Boundedness of the extremal solution for some p-Laplacian problems

Wei, Long

doi:10.2478/s12175-014-0211-0

Cited by 1 publication

(1 citation statement)

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“…For a more general non-linearity, Cabré and Sanchón [5] proved that every semi-stable solution is bounded for a explicit exponent which is optimal for the boundedness of semi-stable solutions and, in particular, it is bigger than the critical Sobolev exponent p * − 1. For general h(s) and p > 1 the interested reader can see [4,10,41,43] for more regularity results about the extremal solution. In [4], Cabré, Capella and Sanchón treated the delicate issue about regularity of extremal solutions u * of (1.7) at λ = λ * when Ω is the unit ball of R N .…”

Section: (12)mentioning

confidence: 99%

Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

Ó¹,

Clemente²

2019

Ann. Acad. Sci. Fenn. Math.

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We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equations involving the p-Laplace Beltrami operator with zero Dirichlet boundary conditions. 1 \B δ F (u) dv g , where F (x, t) =ˆt 0 f (x, s) ds.

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