IJPAM: Volume 67, No. 1 (2011)

QUASILINEAR ELLIPTIC EQUATIONS WITH A DAMPING
TERM VIA PICONE-TYPE IDENTITIES:
OSCILLATORY SOLUTIONS AND UNIQUENESS

Tadie
Institute of Mathematics
5, Universitetsparken
Copenhagen, 2100, DENMARK
e-mail: [email protected]


Abstract.We establish a uniqueness theorem and some oscillation criterion for the equation

\begin{displaymath}Pu:= \nabla \bigg\{ a(x)\Phi(\nabla u)\bigg\} + f(x,u,\nabla u)+ c(x)\phi(u)=0 \quad \text{in } \Bbb R^n .\end{displaymath}

We use solely some selected Picone's type formulas and the known such results for the related half linear equation. The work is an extension of the earlier one on semilinear equations (see [#!t3!#]). This work underlines the fact that such results can follow mainly from some properties of the main coefficients $ a(x) $ and $c(x)$ of the equation. The usual methods based on the Riccati techniques (see [#!m1!#], [#!xu!#]) make our methods quite simple. Equations with $p-$Laplacian displayed above and associated half-linear equations (when $f\equiv 0$) have been widely investigated lately because of the interest in their applications (e.g. in physical and biological problems, see, e.g. [#!d2!#], [#!m1!#] and referrences therein).

Received: December 16, 2010

AMS Subject Classification: 35J60, 35J70

Key Words and Phrases: Picone's identity, quasilinear elliptic equations

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 67
Issue: 1