Existence and nonexistence of global positive solutions for the evolution P-Laplacian equations in exterior domains

Abstract

This paper deals with the existence and nonexistence of global positive solutions for two evolution P-Laplacian equations in exterior domains with inhomogeneous boundary conditions. We demonstrate that q c = n ( p − 1 ) / ( n − p ) is its critical exponent provided 2 n / ( n + 1 ) < p < n . Furthermore, we prove that if max { 1 , p − 1 } < q ≤ q c , then every positive solution of the equations blows up in finite time; whereas for q > q c , the equations admit the global positive solutions for some boundary value f ( x ) and some initial data u 0 ( x ) . We also demonstrate that every positive solution of the equations blows up in finite time provided n ≤ p .