The behavior of the solutions of a nonlinear boundary value problem for a second order elliptic equation in an unbounded domain. (English. Russian original) Zbl 1008.35018
Trans. Mosc. Math. Soc. 2001, 125-147 (2001); translation from Tr. Mosk. Mat. O.-va 62, 136-161 (2001).
This paper is devoted to the behaviour of solutions of
\[
Lu:= \sum^n_{i,j=1} {\partial\over \partial x_i}\left (a_{ij}(x){\partial u\over \partial x_j}\right) =0
\]
in unbounded domains having various structures: cones, cylinders, paraboloids, and others, with a nonlinear boundary condition
\[
{\partial u\over\partial v}+b(x)|u|^{p-1}u=0 \quad\text{or}\quad {\partial u\over \partial v}- b(x)|u|^{p-1} u=0,
\]
where \(p>0\), \({\partial u\over \partial v}= \sum^n_{i,j=1} a_{ij}(x) {\partial u\over\partial x_j}\cos (n,x_i) \), \(b(x)\gtrless 0\), \(n\) is the outward pointing unit normal. The authors show that the behaviour of the solution essentially depends on the sign of the coefficient \(b(x)\). They prove a Phragmèn-Lindelöf type theorem, together with theorems on the existence of positive solutions and theorems on the lane of such solutions.
Reviewer: Messoud Efendiev (Berlin)
MSC:
35J25 | Boundary value problems for second-order elliptic equations |
35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
35J65 | Nonlinear boundary value problems for linear elliptic equations |
35B40 | Asymptotic behavior of solutions to PDEs |