Multidimensional Dirichlet series in the problem of the asymptotics of spectral series of nonlinear elliptic operators (Q1058218)

This substantial paper is a contribution to the use of multidimensional Dirichlet series, and especially of the recent results of Leont'cev, to the construction and study of asymptotic solutions of boundary value problems for nonlinear elliptic equations. An important question is that of ''superposition'' of individual solutions, and the paper shows that, for broad classes of equations, these superposition laws can be expressed in terms of multidimensional Dirichlet series. The paper is divided in three main parts: (1) Dirichlet series in ordinary differential equations; (2) Asymptotic solutions of the generalized nonlinear Steklov problem for elliptic equations with variable coefficients, (3) Complex WKB method in boundary value problems of the type of the Steklov problem for nonlinear equations. Applications are given to equations of electroneutral semiconductors, sine-Gordon equations and Schrödinger equations.