Optimal [formula omitted]-regularity, Pohozhaevʼs identity, and nonexistence of weak solutions to some quasilinear elliptic equations

Abstract

We begin by establishing a sharp (optimal) W loc 2 , 2 -regularity result for bounded weak solutions to a nonlinear elliptic equation with the p-Laplacian, Δ p u = def div ( | ∇ u | p − 2 ∇ u ) , 1 < p < ∞ . We develop very precise, optimal regularity estimates on the ellipticity of this degenerate (for 2 < p < ∞ ) or singular (for 1 < p < 2 ) problem. We apply this regularity result to prove Pohozhaevʼs identity for a weak solution u ∈ W 1 , p ( Ω ) of the elliptic Neumann problem (P) − Δ p u + W ′ ( u ) = f ( x ) in Ω ; ∂ u / ∂ ν = 0 on ∂ Ω . Here, Ω is a bounded domain in R N whose boundary ∂ Ω is a C 2 -manifold, ν ≡ ν ( x 0 ) denotes the outer unit normal to ∂ Ω at x 0 ∈ ∂ Ω , x = ( x 1 , … , x N ) is a generic point in Ω, and f ∈ L ∞ ( Ω ) ∩ W 1 , 1 ( Ω ) . The potential W : R → R is assumed to be of class C 1 and of the typical double-well shape of type W ( s ) = | 1 − | s | β | α for s ∈ R , where α , β > 1 are some constants. Finally, we take an advantage of the Pohozhaev identity to show that problem (P) with f ≡ 0 in Ω has no phase transition solution u ∈ W 1 , p ( Ω ) ( 1 < p ⩽ N ), such that − 1 ⩽ u ⩽ 1 in Ω with u ≡ − 1 in Ω − 1 and u ≡ 1 in Ω 1 , where both Ω − 1 and Ω 1 are some nonempty subdomains of Ω. Such a scenario for u is possible only if N = 1 and Ω − 1 , Ω 1 are finite unions of suitable subintervals of the open interval Ω ⊂ R 1 .