ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXII, 1(2003)
p. 45 – 57

The Application of Picone-type Identity for some Nonlinear Elliptic Differential Equations
G. Bognar and O. Dosly

Abstract.  We established a Picone-type identity for the second order partial differential equation $$\sum_{i=1}^N\frac{\partial}{\partial x_i} \left(r_i(x)\varphi\left(\frac{\partial u}{\partial x_i}\right)\right)+ c(x)\varphi(u)=0,\quad \varphi(u):=|u|^{p-1}u,\ p>0. \leqno{(*)}$$ Using this identity we prove the Leighton-type comparison theorem for a pair of equations of the form (*). Properties of the principal eigenvale of a certain eigenvalue problem associated with (*) are investigated as well.

AMS subject classification:  35 J 20
Keywords:  Picone's identity, half-linear PDE, $p$-Laplacian, eigenvale problem, principal eigenvalue

Acta Mathematica Universitatis Comenianae
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