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

Abstract

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