Semi-stable and extremal solutions of reaction equations involving the $p$-Laplacian

Abstract

We consider nonnegative solutions of −∆pu = f (x, u), where p > 1 and ∆p is the p-Laplace operator, in a smooth bounded domain of R N with zero Dirichlet boundary conditions. We introduce the notion of semistability for a solution (perhaps unbounded). We prove that certain minimizers, or one-sided minimizers, of the energy are semi-stable, and study the properties of this class of solutions.Under some assumptions on f that make its growth comparable to u m , we prove that every semi-stable solution is bounded if m < mcs. Here, mcs = mcs(N, p) is an explicit exponent which is optimal for the boundedness of semistable solutions. In particular, it is bigger than the critical Sobolev exponent p * − 1.We also study a type of semi-stable solutions called extremal solutions, for which we establish optimal L ∞ estimates. Moreover, we characterize singular extremal solutions by their semi-stability property when the domain is a ball and 1 < p < 2.

Estimates and Rigidity for Stable Solutions to Some Nonlinear Elliptic Problems

“…Indeed, they showed that whenever n ≥ p + 4p/(p − 1), the function u(x) = log(|x| −p ) is a W 1,p (B 1 ) singular stable solution to (I.8), with Ω = B 1 and f (u) = p p−1 (n − p)e u . Stable solutions to (I.8) are known to be bounded in the optimal dimension range (I.9) also in the case of power-like nonlinearities and arbitrary domains, thanks to the result of Cabré and Sanchón [41], and in the radial case for every locally Lipschitz nonlinearity, as proved by Cabré, Capella, and Sanchón [33].…”

Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

“…Our first proposition establishes the analog of the classical results for (P λ ) in the Euclidean case. Using some ideas coming from Cabré and Sanchón [5] and Luo, Ye and Zhou [35] we prove the existence of a critical parameter λ * which is related with the resolvability of (P λ ).…”

“…They proved that for every p > 1 and h(s) = e s , the extremal solution u * is an energy solution for every dimension and that it is bounded in some range of dimensions. 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 this paper we investigate similar results in the inhomogeneous context of a Riemannian manifold. We use some ideas of [5], comparison principle for −∆ p (because it is uniformly elliptic) and the positivity of the first eigenvalue (as well the corresponding eigenfunction) of −∆ p on Ω (cf. [1,36,38]).…”

Some Liouville-type results for stable solutions involving the mean curvature operator: The radial case

“…where f ∈ C 1 (R) and N ≥ 2. Over the last two decades stable solutions of nonlinear PDEs has received a lot of attention, especially in the case of problems driven by the p-Laplacian operator (see for instance [1,7,9,10,17,3,12,5,2,4,11,6,16]). However the case involving the mean curvature operator has remained almost inexplored.…”