On the principal eigenvalue of elliptic operators in \(\mathbb R^N\) and applications (Q2502549)

Let \(-L\) be a linear elliptic operator on a bounded and smooth domain \(\Omega \subset \mathbb R^n\) with \[ Lu = a_{ij}\partial _{ij}u+b_i(x)\partial_iu+c(x)u, \] where the usual summation convention is used. It is well-known that the knowledge of good estimates of the first Dirichlet eigenvalue (principal eigenvalue) \(\mu_1\) of the operator is important for many problems, for example for the positiveness or uniqueness of solutions of nonlinear Dirichlet problems. Let \(-L\) be a general elliptic operator defined in a domain \(\Omega \subseteq \mathbb R^n\). The aim of this paper are two different generalizations of the principal eigenvalue for unbounded domains, the first one \(\lambda_1\) was introduced by \textit{H. Berestycki, L. Nirenberg} and \textit{S. R. S. Varadhan} [Commun. Pure Appl. Math. 47, 47--92 (1994; Zbl 0806.35129)]: \[ \lambda_1(-L,\Omega ) = \sup \{ \lambda \;| \;\exists \;\phi \;\in C^2(\Omega) \cap C^1_{loc}(\overline{\Omega}), \phi > 0 \text{ and } (L+\lambda)\phi \leq 0 \text{ in } \Omega \} \] and the second one \(\lambda_1'\) was recently given by \textit{H. Berestycki, F. Hamel} and \textit{L. J. Roques} [Math. Biology 51, 75--113 (2005; Zbl 1066.92047)]: \[ \begin{multlined} \lambda_1'(-L,\Omega ) = \inf \{ \lambda \;| \;\exists \;\phi \;\in C^2(\Omega) \cap C^1_{loc}(\overline{\Omega})\cap W^{2,\infty}(\Omega), \phi > 0 \;\text{and}\\ -(L+\lambda) \phi \leq 0 \;\text{in} \;\Omega, \;\phi =0 \text{ on } \partial \Omega \text{ if } \partial \Omega \;\not= \emptyset \}. \end{multlined} \] The topic of the paper is to describe how \(\lambda_1,\lambda_1'\) are involved in nonlinear problems, to identify classes of operators, for which either equality or an inequality between both holds. Three open problems are also presented and a motivation is given for both the definitions.