A Class of Equations with Three Solutions

Biagio Ricceri
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Biagio Ricceri: Department of Mathematics and Informatics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Mathematics, 2020, vol. 8, issue 4, 1-8

Abstract: Here is one of the results obtained in this paper: Let ? ? R n be a smooth bounded domain, let q > 1 , with q < n + 2 n ? 2 if n ? 3 and let ? 1 be the first eigenvalue of the problem ? ? u = ? u in ? , u = 0 on ? ? . Then, for every ? > ? 1 and for every convex set S ? L ? ( ? ) dense in L 2 ( ? ) , there exists ? ? S such that the problem ? ? u = ? ( u + ? ( u + ) q ) + ? ( x ) in ? , u = 0 on ? ? , has at least three weak solutions, two of which are global minima in H 0 1 ( ? ) of the functional u ? 1 2 ? ? | ? u ( x ) | 2 d x ? ? ? ? 1 2 | u + ( x ) | 2 ? 1 q + 1 | u + ( x ) | q + 1 d x ? ? ? ? ( x ) u ( x ) d x where u + = max { u , 0 } .

Keywords: minimax; multiplicity; global minima (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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