Positive Solution of Extremal Pucci’s Equations with Singular and Sublinear Nonlinearity

Abstract

In this paper, we establish the existence of a positive solution to $$\begin{aligned} \left\{ \begin{array}{ll} -\mathcal {M}^{+}_{\lambda ,\Lambda }(D^{2}u)=\frac{\mu k(x)f(u)}{u^{\alpha }}-\eta h(x)u^{q} &{}\quad \text {in }\;\Omega \\ u=0 &{}\quad \text {on }\;\partial \Omega , \end{array} \right. \end{aligned}$$ where $$\Omega $$ is a smooth bounded domain in $$\mathbb {R}^{n},~n\ge 1.$$ Under certain conditions on $$k,f~\text {and}~h,$$ using viscosity sub- and super solution method with the aid of comparison principle, we establish the existence of a unique positive viscosity solution. This work extends and complements the earlier works on semilinear and singular elliptic equations with sublinear nonlinearity.