J Austral Math Soc Ser A 55 pp386--402, 1993.

Saddle Point Theorem and Nonlinear Scalar Neumann Boundary Value Problems

M. N. Nkashama

Abstract

We are concerned with existence results for nonlinear scalar Neumann boundary value problems u¢¢+ g(x, u) = 0, u¢(0) = u¢(p) = 0 where g(x, u) satisfies Caratheodory conditions and is (possibly) unbounded. On the one hand we only assume that the function (sgn u)g(x, u) is bounded either from above or from below in some function space, and we impose conditions which relate the asymptotic behaviour of the function ò₀^p G(x, u)dx (for |u| large) with the first two eigenvalues of the corresponding linear problem (here G(x, u) = ò₀^u g(x, s)ds is the potential generated by g). On the other hand we consider cases where the function (sgn u)g(x, u) is unbounded. The potential G(x, u) is not necesarily required to satisfy a convexity condition. Our method of proof is variational, we make use of the Saddle Point Theorem.

1991 AMS Subject Classification: 34B15, 34B25

Browse the article

Authors

M. N. Nkashama

University of Alabama at Birmingham, Birmingham, Alabama 35294, U.S.A.