Solutions in Lebesgue spaces to nonlinear elliptic equations with subnatural growth terms

Abstract

We study the existence problem for positive solutions u ∈ L r (R n), 0 < r < ∞, to the quasilinear elliptic equation −∆ p u = σu q in R n in the sub-natural growth case 0 < q < p − 1, where ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian with 1 < p < ∞, and σ is a nonnegative measurable function (or measure) on R n. Our techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators in place of ∆ p such as the A-Laplacian divA(x, ∇u), or the fractional Laplacian (−∆) α on R n , as well as linear uniformly elliptic operators with bounded measurable coefficients div(A∇u) on an arbitrary domain Ω ⊆ R n with a positive Green function.

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 to Ẇ 1,p 0 (R n ), so we can not use the dual of Ẇ 1,p 0 (R n ) to control interaction between σ and µ.…”

“…Remark 3.7. The inequality (3.5) gives a refinement of [23,Corollary 1.2]. Suppose that 1 < p < n, 0 < q < p − 1 and 0 < γ < ∞.…”

Quasilinear elliptic equations with sub-natural growth terms in bounded domains

“…Recently, Verbitsky and his colleagues studied the problem of existence of solutions to elliptic equations related to (1.1) and presented some criteria (see [8,9,10,17,16,31,32,33,34,37,38]). In their study, they treated the cases of Ω = R n , or p = 2.…”