Multiplicity of solutions of a nonlinear boundary problem with homogeneous neumann boundary conditions

Abstract

We arranged the problem μ″ + f(u)=g(x) where f is a real C² smooth function with n simple zeroes z₁,…,z_n and g is a C² smooth function wnich maps [0,π] in R and satisfies a certain smaiiness condition relative to f We show that if n is even problem (∗) has at least n−1 solutions; if n is odd, problem (∗) has at least n solutions if f′(z₁)<0, and at least n−2 solutions if f (Z)>0. We also obtain estimates of the range of these solutions and calculate the stable ones (stable as stationary solutions of an associated diffusion problem) with a simple numerical method

Keywords:

• One-dimensional boundary value problem
• number of solutions associated parabolic equation
• stability
• numerical method

AMS(MOS)::

• 34B15
• 35K60
• 65L10

^∗Supported in part by the SFB 123 of the University of Heidelberg, Rep Fed of Germany

^∗Supported in part by the SFB 123 of the University of Heidelberg, Rep Fed of Germany

Notes

^∗Supported in part by the SFB 123 of the University of Heidelberg, Rep Fed of Germany