The authors extend results of their previous paper [Adv. Differ. Equ. 11, No. 1, 91–119 (2006; Zbl 1132.35427)]. Precisely, define
where is continuous on , being the space of symmetric matrices. Moreover, satisfies, for all real number and all non negative
and for all , , with , ,
Moreover, suitable conditions about the uniform continuity of are assumed. The functions and , are supposed to be continuous and bounded. Additional conditions are assumed to prove specific theorems.
The main object of the work is the following definition of eigenvalue
where the inequality is understood in the viscosity sense. The following results are proved.
(i) If is bounded and continuous in , and if then there exists a nonnegative solution of the problem
(ii) There exists in such that is a viscosity solution of
(iii) is -Hölder continuous for all and locally Lipschitz continuous.
To get the previous results, the authors prove a comparison principle and some boundary estimates which may have a own interest. Also, they prove regularity results which give the required relative compactness used to prove the main existence theorem.