Abstract and Applied Analysis
Volume 6 (2001), Issue 3, Pages 163-189
doi:10.1155/S1085337501000549

Nonexistence theorems for weak solutions of quasilinear elliptic equations

A. G. Kartsatos1 and V. V. Kurta2

1Department of Mathematics, University of South Florida, Tampa 33620-5700, FL, USA
2416 Fourth Street, P.O. Box 8604, Ann Arbor, MI 48107-8604, USA

Received 14 April 2001

Copyright © 2001 A. G. Kartsatos and V. V. Kurta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

New nonexistence results are obtained for entire bounded (either from above or from below) weak solutions of wide classes of quasilinear elliptic equations and inequalities. It should be stressed that these solutions belong only locally to the corresponding Sobolev spaces. Important examples of the situations considered herein are the following: Σi=1n(a (x)|∇u| p−2uxi)=−|u| q−1u,Σi=1n(a (x)| uxi | p−2uxi)xi=−|u| q−1u,Σi=1n(a (x)|∇u| p−2uxi/1+|∇u| 2)xi=−|u| q−1u, where n≥1, p>1, q>0 are fixed real numbers, and a(x) is a nonnegative measurable locally bounded function. The methods involve the use of capacity theory in connection with special types of test functions and new integral inequalities. Various results, involving mainly classical solutions, are improved and/or extended to the present cases.