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Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators. (English) Zbl 1132.35032

The authors extend results of their previous paper [Adv. Differ. Equ. 11, No. 1, 91–119 (2006; Zbl 1132.35427)]. Precisely, define

G(x,u,u,D 2 u)=F(x,u,D 2 u)+b(x)·u|u| α +c(x)|u| α u,

where F is continuous on Ω× N ×S, S being the space of N×N symmetric matrices. Moreover, F satisfies, for all real number t and all non negative μ

F(x,tp,μX)=|t| α μF(x,p,X),

and for all p N {0}, MS, NS with N0, α>-1, 0<aA

a|p| α tr (N)F(x,p,M+N)-F(x,p,M)A|p| α tr (N)·

Moreover, suitable conditions about the uniform continuity of F(x,p,X) are assumed. The functions b:Ω N and c:Ω, 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

λ ¯=sup{λ:φ>0inΩ,G(x,φ,φ,D 2 φ)+λφ 1+α 0},

where the inequality is understood in the viscosity sense. The following results are proved.

(i) If f0 is bounded and continuous in Ω, and if λ<λ ¯ then there exists a nonnegative solution of the problem

G(x,u,u,D 2 u)+λu 1+α =finΩ,u=0onΩ·

(ii) There exists φ>0 in Ω such that φ is a viscosity solution of

G(x,φ,φ,D 2 φ)+λφ 1+α =finΩ,φ=0onΩ·

(iii) φ is γ-Hölder continuous for all γ(0,1) 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.

MSC:
35J60Nonlinear elliptic equations
35P30Nonlinear eigenvalue problems for PD operators; nonlinear spectral theory
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B50Maximum principles (PDE)
35B65Smoothness and regularity of solutions of PDE