Vekua established the existence and uniqueness of solutions to a quasilinear elliptic equation on the bounded domain $ \Omega $ with a sufficiently smooth boundary, with Dirichlet boundary conditions in the Sobolev spaces $ W^{2}_{p}(\Omega) $ for $ p > 2 $, the second-order derivatives of the solution belong to $ L_p(\Omega) $, $ p > 2 $. The goal of this note is to extend his results: the corresponding solvability result in the Besov space $ B^{2+\alpha}_{p, 1}(\Omega)\hookrightarrow C^{1}(\overline{\Omega}) $ for $ 1 < p < 2 $ and $ \alpha = 2/p-1 $ with the second-order derivatives of the solution belonging to $ B^{\alpha}_{p, 1}(\Omega)\hookrightarrow L_2(\Omega) $.
Citation: Nazarbay Bliev, Nurlan Yerkinbayev. Dirichlet problem for two-dimensional quasilinear second-order elliptic equations in Besov spaces[J]. AIMS Mathematics, 2026, 11(9): 28490-28501. doi: 10.3934/math.20261135
Vekua established the existence and uniqueness of solutions to a quasilinear elliptic equation on the bounded domain $ \Omega $ with a sufficiently smooth boundary, with Dirichlet boundary conditions in the Sobolev spaces $ W^{2}_{p}(\Omega) $ for $ p > 2 $, the second-order derivatives of the solution belong to $ L_p(\Omega) $, $ p > 2 $. The goal of this note is to extend his results: the corresponding solvability result in the Besov space $ B^{2+\alpha}_{p, 1}(\Omega)\hookrightarrow C^{1}(\overline{\Omega}) $ for $ 1 < p < 2 $ and $ \alpha = 2/p-1 $ with the second-order derivatives of the solution belonging to $ B^{\alpha}_{p, 1}(\Omega)\hookrightarrow L_2(\Omega) $.
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Nazarbay Bliev, Nurlan Yerkinbayev. Dirichlet problem for two-dimensional quasilinear second-order elliptic equations in Besov spaces[J]. AIMS Mathematics, 2026, 11(9): 28490-28501. doi: 10.3934/math.20261135
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