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Open AccessArticle

A Class of Equations with Three Solutions

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
Keywords: minimax; multiplicity; global minima minimax; multiplicity; global minima
MDPI and ACS Style

Ricceri, B. A Class of Equations with Three Solutions. Mathematics 2020, 8, 478.

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