}$, and $f$ is a nonnegative functional in the dual space of the ambient solution space. If $f \\equiv0$, then we show that \\eqref{main_prob_abstract_1} does not admit any nontrivial weak solution. This phenomenon stands in contrast to the purely local and purely nonlocal cases. On th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.15507","kind":"arxiv","version":3},"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/2510.15507/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"}