Oscillation Theory for Nonlinear Differential Equations and Its Applications to Elliptic Equations

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The purpose of this research project is to study oscillation problems for ordinary differential equations and elliptic equations with p-Laplacian. The main results are as follows: (1) Oscillation and non-oscillation criteria are given for all nontrivial solutions of second-order damped nonlinear differential equations with one-dimensional p-Laplacian. (2) A sufficient condition is obtained for half-linear differential equations with delay nonlinear perturbations to have a non-oscillatory solution. (3) We establish a new comparison theorem on the oscillation of solutions of nonlinear differential equations with one-dimensional p-Laplacian. (4) We consider radial solutions of elliptic equations with p-Laplacian, and give sufficient conditions for the equations to have a positive solution.