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In spite of a possible failure of the Palais-Smale compactness condition, in this article we apply minimax method to obtain the weak solution to such an equation. In particular, in the case of $N-$Laplacian, using the minimization and the Ekeland variational principle, we obtain multiplicity of weak solutions.\n  Finally, we will prove the abov"},"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":"1107.0375","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-07-02T12:13:32Z","cross_cats_sorted":[],"title_canon_sha256":"29fe52d015b0d0e478a64bdd1ef7f6c2eed2346860541de950b8b256282d10fd","abstract_canon_sha256":"cf302e69550848aec7ac8d491e22923bfcfac6a0149de610a782d03d514f90c3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:54.086975Z","signature_b64":"FEIh3uBJX9326SelIltw2RsCH/Q8LFLc431sB09pxL9aYwfolhIWS5BBRFDjVG/feGI0eidbsMe1WCo5Lbg+BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18f9e14640826daf6b293bd8294a800100d006b012ab7a5c5a762c652f0cf73c","last_reissued_at":"2026-05-18T04:18:54.086561Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:54.086561Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence and multiplicity of solutions to equations of $N-$Laplacian type with critical exponential growth in $\\mathbb{R}^{N}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guozhen Lu, Nguyen Lam","submitted_at":"2011-07-02T12:13:32Z","abstract_excerpt":"In this paper, we deal with the existence and multiplicity of solutions to the nonuniformly elliptic equation of the N-Lapalcian type with a potential and a nonlinear term of critical exponential growth and satisfying the Ambrosetti-Rabinowitz condition. 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