Abstract
We study the existence of positive viscosity solutions to Trudinger’s equation for cylindrical domains \({\Omega\times[0, T)}\), where \({\Omega\subset {I\!R}^{n}, n\ge 2,}\) is a bounded domain, T > 0 and \({2\le p < \infty}\). We show existence for general domains \({\Omega,}\) when \({n<p<\infty}\). For \({2\le p\le n}\), we prove existence for domains \({\Omega}\) that satisfy a uniform outer ball condition. We achieve this by constructing suitable sub-solutions and super-solutions and applying Perron’s method.
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An erratum to this article is available at http://dx.doi.org/10.1007/s00030-016-0423-9.
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Bhattacharya, T., Marazzi, L. On the viscosity solutions to Trudinger’s equation. Nonlinear Differ. Equ. Appl. 22, 1089–1114 (2015). https://doi.org/10.1007/s00030-015-0315-4
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DOI: https://doi.org/10.1007/s00030-015-0315-4