Optimal Wloc2,2-regularity, Pohozhaevʼs identity, and nonexistence of weak solutions to some quasilinear elliptic equations

AbstractWe begin by establishing a sharp (optimal) Wloc2,2-regularity result for bounded weak solutions to a nonlinear elliptic equation with the p-Laplacian, Δpu=defdiv(|∇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∈W1,p(Ω) of the elliptic Neumann problem(P)−Δpu+W′(u)=f(x)in Ω;∂u/∂ν=0on ∂Ω. Here, Ω is a bounded domain in RN whose boundary ∂Ω is a C2-manifold, ν≡ν(x0) denotes the outer unit normal to ∂Ω at x0∈∂Ω, x=(x1,…,xN) is a generic point in Ω, and f∈L∞(Ω)∩W1,1(Ω). The potential W:R→R is assumed to be of class C1 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∈W1,p(Ω) (1