On the generalized principal eigenvalue of quasilinear operator: definitions and qualitative properties

Abstract

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 $$\begin{aligned} {\mathcal {K}}_V[u]:=-\,\Delta _p u +Vu^{p-1},\quad u \ge 0. \end{aligned}$$ This operator is a natural generalization of self-adjoint linear operators. If $$\Omega $$ 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 $${\mathcal {K}}_V$$ . Here we investigate the relation between three types of the generalized principal eigenvalue for $${\mathcal {K}}_V$$ 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 $${\mathcal {K}}_V$$ in $$\mathbb {R}^N$$ . 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.