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We show that this problem exhibits a concave-convex nature for $1<q<p-1$. In fact, we prove that there exists a positive value $\\lambda^*$ such that the problem has no positive solution for $\\lambda > \\lambda^*$ and a minimal positive solution for $0<\\lambda < \\lambda^*$. If in addition we assume that $p"},"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":"1703.03376","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-09T18:05:14Z","cross_cats_sorted":[],"title_canon_sha256":"5448365de1545a9930f8e2da6209e18d874d3932649a9ccb99cb2f739f29a679","abstract_canon_sha256":"bd168d889c97fbf278dc4ed32875433f322047447e5b838438979b87c06a0f51"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:00.557483Z","signature_b64":"64rn1zC01fymZjEufty3TkmFxVbbFmLF6qj08f/IG32+ZUKGhdgzAE5P1j+rNoPEFnBpkT7XLX2mTJz2N26jDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b5268cfe2a856cce746d671a81877617b0c9a546b6a15c72e05e9c4b80100cb","last_reissued_at":"2026-05-18T00:49:00.556669Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:00.556669Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A concave-convex problem with a variable operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alexis Molino, Julio D. Rossi","submitted_at":"2017-03-09T18:05:14Z","abstract_excerpt":"We study the following elliptic problem $-A(u) = \\lambda u^q$ with Dirichlet boundary conditions, where $A(u) (x) = \\Delta u (x) \\chi_{D_1} (x)+ \\Delta_p u(x) \\chi_{D_2}(x)$ is the Laplacian in one part of the domain, $D_1$, and the $p-$Laplacian (with $p>2$) in the rest of the domain, $D_2 $. We show that this problem exhibits a concave-convex nature for $1<q<p-1$. In fact, we prove that there exists a positive value $\\lambda^*$ such that the problem has no positive solution for $\\lambda > \\lambda^*$ and a minimal positive solution for $0<\\lambda < \\lambda^*$. 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