Nonexistence of positive solutions for some quasilinear elliptic equations in strip-like domains

The nonexistence of positive solutions is discussed for -△_pu = a(x)u^q-1 in Ω, u|∂Ωn = 0, for the case where a(x) is a bounded positive function and Ω is a strip-like domain such as Ω = Ω_d x ℝ^N-d with Ω_d bounded in ℝ^d. The existence of nontrivial solution of (E) is proved by Schindler for q ∈ (p,p*) where p* is Sobolev's critical exponent. Our method of proofs for nonexistence rely on the "Pohozaev-type inequality" (for q ≥ p*); and on a new argument concerning the simplicity of the first eigenvalue for (generalized) eigenvalue problems combined with translation invariance of the domain (for q ≤ p).