Publication details
On the generalized principal eigenvalue of quasilinear operator: definitions and qualitative properties
| Authors | |
|---|---|
| Year of publication | 2019 |
| Type | Article in Periodical |
| Magazine / Source | CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS |
| MU Faculty or unit | |
| Citation | |
| web | https://doi.org/10.1007/s00526-019-1523-2 |
| Doi | http://dx.doi.org/10.1007/s00526-019-1523-2 |
| Keywords | ELLIPTIC-OPERATORS; POSITIVE SOLUTIONS; REGULARITY; INEQUALITIES; EXISTENCE; MAXIMUM |
| Description | The notions of generalized principal eigenvalue for linear second order elliptic operators in general domains introduced by Berestycki et al. (Commun Pure Appl Math 47:47-92, 1994) and Berestycki and Rossi (J Eur Math Soc (JEMS) 8:195-215, 2006, Commun Pure Appl Math 68:1014-1065, 2015) have become a very useful and important tool in analysis of partial differential equations. This motivates us for our study of various concepts of eigenvalue for quasilinear operator of the form KV[u]:=-Delta(p)u+V-u(p-1), u >= 0. This operator is a natural generalization of self-adjoint linear operators. If is a smooth bounded domain, we already proved in Nguyen and Vo (J Funct Anal 269:3120-3146, 2015) that the generalized principal eigenvalue coincides with the (classical) first eigenvalue of KV. Here we investigate the relation between three types of the generalized principal eigenvalue for KV on general smooth domain (possibly unbounded), which plays an important role in the investigation of their limits with respect to the parameters. We also derive a nice simple condition for the simplicity of the generalized principal eigenvalue and the spectrum of KV in RN. To these aims, we employ new ideas to overcome fundamental difficulties originated from the nonlinearity of p-Laplacian. We also discuss applications of the notions by examining some examples. |