Abstract
In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the main ingredient is to construct appropriate subsolutions and supersolutions as barrier functions. We also prove a nonexistence result to show the convergence rate of the phase functions is optimal.
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Acknowledgements
Corresponding Author J. Bao is supported by the National Key Research and Development Program of China (No. 2020YFA0712900) and the Beijing Natural Science Foundation (No. 1222017). The second author Z. Liu is supported by the China Postdoctoral Science Foundation (No. 2022M720327). The third author C. Wang is supported by “the Fundamental Research Funds for the Central Universities” in UIBE (No. 23QD04).
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Bao, J., Liu, Z. & Wang, C. Existence of Entire Solutions to the Lagrangian Mean Curvature Equations in Supercritical Phase. J Geom Anal 34, 146 (2024). https://doi.org/10.1007/s12220-024-01589-7
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DOI: https://doi.org/10.1007/s12220-024-01589-7