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Also, we assume $g$ behaves like $\\exp (\\beta |u|^4)$ as $|u|\\to \\infty.$ We prove the existence of at least one weak solution $u \\in H^1({\\BR}^2)$ with $u^2 \\in H^1({\\BR}^2).$ Mountain pass in a suitable Orlicz space together with Moser-Trudinger are employed to establish this result. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schr\\\"{o}dinger "},"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":"math/0609245","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2006-09-08T20:45:31Z","cross_cats_sorted":[],"title_canon_sha256":"4520fa9a177d905fad307e5661e683b268169a313427dbf4afcd8c03c2e7942e","abstract_canon_sha256":"75e836199b165d362835c0a77285181c3dbd4e4ea87a4f269f6ae4dd1736737c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:23.799059Z","signature_b64":"9J7Y7J2vStJcKgB6AyCxpxHr6Kx69Ak4jPrMMEebiS6gJ3V7CugQJk4nEbBeDdoTzYvtcJcp/IWSheo6tv8NAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1facb726c6d81cb60e4b381bba0fa66943d2bb2bd2110e6233a41efee76ee977","last_reissued_at":"2026-05-18T01:38:23.798410Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:23.798410Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On a class of periodic quasilinear Schr\\\"{o}dinger equations involving critical growth in ${\\BR}^2$","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Abbas Moameni","submitted_at":"2006-09-08T20:45:31Z","abstract_excerpt":"We consider the equation $- \\Delta u+V(x)u- k(\\Del(|u|^{2}))u=g(x,u), u>0, x \\in {\\BR}^2,$ where $V:{\\BR}^2\\to {\\BR}$ and $g:{\\BR}^2 \\times {\\BR}\\to {\\BR}$ are two continuous $1-$periodic functions. Also, we assume $g$ behaves like $\\exp (\\beta |u|^4)$ as $|u|\\to \\infty.$ We prove the existence of at least one weak solution $u \\in H^1({\\BR}^2)$ with $u^2 \\in H^1({\\BR}^2).$ Mountain pass in a suitable Orlicz space together with Moser-Trudinger are employed to establish this result. 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