Behavior of solutions of first boundary value problem for a second-order elliptic equation in an unbounded domain (Q1074766)

The author considers a boundary value problem \[ (1)\quad Lu\equiv (a^{ij}(x)u_{x_ i})_{x_ j}+b^ k(x)u_{x_ k}+c(x)u=f(x)+(f_ i(x))_{x_ i},\quad u|_{\Gamma}=0, \] L being a uniformly elliptic operator in an unbounded domain \(\Omega \subset R^ n\), \(\Gamma\) \(\subset \partial \Omega\), \(\Gamma\) belongs to a neighborhood of infinity, u being a generalized solution. Let L behave as the Laplace operator when \(| x| \to \infty\), namely, \(a^{ij}(x)\to \delta^{ij}\), \(b^ k(x)| x| \to 0\), \(c(x)| x|^ 2\to 0\), then under conditions on the coefficients, \(\partial \Omega\) and r.h. of (1) the author derives an estimate for the \(\max_{| x| =R} u^ 2(x)\) when R is large enough. An example for \(n=2\) shows the preciseness of the estimate. In this case the estimate may be presented in the form \(\max_{\Omega \cap \{| x| =R\}}| u(x)| \leq C\cdot R^{(-1/2)+\epsilon},\) \(\forall \epsilon >0\), C=constant, supposing that r.h. of (1) equals zero.