Compactness and existence results in weighted Sobolev spaces of radial functions, Part I: Compactness

2014/03/15 by Marino Badiale, Michela Guida, Sergio Rolando · 4 citations
Mathematics · #math.AP #math.FA #msc:35J05 #msc:35J20 #msc:35J60 #msc:46E30 #msc:46E35

paper · pdf · doi:10.1007/s00526-015-0817-2

published as Calculus of Variations and Partial Differential Equations 54 (2015), 1061-1090 · 42 pages, 5 figures

arxiv created 2014/03/15 · arxiv updated 2016/12/08

Abstract

Given two measurable functions V(r)≥ 0 and K(r)> 0, r>0, we define the weighted spaces H_V¹ = \u ∈ D^1,2(ℝ^N): ∫_ℝ^NV(|x|)u²dx < ∞ \, L_K^q = L^q(ℝ^N,K(|x|)dx) and study the compact embeddings of the radial subspace of H_V¹ into L_K^q₁+L_K^q₂, and thus into L_K^q (=L_K^q+L_K^q) as a particular case. Both super- and sub-quadratic exponents q₁, q₂ and q are considered. Our results do not require any compatibility between how the potentials V and K behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately. Applications to existence results for nonlinear elliptic problems like -\triangle u + V(|x|)u = f(|x|,u) inℝ^N, u ∈ H_V¹, will be given in a forthcoming paper.

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