A three solution Theorem for Pucci's extremal operator

Abstract: In this talk we will discuss the existence of three positive solutions to the following boundary value problem:

\(-\mathcal{M}_{\lambda,\Lambda}^+(D^2u)\)  =  \(f(u)\)       \(\rm{in} \)    \(\Omega,\\\)

                       \(u\) = \(0\)            \(\rm{on}\)    \(\partial\Omega,\)

where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^n\) and \(f:[0,\infty]\to[0,\infty]\) is a \(C^{\alpha}\) function with \(f(0)\geq 0\). \(\mathcal{M}_{\lambda,\Lambda}^+\) is the Pucci's maximal operator. The idea of the proof relies on the construction of two pairs of sub-super solutions \((\psi_1,\phi_1)\) and \((\psi_2,\phi_2)\) where \(\psi_1\leq \psi_2\leq \phi_1\), \(\psi_1\leq\phi_2\leq \phi_1\) with \(\psi_2\nless\phi_2 \), and \(\psi_2, \phi_2\) are strict sub and supersolutions, then we establish the existence of three solutions \(u_1, u_2\) and \(u_3\)for the above boundary value problem such that \(u_1\in[\psi_1,\phi_2\), \(u_2\in[\psi_2,\phi_1]\) and \(u_3\in [\psi_1,\phi_1]\setminus[\psi_1,\phi_2]\cap[\psi_2,\phi_1]\).