On the Asymptotic Behavior of Solutions of a Semilinear Elliptic Boundary Problem in Unbounded Domains

We consider solutions of an elliptic linear equation $Lu=0$ of second order in an unbounded domain $Q$ in $\mathbb R^n$ supposing that $Q\subset\{x=(x',x_n)\colon 0<x_n<\infty,\, |x'|<\gamma(x_n)\}$, where $1\le \gamma(t)\le At+B$, and that $u$ satisfies the nonlinear boundary condition $\frac{\partial u}{\partial N}+k(x)u+b(x)|u(x)|^{p-1}u(x)=0$ on the part of the boundary of $Q$ where $x_n>0$. We show that any such solution $u$ growing moderately at infinity tends to $0$ as $|x|\to\infty$. Earlier we showed this theorem for the case $\gamma(x_n)=B$, i.e. for a cylindrical domain $Q=\Omega\times (0,\infty)$, $\Omega\subset\mathbb R^{n-1}$, and for the case when $A\le A_0$ with a constant $A_0$ sufficiently small. Here we admit any value of $A_0$. Our theorem is true even for the domain which is an outer part of a cone, and for the half-space $x_n>0$. Besides, we consider here more general operators $L$ with lower order terms. Notice that the new proof is quite different from those in our earlier works.