Existence of solutions of certain quasilinear elliptic equations in ℝN without conditions at infinity

Abstract

This paper deals with conditions for the existence of solutions of the equations $$ - \sum\limits_{i = 1}^n {D_i A_i } (x,u,Du) + A_0 (x,u) = f(x), x \in \mathbb{R}^n ,$$ considered in the whole space ℝn, n ≥ 2. The functions A i (x, u, ξ), i = 1,…, n, A 0(x, u), and f(x) can arbitrarily grow as |x| → ∞. These functions satisfy generalized conditions of the monotone operator theory in the arguments u ∈ ℝ and ξ ∈ ℝn. We prove the existence theorem for a solution u ∈ W 1,ploc (ℝn) under the condition p > n.