Nonexistence theorems for weak solutions of quasilinear elliptic equations

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: ${\Sigma }_{i=1}^n\left(a\left(x\right){\left|\nabla u\right|}^{p-2}{u}_{x_i}\right)=-{\left|u\right|}^{q-1}u$,${\Sigma }_{i=1}^n{\left(a\left(x\right){\left|u_{x_i}\right|}^{p-2}{u}_{x_i}\right)}_{x_i}=-{\left|u\right|}^{q-1}u$,${\Sigma }_{i=1}^n{\left(a\left(x\right){\left|\nabla u\right|}^{p-2}{u}_{x_i}/\sqrt{1+{\left|\nabla u\right|}^2}\right)}_{x_i}=-{\left|u\right|}^{q-1}u$, where $n\ge 1,p>1,q>0$ are fixed real numbers, and $a\left(x\right)$ 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.