Relative Morse index and multiple solutions for a non-periodic Dirac equation with external fields

Abstract

This paper is concerned with the stationary solutions of the Dirac equation −i∑k=13αk∂ku+aβu+ωu+V(x)u=Gu(x,u),where G is asymptotically quadratic and is not assumed to be C2. We present a new approach to construct an index theory for the associated linear Dirac equation and define the relative Morse index to measure the difference between the nonlinearity at the origin and at infinity. By building upon the idea of combining the index theory and a generalized linking theorem, we obtain existence and multiplicity of stationary solutions.