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We will prove that if there is at least one eigenvalue of the $p$-Laplace operator between $\\lim_{u\\to 0} f(x,u)/|u|^{p-2}u$ and $\\lim_{|u|\\to +\\infty} f(x,u)/|u|^{p-2}u$, then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are obtained b"},"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":"1603.06718","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-03-22T10:11:17Z","cross_cats_sorted":[],"title_canon_sha256":"fc39d28efdcb18c798eaa72d9eba11698f27a2bf867eeff6de55b57aa17d84f8","abstract_canon_sha256":"b43371b3a2a9fe6d1f5bb13509436e9d248478766a01ad95abc422b6d867e526"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:18:34.867340Z","signature_b64":"QF7k/A1NxT8aB29fW9NnGhRY5DMcEyDj7IqneKwEUAL4tryMeEqUqNZt6GjssQYcr63HzapNFrzvqo2A8rbtCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c31daeafba3b9ef77ea878c6b4236a4ba0c9552205b3f06f489f70ea008f9a8","last_reissued_at":"2026-05-18T01:18:34.866875Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:18:34.866875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stationary solutions and connecting orbits for $p$-Laplace equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aleksander Cwiszewski, Mateusz Maciejewski","submitted_at":"2016-03-22T10:11:17Z","abstract_excerpt":"We deal with one dimensional $p$-Laplace equation of the form $$ u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \\ x\\in (0,l), \\ t>0, $$ under Dirichlet boundary condition, where $p>2$ and $f\\colon [0,l]\\times \\mathbb{R}\\to \\mathbb{R}$ is a continuous function with $f(x,0)=0$. We will prove that if there is at least one eigenvalue of the $p$-Laplace operator between $\\lim_{u\\to 0} f(x,u)/|u|^{p-2}u$ and $\\lim_{|u|\\to +\\infty} f(x,u)/|u|^{p-2}u$, then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. 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