Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms

Abstract

Abstract We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation - Δ p ⁢ u = σ ⁢ u q + μ on ⁢ ℝ n -\Delta_{p}u=\sigma u^{q}+\mu\quad\text{on }\mathbb{R}^{n} in the sub-natural growth case 0 < q < p - 1 {0<q<p-1} , where Δ p {\Delta_{p}} ( 1 < p < ∞ {1<p<\infty} ) is the p-Laplacian, and σ, μ are positive Borel measures on ℝ n {\mathbb{R}^{n}} . Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case 0 < q < 1 {0<q<1} are treated for the fractional Laplace operator ( - Δ ) α {(-\Delta)^{\alpha}} in place of - Δ p {-\Delta_{p}} , on ℝ n {\mathbb{R}^{n}} for 0 < α < n 2 {0<\alpha<\frac{n}{2}} , and on an arbitrary domain Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} with positive Green’s function in the classical case α = 1 {\alpha=1} .