{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T12:11:06Z","timestamp":1782735066738,"version":"3.54.5"},"reference-count":18,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T00:00:00Z","timestamp":1717459200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NNSFs of China","award":["12061062"],"award-info":[{"award-number":["12061062"]}]},{"name":"NNSFs of China","award":["12161080"],"award-info":[{"award-number":["12161080"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper concerns with the existence of radial solutions of the biharmonic elliptic equation \u25b52u=f(|x|,u,|\u2207u|,\u25b5u) in an annular domain \u03a9={x\u2208RN:r1&lt;|x|&lt;r2}(N\u22652) with the boundary conditions u|\u2202\u03a9=0 and \u25b5u|\u2202\u03a9=0, where f:[r1,r2]\u00d7R\u00d7R+\u00d7R\u2192R is continuous. Under certain inequality conditions on f involving the principal eigenvalue \u03bb1 of the Laplace operator \u2212\u25b5 with boundary condition u|\u2202\u03a9=0, an existence result and a uniqueness result are obtained. The inequality conditions allow for f(r,\u03be,\u03b6,\u03b7) to be a superlinear growth on \u03be,\u03b6,\u03b7 as |(\u03be,\u03b6,\u03b7)|\u2192\u221e. Our discussion is based on the Leray\u2013Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates.<\/jats:p>","DOI":"10.3390\/axioms13060383","type":"journal-article","created":{"date-parts":[[2024,6,4]],"date-time":"2024-06-04T11:48:50Z","timestamp":1717501730000},"page":"383","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5193-030X","authenticated-orcid":false,"given":"Yongxiang","family":"Li","sequence":"first","affiliation":[{"name":"Department of Mathematics, Northwest Normal University, Lanzhou 730070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yanyan","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Northwest Normal University, Lanzhou 730070, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1137\/0150041","article-title":"Traveling waves in a suspension bridge","volume":"50","author":"McKenna","year":"1990","journal-title":"SIAM J. Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1006\/jdeq.1996.3155","article-title":"Traveling waves in a nonlinear suspension beam: Theoretical results and numerical observations","volume":"135","author":"Chen","year":"1997","journal-title":"J. Differ. Equ."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Gazzola, F., Grunau, H., and Sweers, G. (2010). Polyharmonic Boundary Value Problems, Lectures Notes in Mathematics, Springer.","DOI":"10.1007\/978-3-642-12245-3"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1155\/S0161171290000692","article-title":"Biharmonic eigenvalue problems and Lp estimates","volume":"13","author":"Gupta","year":"1990","journal-title":"Int. J. Math. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/s00526-002-0182-9","article-title":"Existence and nonexistence results for critical growth biharmonic elliptic equations","volume":"18","author":"Gazzola","year":"2003","journal-title":"Calc. Var."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"241518","DOI":"10.1155\/2010\/241518","article-title":"Multiple solutions for biharmonic equations with asymptotically linear nonlinearities","volume":"2010","author":"Pei","year":"2010","journal-title":"Bound. Value Probl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/S0252-9602(07)60055-1","article-title":"Biharmonic equations with asymptotically linear nonlinearities","volume":"27B","author":"Liu","year":"2007","journal-title":"Acta Math. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/s00009-023-02513-z","article-title":"Positive solutions for biharmonic equations: Existence, uniqueness and multiplicity","volume":"20","author":"Feng","year":"2023","journal-title":"Mediterr. J. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"108687","DOI":"10.1016\/j.aml.2023.108687","article-title":"Positive solutions for a class of biharmonic equations: Existence and uniqueness","volume":"143","author":"Feng","year":"2023","journal-title":"Appl. Math. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3325","DOI":"10.1016\/j.na.2007.03.028","article-title":"Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation","volume":"68","author":"An","year":"2008","journal-title":"Nonlinear Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2271","DOI":"10.1016\/j.na.2009.11.001","article-title":"On sign-changing solution for a fourth-order asymptotically linear elliptic problem","volume":"72","author":"Liu","year":"2010","journal-title":"Nonlinear Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.jmaa.2011.05.030","article-title":"Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin","volume":"383","author":"Zhang","year":"2011","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"14704","DOI":"10.3934\/math.2023752","article-title":"Multiplicity results for some fourth-order elliptic equations with combined nonlinearities","volume":"8","author":"Pei","year":"2023","journal-title":"AIMS Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1090\/S0002-9939-99-05430-1","article-title":"On fourth-order elliptic boundary value problems","volume":"128","author":"Pao","year":"2000","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/j.jmaa.2010.07.027","article-title":"Nonlinear fourth-order elliptic equations with nonlocal boundary conditions","volume":"372","author":"Pao","year":"2010","journal-title":"J. Math. Anal. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmaa.2004.09.063","article-title":"On fourth-order elliptic boundary value problems with nonmonotone nonlinear function","volume":"307","author":"Wang","year":"2005","journal-title":"J. Math. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1080\/17476933.2017.1292261","article-title":"Positive radial solutions for elliptic equations with nonlinear gradient terms in an annulus","volume":"63","author":"Li","year":"2018","journal-title":"Complex Var. Elliptic Equ."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Deimling, K. (1985). Nonlinear Functional Analysis, Springer.","DOI":"10.1007\/978-3-662-00547-7"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/383\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:53:42Z","timestamp":1760108022000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/6\/383"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,4]]},"references-count":18,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["axioms13060383"],"URL":"https:\/\/doi.org\/10.3390\/axioms13060383","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,4]]}}}