Quasilinear equations with natural growth in the gradients in spaces of Sobolev multipliers

Document Type

Article

Publication Date

6-1-2018

Abstract

We study the existence problem for a class of nonlinear elliptic equations whose prototype is of the form - Δ pu= | ∇ u| p+ σ in a bounded domain Ω ⊂ Rn. Here Δ p, p> 1 , is the standard p-Laplacian operator defined by Δpu=div(|∇u|p-2∇u), and the datum σ is a signed distribution in Ω. The class of solutions that we are interested in consists of functions u∈W01,p(Ω) such that | ∇ u| ∈ M(W1,p(Ω) → Lp(Ω)) , a space pointwise Sobolev multipliers consisting of functions f∈ Lp(Ω) such that∫Ω|f|p|φ|pdx≤C∫Ω(|∇φ|p+|φ|p)dxC,for some C> 0. This is a natural class of solutions at least when the distribution σ is nonnegative and compactly supported in Ω. We show essentially that, with only a gap in the smallness constants, the above equation has a solution in this class if and only if one can write σ=divF for a vector field F such that |F|1p-1∈M(W1,p(Ω)→Lp(Ω)). As an important application, via the exponential transformation u↦v=eup-1, we obtain an existence result for the quasilinear equation of Schrödinger type -Δpv=σvp-1, v≥ 0 in Ω , and v= 1 on ∂Ω , which is interesting in its own right.

Publication Source (Journal or Book title)

Calculus of Variations and Partial Differential Equations

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