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In contrast, much less information is available in the planar case $d=2$ which is the focus of the present paper. It has been observed by Cingolani and the second author \\cite{Cingolani-Weth-2016} that the variational structure of $("},"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":"1703.05090","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-15T11:36:03Z","cross_cats_sorted":[],"title_canon_sha256":"7357b8c4c8929e3e4e2db4a4c553090c52ada5895eb677464976f27dc0580c3a","abstract_canon_sha256":"fd80960cae2f7621ff49a3eb1fcfe9c5c7a3beb6c46f489678dd3c99a3e49122"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:38.423304Z","signature_b64":"6uPRh2tt3RRnhZyQSP5gPhiIzKtNez0LZR2VhwRf68WheWQpNdbR5RK0TIjFuBfMTdBrXA9nPybXLYUs4EM1Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0f08e87ac65b9ec6bd97d389d7813c9b19ae2a65233e1de6e867ea6f60027eda","last_reissued_at":"2026-05-18T00:48:38.422811Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:38.422811Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ground states and high energy solutions of the planar Schr\\\"odinger-Poisson system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Miao Du, Tobias Weth","submitted_at":"2017-03-15T11:36:03Z","abstract_excerpt":"In this paper, we are concerned with the Schr\\\"{o}dinger-Poisson system \\begin{equation} (0.1)\\qquad -\\Delta u + u +\\phi u = |u|^{p-2}u \\quad \\text{in}\\ \\mathbb{R}^{d},\\qquad \\Delta \\phi= u^{2} \\quad \\text{in}\\ \\mathbb{R}^{d}. \\end{equation} Due to its relevance in physics, the system has been extensively studied and is quite well understood in the case $d \\ge 3$. In contrast, much less information is available in the planar case $d=2$ which is the focus of the present paper. It has been observed by Cingolani and the second author \\cite{Cingolani-Weth-2016} that the variational structure of $("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05090","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1703.05090","created_at":"2026-05-18T00:48:38.422881+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.05090v1","created_at":"2026-05-18T00:48:38.422881+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.05090","created_at":"2026-05-18T00:48:38.422881+00:00"},{"alias_kind":"pith_short_12","alias_value":"B4EOQ6WGLOPM","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"B4EOQ6WGLOPMNPMX","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"B4EOQ6WG","created_at":"2026-05-18T12:31:08.081275+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/B4EOQ6WGLOPMNPMX2OE5PAJ4TM","json":"https://pith.science/pith/B4EOQ6WGLOPMNPMX2OE5PAJ4TM.json","graph_json":"https://pith.science/api/pith-number/B4EOQ6WGLOPMNPMX2OE5PAJ4TM/graph.json","events_json":"https://pith.science/api/pith-number/B4EOQ6WGLOPMNPMX2OE5PAJ4TM/events.json","paper":"https://pith.science/paper/B4EOQ6WG"},"agent_actions":{"view_html":"https://pith.science/pith/B4EOQ6WGLOPMNPMX2OE5PAJ4TM","download_json":"https://pith.science/pith/B4EOQ6WGLOPMNPMX2OE5PAJ4TM.json","view_paper":"https://pith.science/paper/B4EOQ6WG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.05090&json=true","fetch_graph":"https://pith.science/api/pith-number/B4EOQ6WGLOPMNPMX2OE5PAJ4TM/graph.json","fetch_events":"https://pith.science/api/pith-number/B4EOQ6WGLOPMNPMX2OE5PAJ4TM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/B4EOQ6WGLOPMNPMX2OE5PAJ4TM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/B4EOQ6WGLOPMNPMX2OE5PAJ4TM/action/storage_attestation","attest_author":"https://pith.science/pith/B4EOQ6WGLOPMNPMX2OE5PAJ4TM/action/author_attestation","sign_citation":"https://pith.science/pith/B4EOQ6WGLOPMNPMX2OE5PAJ4TM/action/citation_signature","submit_replication":"https://pith.science/pith/B4EOQ6WGLOPMNPMX2OE5PAJ4TM/action/replication_record"}},"created_at":"2026-05-18T00:48:38.422881+00:00","updated_at":"2026-05-18T00:48:38.422881+00:00"}