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
google scholar link
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