{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T20:53:34Z","timestamp":1772571214649,"version":"3.50.1"},"reference-count":78,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T00:00:00Z","timestamp":1744243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"INdAM\u2013GNAMPA Project \u201cProblemi Differenziali Non Lineari: Esistenza e Moltepplcit\u00e0 di soluzioni\u201d","award":["CUP E5324001950001"],"award-info":[{"award-number":["CUP E5324001950001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we establish the existence of positive solutions for a problem driven by a multi-phase operator composed of two distinct exponent Laplacian-type operators and a generalised lower-order term, which ensures asymmetric behaviour across three subregions of the domain under consideration. The reaction term involves a mild singularity at zero and includes a possibly sign-changing perturbation function. Under additional restrictive conditions, we also obtain a uniqueness result for the problem. Our existence result is based on pseudomonotone operator theory. Moreover, a detailed analysis, combined with a D\u00edaz\u2013Sa\u00e1-type argument, allows us to also establish a uniqueness theorem. To the best of our knowledge, this is the first work addressing such a generalisation of the multi-phase operator. These novel results can serve as a foundation for more general physical and engineering models.<\/jats:p>","DOI":"10.3390\/sym17040573","type":"journal-article","created":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T07:41:51Z","timestamp":1744270911000},"page":"573","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Existence and Uniqueness of Positive Solutions for Singular Asymmetric Multi-Phase Equations"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-7336-4410","authenticated-orcid":false,"given":"Giuseppe","family":"Failla","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Sciences, Physical Sciences and Earth Sciences (MIFT), University of Messina, Viale Ferdinando Stagno d\u2019Alcontres, 98166 Messina, Italy"}]},{"given":"Leszek","family":"Gasi\u0144ski","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the National Education Commission, ul. Podchor\u0105\u017cych 2, 30-084 Krak\u00f3w, Poland"}]},{"given":"Anna","family":"Petiurenko","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of the National Education Commission, ul. Podchor\u0105\u017cych 2, 30-084 Krak\u00f3w, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,10]]},"reference":[{"key":"ref_1","first-page":"043","article-title":"Optimal gradient estimates for multi-phase integrals","volume":"4","year":"2022","journal-title":"Math. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1631","DOI":"10.1016\/j.jde.2019.02.015","article-title":"Regularity for multi-phase variational problems","volume":"267","author":"Oh","year":"2019","journal-title":"J. Differ. Equations"},{"key":"ref_3","first-page":"675","article-title":"Averaging of functionals of the calculus of variations and elasticity theory","volume":"50","author":"Zhikov","year":"1986","journal-title":"Izv. Akad. Nauk SSSR Ser. 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