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

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 _qc=n(p−1)/(n−p)$q_c=n(p-1)/(n-p)$ is its critical exponent provided 2n/(n+1)2n/(n+1)<p<n. Furthermore, we prove that if max{1,p−1}q_c$max\{1,p-1\}<q\le q_c$, then every positive solution of the equations blows up in finite time; whereas for q>_qc$q>q_c$, the equations admit the global positive solutions for some boundary value f(x)$f\left(x\right)$ and some initial data _u0(x)$u_0\left(x\right)$. We also demonstrate that every positive solution of the equations blows up in finite time provided n≤p$n\le p$.