IJPAM: Volume 5, No. 3 (2003)

SOLUTIONS OF SOME ANISOTROPIC EQUATIONS

Tadie
Institute of Mathematics
Universitetsparken 5
2100, Copenhagen, DENMARK
e-mail: tad@math.ku.dk


Abstract.For the problem (Qc) below, in [#!cite!#]7 we established the following results:


Theorem A. If there are two functions $\phi$ and $\psi$, which are (Qc)-compatible in $\Omega$ with all their second derivatives positive, then:

i) any classical solution $u$ of (Qc) satisfies $ \; \phi \leq u \leq \psi \;
$ in $\Omega$. Such a solution is unique if the $\mu_i$ depend only on $x$;

ii) if it exists, the solution of (Qc), which has all its second derivatives positive is unique.


Theorem B. Under the hypotheses (h1) - (h5),
1) If there are two functions $\phi$ and $\psi$, which are (Qc)-compatible in $\Omega$ with all their second derivatives positive, then:

i) if the $\mu_i$ are independent of $u$, (Qc) has a unique solution $u
\in C^2(\overline{\Omega})$ such that $ \; \phi \leq u \leq \psi \;
$ in $\Omega$;

ii) the same conclusion holds when the $\mu_i$ depend also on $u$ but the uniqueness holds only for solutions, whose all second derivatives are all positive.


The object of this note is to complete the work with some extension to the large solutions (as in (Q$\infty$)).

Received: January 10, 2003

AMS Subject Classification: 35J60, 34B15

Key Words and Phrases: (Qc)-compatible, existence and uniqness of solution, partial differential equations

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2003
Volume: 5
Issue: 3