Growth of entire \({\mathcal A}\)-subharmonic functions. (English) Zbl 1018.35027

The authors investigate the growth of entire \(\mathcal A\)- subharmonic functions. Based on the results of the pointwise potential estimate for the \(\mathcal A\)-superharmonic functions in [T. Kilpeläinen and J. Malý, Acta Math. 172, 137-161 (1994; Zbl 0820.35063)] and on the method by A. Eremenko and J. L. Lewis [Ann. Acad. Sci. Fenn., Ser. A I, Math. 16, 361-376 (1991; Zbl 0727.35022)], the authors give an estimate of the growth of a nonnegative \(\mathcal A\)-subharmonic function in \(\mathbb{R}^n\) in terms of the Wolff potential of its Riesz measure. The results extend the Nevanlinna’s first fundamental theorem for subharmonic functions to the nonlinear setting. As its corollaries, the authors prove that a nonnegative \(\mathcal A\)-subharmonic function has the same order as the Wolff potential of its Riesz measure. These results generalize the classical results for the Laplacian in [W. K. Hayman and P. B. Kennedy, Subharmonic Functions, Vol. I, London Mathematical Society Monographs, No. 9, Academic Press, London, (1976; Zbl 0419.31001)].


35J60 Nonlinear elliptic equations
31C45 Other generalizations (nonlinear potential theory, etc.)
35J70 Degenerate elliptic equations
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