**T. Chantladze, N. Kandelaki, A. Lomtatidze, and D. Ugulava**

##
The *P*-Laplacian and Connected Pairs of Functions

**abstract:**

The present investigation is stimulated by the works [1], [2] and [3]
in which the authors study oscillatory properties of half-linear ordinary
differential equations, of the so-called $P$-Laplacian. Here we consider its
two-dimensional version. Moreover, it turns out that the fundamentals of
the $P$-Laplacian theory can be successfully involved in a more
general scheme which we call the theory of connected pairs. We will see that
it contains, on the one hand, the notion of analyticity and, on the other
hand, it allows one to apply a very important transformation of the analysis,
the Legendre transformation to the investigation of the $P$-Laplacian.

**Mathematics Subject Classification:**
34K15, 34K40.

**Key words and phrases:**
Half-linear ordinary differential equations, the P-Laplacian, connected pairs.