Implicit elliptic equations via Krasnoselskii-Schaefer type theorems


Existence of solutions to the Dirichlet problem for implicit elliptic equations is established by using Krasnoselskii–Schaefer type theorems owed to Burton–Kirk and Gao–Li–Zhang. The nonlinearity of the equations splits into two terms: one term depending on the state, its gradient and the elliptic principal part is Lipschitz continuous, and the other one only depending on the state and its gradient has a superlinear growth and satisfies a sign condition. Correspondingly, the associated operator is a sum of a contraction with a completely continuous mapping. The solutions are found in a ball of a Lebesgue space of a sufficiently large radius established by the method of a priori bounds.


Radu Precup
Babes-Bolyai University, Cluj-Napoca, Romania


 implicit elliptic equation; fixed point; Krasnoselskii theorem for the sum of two operators

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R. Precup, Implicit elliptic equations via Krasnoselskii-Schaefer type theorems, Electron. J. Qual. Theor. Diff. Eqns. 2020, no. 87, 1-9,



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Electronic Journal of Qualitative Theory of Differential Equations

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Bolyai Institute, University of Szeged and the Hungarian Academy of Sciences

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