Regularity of minimizers of semilinear elliptic problems up to dimension four

2009/09/25 by Xavier Cabré, Cabre, Xavier
Computer Science · Mathematics · #35B45 #35J61 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.0909.4696

openalex publication_date 2009/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the class of semi-stable solutions to semilinear equations -Δu=f(u) in a bounded smooth domain Ω of Rⁿ (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an priori L^∞ bound which holds for every positive semi-stable solution and every nonlinearity f. This estimate leads to the boundedness of all extremal solutions when n=4 and Ω is convex. This result was previously known only in dimensions n≤ 3 by a result of G. Nedev. In dimensions 5 ≤ n ≤ 9 the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case Ω=B_R by a result of A. Capella and the author.

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