Boundary blow-up solutions and their applications in quasilinear elliptic equations
It is considered the quasilinear elliptic equation \[ -\Delta_p u=a| u| ^{p-2}u-b(x)| u| ^{q-1}u\quad\text{ in}\;\Omega,\quad u=0\;\text{ on}\;\partial\Omega, \] where \(\Delta_pu=\text{ div\,}(| Du| ^{p-2}Du),\) \(1<p<\infty,\) \(q>p-1,\) \(a\) is a real parameter, \(b(x)\geq 0\) is in \(C^\mu(\bar\Omega)\) \((1>\mu>0),\) and \(\Omega\) is a bounded smooth domain in \({\mathbb R}^N\) \((N\geq 2).\) It is also considered the problem \[ -\Delta_p u=a| u| ^{p-2}u-b| u| ^{q-1}u\quad\text{ in}\;{\mathbb R}^N, \] where \(b>0\) is a constant. Based on a comparison principle it is studied the existence, uniqueness and asymptotic behavior of various boundary blow-up solutions.
scientific article; zbMATH DE number 1997291
scientific article