Multiple solutions of nonlinear integro-differential equations

Abstract

The paper deals with the boundary value problem (1) $-\\Delta u=f(x,u,K(u))$ in $\\Omega$, (2) $u=g$ on $\\partial\\Omega$, where $\\Omega\\subset{\\bf R}^n$, $n\\geq 2$, is a bounded domain and $K(u)$ a nonlinear integral operator. The main attention of the author is concentrated on the existence of multiple solutions of the problem considered. By the constructive method the existence of a maximal (minimal) solution is proved, by an application of a variational equation the existence of multiple solutions is proved, and by an application of the topological degree arguments the existence of a multiple positive solution is proved. The paper contains some misprints and unclear statements.