2014/03/07 by Cao Tien Dat, Igor E. Verbitsky
Mathematics · #Domain (mathematical analysis) #Elliptic curve #Energy (signal processing) #Energy method #Function (biology) #Geometric Analysis and Curvature Flows #Integrable system #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.1007/s00526-014-0722-0
19 pages, Calc. Var. Partial Differential Equations (2014)
openalex publication_date 2014/03/07 · arxiv created 2014/09/14 · arxiv updated 2014/09/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study finite energy solutions to quasilinear elliptic equations of the type -Δpu=σ uq in ℝn, where Δp is the p-Laplacian, p>1, and σ is a nonnegative function (or measure) on ℝn, in the case 0<q < p-1 ( below the "natural growth" rate q=p-1 ). We give an explicit necessary and sufficient condition on σ which ensures that there exists a solution u in the homogeneous Sobolev space L01,p(ℝn), and prove its uniqueness. Among our main tools are integral inequalities closely associated with this problem, and Wolff potential estimates used to obtain sharp bounds of solutions. More general quasilinear equations with the A-Laplacian div A(x,∇ ⋅) in place of Δp are considered as well.