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

Abstract

We study the existence of positive solutions to quasilinear elliptic equations of the typein the sub-natural growth case 0 < q < p − 1, where ∆ p u = ∇ · (|∇u| p−2 ∇u) is the p-Laplacian with 1 < p < n, and σ and µ are nonnegative Radon measures on R n . We construct minimal generalized solutions under certain generalized energy conditions on σ and µ. To prove this, we give new estimates for interaction between measures. We also construct solutions to equations with several sub-natural growth terms using the same methods.✩ This is a pre-print of an article published in Nonlinear Analysis. The final authenticated version is available online at: https://doi.

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

“…Proof. As in the proof of [19,Corollary 3.6], using (6.1) and the Havin-Maz'ya potential estimate (see [20]) we find that Ω (W p,1 (θ dx)) . Combining (6.3) and (6.4), we obtain (6.2).…”

“…Hence u = W p,w µ ∈ H 1,p 0 (Ω; w). As in the proof of [19,Lemma 3.3], the Picone-type inequality in [1,7] yields the following estimate.…”

“…Let us define generalized p-energy by using minimal p-superharmonic solutions. For p = 2 or Ω = R n , we refer to [34,19].…”

“…, J. See the proof of[19, Corollary 6.3] for detail. Quasilinear PDE with L s,t coefficients Let us assume that µ ∈ M + 0 (Ω) is finite.…”

Trace inequalities of the Sobolev type and nonlinear Dirichlet problems

“…Their work shows a connection between the existence of positive solutions to (1.4), energy conditions of the type of (4.1) (or equivalent (1.2)) and certain weighted norm inequalities. An extension of their result to (p, w)-Laplace equations on bounded domains was given by the author [24] (see also [25]). The counterpart of the Cascante-Ortega-Verbitsky theorem was also presented.…”