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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 $$G(x,u,\nabla u, D^2u)=F(x,\nabla u, D^2u)+b(x)\cdot \nabla u\vert \nabla u\vert ^\alpha+c(x)\vert u\vert ^\alpha u,$$ where $F$ is continuous on $\Omega\times \bbfR^N\times S$, $S$ being the space of $N\times N$ symmetric matrices. Moreover, $F$ satisfies, for all real number $t$ and all non negative $\mu$ $$F(x,t p,\mu X)=\vert t\vert ^\alpha\mu F(x,p,X),$$ and for all $p\in \bbfR^N\setminus \{0\}$, $M\in S$, $N\in S$ with $N\ge 0$, $\alpha>-1$, $0<a\le A$ $$a\vert p\vert ^\alpha \mathrm{tr}(N)\le F(x,p,M+N)-F(x,p,M)\le A\vert p\vert ^\alpha \mathrm{tr}(N).$$ Moreover, suitable conditions about the uniform continuity of $F(x,p,X)$ are assumed. The functions $b:\Omega\to \bbfR^N$ and $c:\Omega\to \bbfR$, 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 $$\overline\lambda=\sup\{\lambda\in \bbfR:\exists \phi>0 \text{ in } \Omega,\ G(x,\phi,\nabla \phi,D^2\phi)+\lambda \phi^{1+\alpha}\le 0\},$$ where the inequality is understood in the viscosity sense. The following results are proved. (i) If $f\le 0$ is bounded and continuous in $\Omega$, and if $\lambda<\overline\lambda$ then there exists a nonnegative solution of the problem $$G(x,u,\nabla u, D^2u)+\lambda u^{1+\alpha}=f \text{ in }\ \Omega,\ u=0 \text{ on } \ \partial\Omega.$$ (ii) There exists $\phi>0$ in $\Omega$ such that $\phi$ is a viscosity solution of $$G(x,\phi,\nabla \phi, D^2\phi)+\lambda \phi^{1+\alpha}=f \text{ in }\ \Omega,\ \phi=0\ \text{ on }\ \partial\Omega.$$ (iii) $\phi$ is $\gamma$-Hölder continuous for all $\gamma\in (0,1)$ 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.

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
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