On the asymptotic behavior of solutions of quasilinear elliptic equations

Abstract

The asymptotic behavior of solutions of second-order quasilinear elliptic and nonhyperbolic partial differential equations defined on unbounded domains in R n contained in {(Xl, ..., Xn): Ix,~ I < ~(~/x~ +... + x~_ 1)} for certain sublinear functions )~ is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data has appropriate asymptotic behavior at infinity. We prove Phragm~n-LindelSf theorems for large classes of nonhyperbolic operators, without ,lower order terms-, including uniformly elliptic operators and operators with well-defined genre, using special barrier functions which are constructed by considering an operator associated to our original operator. We also estimate the rate at which a solution converges to its limiting function at infinity in terms of properties of the top order coefficient ann of the operator and the rate at which the boundary values converge to their limiting function; these results are proven using appropriate barrier functions which depend on the behavior of the coefficients of the operator and the rate of convergence of boundary values.