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 equationin the sub-natural growth case 0 < q < p − 1, where ∆ p (1 < p < ∞) is the p-Laplacian, and σ, µ are positive Borel measures on 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 (−∆) α in place of −∆ p , on R n for 0 < α < n 2 , and on an arbitrary domain Ω ⊂ R n with positive Green's function in the classical case α = 1.

Existence of minimal solutions to quasilinear elliptic equations with several sub-natural growth terms

“…Cao and Verbitsky [11] constructed weak solutions in W 1,p loc (R n ) under a certain capacity condition and gave two-sided pointwise estimates of such solutions. Seesanea and Verbitsky [23] gave a sufficient condition for the existence of L r -integrable p-superharmonic solutions. From existence of such solutions, behavior of the potentials is derived conversely.…”

“…The authors do not know the same statement even if σ = 0. Theorem 1.1 includes the existence theorems in [23] and [25] as the special cases µ = 0 and γ = 1. In general, our generalized solutions do not belong toẆ 1,p 0 (R n ), so we can not use the dual ofẆ 1,p 0 (R n ) to control interaction between σ and µ.…”

Solutions to sublinear elliptic equations with finite generalized energy

“…Moreover, arguing by induction, we see that minimality of such a solution follows immediately from its construction in Theorem 4.2 (cf. [19,Lemma 5.5]).…”

Wolff’s inequality for intrinsic nonlinear potentials and quasilinear elliptic equations

“…However, a similar problem for solutions u ∈ L r (R n ) with r < ∞ turned out to be more complicated. Some sharp sufficient conditions for that were established recently in [SV3] (see also [SV1], [SV2] where finite energy solutions and their generalizations are treated).…”