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Title: Nonlinear Elliptic Equations in Unbounded Domains
Authors: 蔡隆義
Tsai, Long-Yi
Contributors: 應數系
Date: 1990-03
Issue Date: 2018-09-25 16:21:56 (UTC+8)
Abstract: The author considers nonlinear elliptic second-order integro-differ- ential equations of the form $$ -\sum_{i=1}^N (\partial/\partial x_i) A_i (x,u(x),\nabla u(x))+F (x,u(x), (Ku)(x))=f(x) $$

in an exterior domain $G$ under Dirichlet boundary conditions. The boundary $\partial G$ is smooth and $\{A_1,\cdots, A_N\}$ satisfy the Leray-Lions conditions in the case $p=2$. The operator $K\: L_2 (G)\to L_2(G)$ is nonlinear, bounded, continuous and has a Fréchet derivative which is bounded on bounded subsets of $L_2(G)$. The function $f$ is assumed to belong to the dual space $H^{-1} (G)$. The author establishes the existence of weak solutions using a concept of weak $\varepsilon$-upper and $\varepsilon$-lower solutions. Examples are given in which the operator $K$ has the form $\int_G \varphi(x,y,u(y))\,dy$. This work represents a continuation of the author's previous papers [same journal 11 (1983), no. 1, 75–84; MR0692993; ibid. 14 (1986), no. 3, 163–177; MR0867950]. Mention must also be made of a paper by P. Hartman and G. Stampacchia [Acta Math. 115 (1966), 271–310; MR0206537] in which existence and regularity for these types of equations are studied using different methods.
Relation: Chinese Journal of Mathematics , Vol. 18, No. 1 , pp. 21-44
AMS MathSciNet:MR1052498
Data Type: article
Appears in Collections:[應用數學系] 期刊論文

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