Behavior of solutions of nonlinear second-order elliptic inequalities (Q1586668)

The author investigates the nonnegative solutions of the inequality \(\mathcal L u\geq F(x,u)\) in \(Q_R\setminus Z_{R_0}\) (\(Q_R\) -- open ball in \(\mathbb{R}^n\) of radius \(R\) center \(O\), \(Z_{R_0}\) -- closed ball of radius \(R_0\), \(0\leq R_0< R\leq \infty \)), where \(\mathcal L = \sum_{i,j=1}^na_{ij}(x)\partial^2/\partial x_i\partial x_j \)+\(\sum_{i=1}^nb_i(x)\partial /\partial x_i\), \(b_i\) are locally bounded functions and \(a_{ij}\) satisfy the ellipticity condition \(0<\sum_{i,j=1}^na_{ij}(x)\xi_i\xi_j\leq C|\xi|^2\) with some constant \(C>0\) for all \(x=(x_1,x_2,\dots ,x_n)\in \Omega \) and \(\xi =(\xi_1,\xi_2,\dots,\xi_n)\in \mathbb{R}^n\). The solution \(u\) of the inequality satisfies \(u_{Q_R\cap \partial\Omega\setminus Z_{R_0}}=0\). It is shown an estimate of \(M(r;u)=\max_{\Omega \cap S_r}u\) by solutions of the ordinary differential equation that is the radial part of the Laplace operator \(\mathcal L = \triangle\). There exist two well-known ways to do this. Using the spherical coordinates in \(\mathbb{R}^n\), either to construct barrier functions or to average solutions of the considered inequality on the unit sphere. However, these methods are not admissible for \(\mathcal L\) of general form. The difficulties are overcome by an interesting statement proposed by the author.