On blow-up solutions and dead zones in semilinear equations

Presented by Corresponding Member of the NAS of Ukraine V.Ya. Gutlyanskii We study semilinear elliptic equations of the form $div(A(z)\nabla u) = f (u)$ in $\Omega \subset \mathbb{C}$, where $A(z)$ stands for a sym metric $2×2$ matrix function with measurable entries, $detA =1$, and such that $1/ K |\xi|^{2} \leqslant \left \langle A(z)\xi, \xi \right \rangle\leqslant K|\xi|^{2}, \xi \in R^{2}, 1\leqslant  K < \infty$. Making use of our Factorization theorem, we give some explicit solutions for the above equation if $f = e^{u}$ or $f = u^{q}$, when matrices $A(z)$ are chosen in an appropriate form.