Implicit elliptic equations via Krasnoselskii-Schaefer type theorems

Abstract

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.

Authors

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

Keywords

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

Paper coordinates

R. Precup, Implicit elliptic equations via Krasnoselskii-Schaefer type theorems, Electron. J. Qual. Theor. Diff. Eqns. 2020, no. 87, 1-9, https://doi.org/10.14232/ejqtde.2020.1.87

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About this paper

Journal

Electronic Journal of Qualitative Theory of Differential Equations

Publisher Name

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

Print ISSN
Online ISSN

1417-3875

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