Note on the PWB-method in the nonlinear case.

*(English)*Zbl 0633.31004The authors continue their series of papers on nonlinear potential theory and the theory of quasiregular mappings in n-space. A very nice survey of their work is contained in three separate articles by these authors in “Summer School in Potential Theory”, I. Laine (ed.) and O. Martio (ed.) (Joensuu/Finland 1983), where also the results of the present paper are reviewed.

The main result of the present paper is to show that the Perron-Wiener- Brelot (PWB) method applies to the present nonlinear situation. An earlier generalization of the PWB-method to a different nonlinear situation due to E. F. Beckenback and L. K. Jackson [Pac. J. Math. 3, 291-313 (1953; Zbl 0050.101)] made use of the fact that the difference of two solutions of their nonlinear equation satisfies a strong maximum principle. This information is not available in the present context and the authors use their own methods in a skillful way in order to overcome this difficulty.

The main result of the present paper is to show that the Perron-Wiener- Brelot (PWB) method applies to the present nonlinear situation. An earlier generalization of the PWB-method to a different nonlinear situation due to E. F. Beckenback and L. K. Jackson [Pac. J. Math. 3, 291-313 (1953; Zbl 0050.101)] made use of the fact that the difference of two solutions of their nonlinear equation satisfies a strong maximum principle. This information is not available in the present context and the authors use their own methods in a skillful way in order to overcome this difficulty.

Reviewer: M.Vuorinen

##### MSC:

31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |