# A Class of Equations with Three Solutions

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 $\mathsf{\Omega}\subset {\mathbf{R}}^{n}$ be a smooth bounded domain, let $q>1$ , with $q<\frac{n+2}{n-2}$ if $n\ge 3$ and let ${\lambda}_{1}$ be the first eigenvalue of the problem $-\Delta u=\lambda u$ in $\mathsf{\Omega}$ , $u=0$ on $\partial \mathsf{\Omega}$ . Then, for every $\lambda >{\lambda}_{1}$ and for every convex set $S\subseteq {H}_{0}^{1}\left(\mathsf{\Omega}\right)$ dense in ${H}_{0}^{1}\left(\mathsf{\Omega}\right)$ , there exists $\alpha \in S$ such that the problem $-\Delta u=\lambda ({u}^{+}-{\left({u}^{+}\right)}^{q})+\alpha \left(x\right)$ in $\mathsf{\Omega}$ , $u=0$ on $\partial \mathsf{\Omega}$ , has at least three weak solutions, two of which are global minima in ${H}_{0}^{1}\left(\mathsf{\Omega}\right)$ of the functional $u\to \frac{1}{2}{\int}_{\mathsf{\Omega}}{|\nabla u\left(x\right)|}^{2}dx-\lambda {\int}_{\mathsf{\Omega}}\left(\frac{1}{2}|{u}^{+}{\left(x\right)|}^{2}-\frac{1}{q+1}{\left|{u}^{+}\left(x\right)\right|}^{q+1}\right)dx-{\int}_{\mathsf{\Omega}}\alpha \left(x\right)u\left(x\right)dx$ where ${u}^{+}=max\{u,0\}$ .
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*Keywords:*minimax; multiplicity; global minima

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**MDPI and ACS Style**

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

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