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

We 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), 11α,β>1 are some constants. Finally, we take an advantage of the Pohozhaev identity to show that problem (P) with f≡0f≡0 in Ω has no phase transition solution  u∈W1,p(Ω)u∈W1,p(Ω) (1