Fucik Spectrum, Sign-Changing, and Multiple Solutions for Semilinear Elliptic Boundary Value Problems with Resonance at Infinity1

In this paper, we use Fucik spectrum, ordinary differential equation theory of Banach spaces to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity, especially with resonance at infinity, and obtain new results on the existence of multiple solutions and sign-changing solutions with a weakening of the P.S. condition. In one case we get up to seven nontrivial solutions. The techniques have independent interest.