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The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of J. Serrin and H. Zou, originally used for the Lane-Emden system, yields another proof but only in lower dimensions. Motivated by this, we further refine this approach to find an alternative proof of the Liouville "},"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":"1608.07592","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-08-26T20:15:52Z","cross_cats_sorted":[],"title_canon_sha256":"96c21e91f96f2fbf938368fe91f4ff65ef586ec036f2b9896ad525fd8bc50ead","abstract_canon_sha256":"3fb18e4ca886f069076c0043221b471a821961d085ead7c9374e16a417280fa4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:07.122345Z","signature_b64":"ecpmUrEqD1fn2S4BYVzyIc/crrz+D7VpYJkQzv0zYkYZ5n842sozgVsniB2Fr8G3ZKUXEoBR18a+R70ecy6ABw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59c7f5e108694b0450c30a2690ffbceea00aa2211c4fa16b44317fcd65cace27","last_reissued_at":"2026-05-18T00:51:07.121827Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:07.121827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A refined approach for non-negative entire solutions of $\\Delta u + u^p = 0$ with subcritical Sobolev growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"John Villavert","submitted_at":"2016-08-26T20:15:52Z","abstract_excerpt":"Let $N\\geq 2$ and $1 < p < (N+2)/(N-2)_{+}$. Consider the Lane-Emden equation $\\Delta u + u^p = 0$ in $\\mathbb{R}^N$ and recall the classical Liouville type theorem: if $u$ is a non-negative classical solution of the Lane-Emden equation, then $u \\equiv 0$. The method of moving planes combined with the Kelvin transform provides an elegant proof of this theorem. A classical approach of J. Serrin and H. Zou, originally used for the Lane-Emden system, yields another proof but only in lower dimensions. 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