Existence of multiple ordered solutions for a semilinear elliptic boundary value problem

Abstract

We consider the problem of existence of multiple ordered solutions for the semi-linear elliptic boundary value problem

where L is a second order uniformly elliptic operator. We show that for the case where f exhibits a certain type of oscillatory behavior and g is related to f by a boundedness condition, there exist multiple ordered solutions of the Neumann problem, related to the falling zeroes of the function f. Imposing a monotonicity condition on f we obtain inclusion results for the solutions of the problem, both in the autonomous case with general boundary conditions and in the nonautonomous Neumann case. The proof of our result is based on the maximum principle and on some theorems of Matano [7, 8]

Keywords:

• Multiple ordered solutions
• inclusion results
• falling zeroes

Additional information

Notes on contributors

Melitta Fiebig-wittmaack

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