On the spectrum of a Baouendi–Grushin type operator: an Orlicz–Sobolev space setting approach

Abstract

We show that the spectrum of a nonhomogeneous Baouendi–Grushin type operator subject with a homogeneous Dirichlet boundary condition is exactly the interval \({(0,\infty)}\) . This is in sharp contrast with the situation when we deal with the “classical” Baouendi–Grushin operator (i.e., an operator of type \({-\Delta_x-|x|^{\xi}\Delta_y}\)) when the spectrum is an increasing and unbounded sequence of positive real numbers. Our proofs rely on a symmetric mountain-pass argument due to Kajikiya. In addition, we can show that for each eigenvalue there exists a sequence of eigenfunctions converging to zero.