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| NONLINEAR BOUNDARY PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS |
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Citation： |
Zheng Songmu.NONLINEAR BOUNDARY PROBLEMS WITH NONLOCAL BOUNDARY CONDITIONS[J].Chinese Annals of Mathematics B,1983,4(2):177~186 |
| Page view： 693
Net amount： 678 |
Authors： |
Zheng Songmu; |
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| Abstract： |
By means of the supersolutioh and subsolution method and monotone iteration technique, the following nonlinear elliptic boundary problem with the nonlocal boundary
conditions
$$\[\left\{ {\begin{array}{*{20}{c}}
{Lu = - \sum\limits_{i,j = 1}^n {\frac{\partial }{{\partial {x_i}}}({a_{ij}}(x)\frac{{\partial u}}{{\partial {x_j}}}) = f(x,u)} }\{u{|_\Gamma } = const(unknown), - \int { - \sum\limits_{i,j = 1}^n {{a_{ij}}(x)\frac{{\partial u}}{{\partial {x_j}}}\cos (n,{x_i})ds = 0} } }
\end{array}} \right.\]$$
is considerd. The sufficient conditions which ensure at least one solution are given. Furthermore, the estimate of the first nonzero eigenvalue for the following linear eigenproblem
$$\[\left\{ {\begin{array}{*{20}{c}}
{L\varphi = \lambda \varphi }\{\varphi {|_\Gamma } = const(unknown), - \int {\sum\limits_{i,j = 1}^n {{a_{ij}}(x)\frac{{\partial u}}{{\partial {x_j}}}\cos (n,{x_i})ds} = 0} }
\end{array}} \right.\]$$
is obtained, that is
$$\[{\lambda _1} \ge \frac{{2\alpha }}{{n{d^2}}}\]$$ |
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