Nonlinear Elliptic Equations in Unbounded Domains

Abstract

The author considers nonlinear elliptic second-order integro-differ- ential equations of the form $$ -\\sum_{i=1}^N (\\partial/\\partial x_i) A_i (x,u(x),\\nabla u(x))+F (x,u(x), (Ku)(x))=f(x) $$\n\n 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.