Boundedness of the extremal solutions in dimension 4

2012/06/27 by Villegas, Salvador
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1206.6233

Abstract

In this paper we establish the boundedness of the extremal solution u^* in dimension N=4 of the semilinear elliptic equation -Δu=λf(u), in a general smooth bounded domain Omega of R^N, with Dirichlet data u|_{∂ Ω}=0, where f is a C¹ positive, nondecreasing and convex function in [0,∞) such that f(s)/s→∞ as s→∞. In addition, we prove that, for N>=5, the extremal solution u^*∈ W^2,(N)/(N-2). This gives u^∗∈ L^(N)/(N-4), if N>=5 and u^*∈ H₀¹, if N=6.

Related

• Regularity of extremal solutions of semilinear elliptic problems with\n non-convex nonlinearities on general domains
• Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity
• Regularity of minimizers of semilinear elliptic problems up to dimension four
• Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities
• A new proof of the boundedness results for stable solutions to\n semilinear elliptic equations