Spurious behavior for a numerical scheme of nonlinear elliptic equations (Q1902682)

The author approximates the equation \(u_{rr} + (N - 1)u_r/r + \lambda(u + u^p) = 0\) in \(0 < r < 1\), \(u_r(0) = u(1) = 0\), representing the radial solution of an elliptic equation on the unit \(N\)-ball, by the difference scheme \((u_{i + 1} - 2u_i + u_{i - 1})/h^2 + (N -1)(u_{i + 1} - u_{i -1})/2ih^2 + \lambda (u_i + u^p_i) = 0\), \(i = 1, \dots, M\), \(u_{M + 1} = 0\), \(u_0 = u_1\). He shows in particular that the case \(N = 3\), \(p = 7\) has spurious solutions.