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We show that this problem has a solution for all $f$ in a suitable space of distributions. Then we apply this result to some classes of functions $V$ which in particular include the Hardy potential and the potential $V(x)=\\lambda_{1,p}(\\Omega)$, where $\\lambda_{1,p}(\\Omega)$ is the Poin"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1709.05187","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-09-15T13:19:30Z","cross_cats_sorted":[],"title_canon_sha256":"dbe895fd417e8ffb81ee330765f13878646f584977374f340c3e8b9f9d8f88fa","abstract_canon_sha256":"f62803c82dbc36edceb78b84f1080d35f9ad0daaebf2989d25b8a314bff62100"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:06.466866Z","signature_b64":"LPRDIGHCKOvKMhGu9bKGqEP/VnuOuvsS7BAHuYBCaq1RXigKk5VT122MMFgi8mMHu6MQZLrLSMWJY9uMEFN+AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b038ed26ee69e98290624c973802dafde46df99e9e12ef23462a253ded84305b","last_reissued_at":"2026-05-18T00:35:06.466216Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:06.466216Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some weakly coercive quasilinear problems with forcing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Andrzej Szulkin, Michel Willem","submitted_at":"2017-09-15T13:19:30Z","abstract_excerpt":"We consider the forced problem $-\\Delta_p u - V(x)|u|^{p-2} u = f(x)$, where $\\Delta_p$ is the $p$-Laplacian ($1<p<\\infty$) in a domain $\\Omega\\subset \\mathbb{R}^N$, $V\\ge 0$ and $Q_V (u) := \\int_\\Omega |\\nabla u|^p\\, dx - \\int_\\Omega V|u|^p\\,dx$ satisfies the condition (A) stated at the beginning of the paper. We show that this problem has a solution for all $f$ in a suitable space of distributions. 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