Multiplicity results for system of Pucci’s extremal operator

Abstract

This article deals with the existence of multiple positive solutions to the following system of nonlinear equations involving Pucci’s extremal operators:

$$\begin{aligned} \left\{ \begin{aligned} -\mathcal {M}_{\lambda _1,\Lambda _1}^+(D^2u_1)&=f_1(u_1,u_2,\dots ,u_n)~~~{} & {} \textrm{in}~~\Omega ,\\ -\mathcal {M}_{\lambda _2,\Lambda _2}^+(D^2u_2)&=f_2(u_1,u_2,\dots ,u_n)~~~{} & {} \textrm{in}~~\Omega ,\\ ~~~~~~~~\vdots&=~~~~~~~~~~~~ \vdots \\ -\mathcal {M}_{\lambda _n,\Lambda _n}^+(D^2u_n)&=f_n(u_1,u_2,\dots ,u_n)~~~{} & {} \textrm{in}~~\Omega ,\\ u_1=u_2=\dots =u_n&=0~~{} & {} \textrm{on}~~\partial \Omega , \end{aligned} \right. \end{aligned}$$

where $\Omega$ is a smooth and bounded domain in $\mathbb {R}^N$ and $f_i:[0,\infty )\times [0,\infty )\dots \times [0,\infty )\rightarrow [0,\infty )$ are $C^{\alpha }$ functions for $i=1,2,\dots ,n$. The multiplicity result in this work is motivated by the work Amann (SIAM Rev 18(4):620–709, 1976), and Shivaji (Nonlinear analysis and applications (Arlington, Tex., 1986), Dekker, New York, 1987), where the three solutions theorem (multiplicity) has been proved for linear equations. Later on, it was extended for a system of equations involving the Laplace operator by Shivaji and Ali (Differ Integr Equ 19(6):669–680, 2006). Thus, the results here can be considered as a nonlinear analog of the results mentioned above. We also have applied the above results to show the existence of three positive solutions to a system of nonlinear elliptic equations having combined sublinear growth by explicitly constructing two ordered pairs of sub and supersolutions.