On blow-up solutions and dead zones in semilinear equations

1Gutlyanskii, VYa., 1Nesmelova, OV, 1Ryazanov, VI
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
Dopov. Nac. akad. nauk Ukr. 2018, 4:9-15
Section: Mathematics
Language: English
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.
Keywords: blowup solutions, quasiconformal mappings, semilinear PDE
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