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

• Published in: Nonlinear differential equations and applications Vol. 22; no. 5; pp. 1067 - 1087
• Main Authors: Mihăilescu, Mihai, Stancu-Dumitru, Denisa, Varga, Csaba
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.10.2015
• Subjects: Analysis; Mathematics; Mathematics and Statistics; 47J10; 47J30; 35H10; Baouendi–Grushin type operator; Orlicz–Sobolev space; Variational methods; 49R05; 46E30; Hypoelliptic equation; Nonlinear eigenvalue problem
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-015-0314-5

We show that the spectrum of a nonhomogeneous Baouendi–Grushin type operator subject with a homogeneous Dirichlet boundary condition is exactly the interval ( 0 , ∞ ) . This is in sharp contrast with the situation when we deal with the “classical” Baouendi–Grushin operator (i.e., an operator of type - Δ x - | x | ξ Δ 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.