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# A Class of Equations with Three Solutions

by Biagio Ricceri
Department of Mathematics and Informatics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy
Mathematics 2020, 8(4), 478; https://doi.org/10.3390/math8040478
Received: 5 March 2020 / Revised: 26 March 2020 / Accepted: 29 March 2020 / Published: 1 April 2020
(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications)
Here is one of the results obtained in this paper: Let $Ω ⊂ R n$ be a smooth bounded domain, let $q > 1$ , with $q < n + 2 n − 2$ if $n ≥ 3$ and let $λ 1$ be the first eigenvalue of the problem $− Δ u = λ u$ in $Ω$ , $u = 0$ on $∂ Ω$ . Then, for every $λ > λ 1$ and for every convex set $S ⊆ H 0 1 ( Ω )$ dense in $H 0 1 ( Ω )$ , there exists $α ∈ S$ such that the problem $− Δ u = λ ( u + − ( u + ) q ) + α ( x )$ in $Ω$ , $u = 0$ on $∂ Ω$ , has at least three weak solutions, two of which are global minima in $H 0 1 ( Ω )$ of the functional $u → 1 2 ∫ Ω | ∇ u ( x ) | 2 d x − λ ∫ Ω 1 2 | u + ( x ) | 2 − 1 q + 1 | u + ( x ) | q + 1 d x − ∫ Ω α ( x ) u ( x ) d x$ where $u + = max { u , 0 }$ . View Full-Text
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Ricceri, B. A Class of Equations with Three Solutions. Mathematics 2020, 8, 478.