On Asymptotics of Solutions of Nonlinear Second Order Elliptic Equations in Cylindrical Domains

Abstract

We consider the equation

$$ L(u) - {a_0}{\left| u \right|^{{p - 1}}}u \equiv \sum\limits_{{i,j = 1}}^n {\frac{\partial }{{\partial {x_i}}}\left( {{a_{{ij}}}\left( {x'} \right)\frac{{\partial u}}{{\partial {x_i}}}} \right) + \sum\limits_{{j = 1}}^n {{a_j}\left( {x'} \right)\frac{{\partial u}}{{\partial {x_j}}} - {a_0}{{\left| u \right|}^{{p - 1}}}u = 0} } $$

(1)

where x=(x ₁, ..., x _n),x’ = (x ₁, ..., x _n-1),

$$ {m_1}\left| \xi \right| \leqslant \sum\limits_{{i,j = 1}}^n {{a_{{ij}}}} \left( {x'} \right){\xi_i}{\xi_j} \leqslant {m_2}{\left| \xi \right|^2},\;{m_1},{m_2} = const >0 $$

,

$$ {a_{{ij}}}\left( {x'} \right) = {a_{{ij}}},\;{a_{{nn}}} \equiv 1,\;{a_{{in}}} \equiv 0 $$

for

$$ i<n,\;{a_0} = connst >1,\;p = const >1 $$

.