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Trace inequalities of the Sobolev type and nonlinear Dirichlet problems

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Abstract

We discuss the solvability of nonlinear Dirichlet problems of the type \(- \Delta _{p, w} u = \sigma \) in \(\Omega \); \(u = 0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \({\mathbb {R}}^{n}\), \(\Delta _{p, w}\) is a weighted (p, w)-Laplacian and \(\sigma \) is a nonnegative locally finite Radon measure on \(\Omega \). We do not assume the finiteness of \(\sigma (\Omega )\). We revisit this problem from a potential theoretic perspective and provide criteria for the existence of solutions by \(L^{p}(w)-L^{q}(\sigma )\) trace inequalities or capacitary conditions. Additionally, we apply the method to the singular elliptic problem \(- \Delta _{p, w} u = \sigma u^{- \gamma }\) in \(\Omega \); \(u = 0\) on \(\partial \Omega \) and derive connection with the trace inequalities.

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Acknowledgements

The author would like to thank Professor Verbitsky for providing useful information on the contents of [56]. This work was supported by JST CREST Grant Number JPMJCR18K3 and JSPS KAKENHI Grant Number JP18J00965 and JP17H01092.

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  1. Graduate School of Information Science and Technology, Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido, 060-0814, Japan

    Takanobu Hara

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Hara, T. Trace inequalities of the Sobolev type and nonlinear Dirichlet problems. Calc. Var. 61, 216 (2022). https://doi.org/10.1007/s00526-022-02339-9

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