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
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