On positive radial symmetric solution of the Dirichlet problem for a certain nonlinear equation and a numerical method for its evaluation (Q1390465)

In the unit disk \(K= \{(x,y)\in \mathbb{R}^2\), \(r^2= x^2+ y^2<1\}\) with boundary \(\Gamma\) we consider the Dirichlet problem \[ \Delta u+ar^m u^{2k} =0, \quad (x,y)\in K, \quad u|_\Gamma =0. \tag{1} \] Here \(m\geq 0\) and \(k\geq 1\) are integers, \(a\equiv \text{const} >0\). We prove the existence of a unique positive solution of problem (1) in the class \(C^2 (\overline K)\) and propose a numerical method for evaluation of this solution.