2016/09/18 by Marino Badiale, Michela Guida, Sergio Rolando · 2 citations
Mathematics · #math.AP #msc:35J92 #msc:35J20 #msc:46E35 #msc:46E30
paper · pdf · doi:10.1016/j.jmaa.2017.02.011
published as Journal of Mathematical Analysis and Applications 451 (2017), 345-370 · This document is an expanded and complementary version of arXiv:1510.03879, and continues the work of arXiv:1403.3803 and arXiv:1506.00056
arxiv created 2016/09/18 · arxiv updated 2018/06/05
Given 1<p<N and two measurable functions V( r) ≥ 0 and K( r) >0, r>0, we define the weighted spaces W=\ u∈ D1,p(ℝN):∫ℝNV( | x| ) | u| pdx<∞ \ , LKq=Lq(ℝN,K( | x| ) dx) and study the compact embeddings of the radial subspace of W into LK^q1+LK^q2, and thus into LKq (=LKq+LKq) as a particular case. We consider exponents q1,q2,q that can be greater or smaller than p. 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. We then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear p-Laplace equations of the form -\triangle pu+V( | x| ) |u|p-1u=g( | x| ,u) in ℝN, 1<p<N, where V and g( | ⋅ | ,u) with u fixed may be vanishing or unbounded at zero or at infinity. Both the cases of g super and sub p-linear in u are studied and, in the sub p-linear case, nonlinearities with g( | ⋅ | ,0) ≠ 0 are also considered.