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2021 A three solutions theorem for Pucci's extremal operator and its application
Mohan Mallick, Ram Baran Verma
Topol. Methods Nonlinear Anal. 58(1): 161-179 (2021). DOI: 10.12775/TMNA.2020.066

Abstract

In this article we prove a three solution type theorem for the following boundary value problem: \begin{equation*} \label{abs} \begin{cases} -\mathcal{M}_{\lambda,\Lambda}^+(D^2u) =f(u)& \text{in }\Omega,\\ u =0& \text{on }\partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ and $f\colon [0,\infty]\to[0,\infty]$ is a $C^{\alpha}$ function. This is motivated by the work of Amann [3] and Shivaji [27], where a three solutions theorem has been established for the Laplace operator. Furthermore, using this result we show the existence of three positive solutions to above boundary value by explicitly constructing two ordered pairs of sub and supersolutions when $f$ has a sublinear growth and $f(0)=0$.

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Mohan Mallick. Ram Baran Verma. "A three solutions theorem for Pucci's extremal operator and its application." Topol. Methods Nonlinear Anal. 58 (1) 161 - 179, 2021. https://doi.org/10.12775/TMNA.2020.066

Information

Published: 2021
First available in Project Euclid: 21 September 2021

Digital Object Identifier: 10.12775/TMNA.2020.066

Rights: Copyright © 2021 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.58 • No. 1 • 2021
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