Oscillation criteria of Kamenev-type for non-linear partial differential equations

Abstract

Using the technique of weighted integral averages, known in the oscillation theory of ordinary differential equations, we obtain new oscillation criteria for the non-linear partial differential equation $\begin{array}{rl}& \mathrm{div}\left(\frac{{\Vert \mathrm{\nabla }y\Vert }^{p-2}\mathrm{\nabla }y}{{(1+{\Vert \mathrm{\nabla }y\Vert }^p)}^{\frac{p-1}{p}}}\right)+f\left(u\right)\frac{{\left|y\right|}^{p-2}y}{{(1-{\left|y\right|}^p)}^{\frac{p-1}{p}}}\\ & =0,p>1\mathrm{and}u\ge u_0,\end{array}$ on unbounded domains.

Keywords:

• The p-prescribed curvature and p-relativistic operators
• oscillatory solution
• Riccati-type inequality
• non-linear partial differential equation

AMS Subject Classifications:

• 35B05
• 35J60
• 34C10

Acknowledgments

The authors thank the anonymous referee for his/her valuable suggestions and comments that helped to improve the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).