Harnack’s inequality for a nonlinear eigenvalue problem on metric spaces. (English) Zbl 1160.35406
Summary: We prove Harnack’s inequality for first eigenfunctions of the \(p\)-Laplacian in metric measure spaces. The proof is based on the famous Moser iteration method, which has the advantage that it only requires a weak (\(1,p\))-Poincaré inequality. As a by-product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35B45 | A priori estimates in context of PDEs |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
Keywords:
Caccioppoli estimate; doubling measure; first eigenvalue; first eigenfunction; Harnack’s inequality; metric space; minimizer; Newtonian space; nonlinear eigenvalue problem; Poincaré inequality; Rayleigh quotient; superminimizer; Sobolev spaceReferences:
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