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$C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$

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  • In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.
    Mathematics Subject Classification: Primary: 35J60; Secondary: 35J70.

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