Existence of multiple solutions to elliptic equations satisfying a global eigenvalue-crossing condition

Abstract

We study the multiplicity of solutions to the elliptic equation $Delta u+ f(x,u)=0$, under the assumption that f(x,u)/u crosses globally but not pointwise any eigenvalue for every x in a part of the domain, when u varies from $-infty$ to $infty$. Also we relax the conditions on uniform convergence of f(x,s)/s, which are essential in many results on multiplicity for asymptotically linear problems.

Keywords

• Index of critical points
• mountain pass type
• nonlinear elliptic equations
• multiplicity of solutions