An application of generalized Morrey spaces to unique continuation property of the quasilinear elliptic equations

Nicky K. Tumalun; Philotheus E. A. Tuerah; Marvel G. Maukar; Anetha L. F. Tilaar; Patricia V. J. Runtu; Nicky K. Tumalun; Philotheus E. A. Tuerah; Marvel G. Maukar; Anetha L. F. Tilaar; Patricia V. J. Runtu

doi:10.3934/math.20231325

AIMS Mathematics

2023, Volume 8, Issue 11: 26007-26020. doi: 10.3934/math.20231325

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Research article

An application of generalized Morrey spaces to unique continuation property of the quasilinear elliptic equations

Department of Mathematics, Universitas Negeri Manado, Tondano Selatan 95618, Indonesia

Received: 24 June 2023 Revised: 25 July 2023 Accepted: 22 August 2023 Published: 08 September 2023
MSC : 26D10, 35J62, 46E30

In this paper, we study nonnegative weak solutions of the quasilinear elliptic equation $ \text{div}(A(x, u, \nabla u)) = B(x, u, \nabla u) $, in a bounded open set $ \Omega $, whose coefficients belong to a generalized Morrey space. We show that $ \log (u + \delta) $, for $ u $ a nonnegative solution and $ \delta $ an arbitrary positive real number, belongs to $ \text{BMO}(B) $, where $ B $ is an open ball contained in $ \Omega $. As a consequence, this equation has the strong unique continuation property. For the main proof, we use approximation by smooth functions to the weak solutions to handle the weak gradient of the composite function which involves the weak solutions and then apply Fefferman's inequality in generalized Morrey spaces, recently proved by Tumalun et al. ^[1].
- quasilinear elliptic equations,
- generalized Morrey spaces,
- Fefferman's inequality,
- bounded mean oscillation,
- strong unique continuation property
Citation: Nicky K. Tumalun, Philotheus E. A. Tuerah, Marvel G. Maukar, Anetha L. F. Tilaar, Patricia V. J. Runtu. An application of generalized Morrey spaces to unique continuation property of the quasilinear elliptic equations[J]. AIMS Mathematics, 2023, 8(11): 26007-26020. doi: 10.3934/math.20231325

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Abstract

In this paper, we study nonnegative weak solutions of the quasilinear elliptic equation $ \text{div}(A(x, u, \nabla u)) = B(x, u, \nabla u) $, in a bounded open set $ \Omega $, whose coefficients belong to a generalized Morrey space. We show that $ \log (u + \delta) $, for $ u $ a nonnegative solution and $ \delta $ an arbitrary positive real number, belongs to $ \text{BMO}(B) $, where $ B $ is an open ball contained in $ \Omega $. As a consequence, this equation has the strong unique continuation property. For the main proof, we use approximation by smooth functions to the weak solutions to handle the weak gradient of the composite function which involves the weak solutions and then apply Fefferman's inequality in generalized Morrey spaces, recently proved by Tumalun et al. ^[1].

References

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