Some new oscillatory behavior of higher-order elliptic partial differential equations

The main objective of this study is to investigate the new adequate conditions for oscillation of higher-order elliptic partial differential equations by using the Riccati transformation and integral average method. The Riccati transformation converts a nonlinear first order Riccati differential equation into a second order linear ordinary differential equation, enabling solution via standard linear methods followed by inversion. Our plan of action is to reduce the multidimensional problem to an ordinary differential problem by using Jensen's inequality. Elliptic partial differential equations are used in almost every field of mathematics and physics, including Lie theory, geometry, and harmonic analysis. An elliptic PDE is fundamentally shown by the Laplacian equation and the Poisson equation. Appropriate examples are provided to highlight our results.