Entire solutions of sublinear elliptic equations in anisotropic media

Abstract

We study the nonlinear elliptic problem − Δ u = ρ ( x ) f ( u ) in R N ( N ⩾ 3 ), lim | x | → ∞ u ( x ) = ℓ , where ℓ ⩾ 0 is a real number, ρ ( x ) is a nonnegative potential belonging to a certain Kato class, and f ( u ) has a sublinear growth. We distinguish the cases ℓ > 0 and ℓ = 0 and prove existence and uniqueness results if the potential ρ ( x ) decays fast enough at infinity. Our arguments rely on comparison techniques and on a theorem of Brezis and Oswald for sublinear elliptic equations.