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By introducing some new ideas and using the well-known results of the problem $(\\mathcal{P})$ in the cases of $a=\\mu=1$ and $b=0$, we obtain some"},"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":"1507.05392","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-20T06:30:29Z","cross_cats_sorted":[],"title_canon_sha256":"800ff63ddc4ea96ed8498e94f723bc65a8156301ad45a0f29fef751d6b4c522a","abstract_canon_sha256":"b9df00e382f9759cd416a367f85d682a5fb0991d9434193d64e8ad0d18692aca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:36.931945Z","signature_b64":"92fdcwnmr0bq/Z1xFVkh+zW4Hu1ofUiulyMcN/Ax1kfrgNmStLmaO/u6/H7C4uheIjivwAsOzJvDV3x99/snAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ae12e06f2ca21103c9829b2e5458ab1dc1fa29825e73ac12a7796e72359891d","last_reissued_at":"2026-05-18T01:36:36.931302Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:36.931302Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On finding solutions of a Kirchhoff type problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yisheng Huang, Yuanze Wu, Zeng Liu","submitted_at":"2015-07-20T06:30:29Z","abstract_excerpt":"Consider the following Kirchhoff type problem $$ \\left\\{\\aligned -\\bigg(a+b\\int_{\\mathbb{B}_R}|\\nabla u|^2dx\\bigg)\\Delta u&= \\lambda u^{q-1} + \\mu u^{p-1}, &\\quad \\text{in}\\mathbb{B}_R, \\\\ u&>0,&\\quad\\text{in}\\mathbb{B}_R,\\\\ u&=0,&\\quad\\text{on}\\partial\\mathbb{B}_R, \\endaligned \\right.\\eqno{(\\mathcal{P})} $$ where $\\mathbb{B}_R\\subset \\bbr^N(N\\geq3)$ is a ball, $2\\leq q<p\\leq2^*:=\\frac{2N}{N-2}$ and $a$, $b$, $\\lambda$, $\\mu$ are positive parameters. 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