Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δ p u = | u | q − 1 u , where p > 1 . If q > p − 1 then u ≡ 0 . On the other hand, if 0 < q < p − 1 and u ( x ) = o ( | x | p / ( p − q − 1 ) ) as | x | → ∞ , then again u ≡ 0 . If q = p − 1 then u ≡ 0 for all solutions with at most algebraic growth at infinity.