Least Extremal Solutions of Ill-Posed Neumann Boundary Value Problem for Semilinear Elliptic Equations in L p

In this paper, by the methods of metric generalized inverse in Banach space and Schauder fixed point theorem, we prove the existence of the least extremal solution of an ill-posed Neumann boundary value problem for semilinear elliptic equations in L p and give a necessary and sufficient condition for a function to be the least extremal solution of the ill-posed Neumann boundary value problem.