Wolff's inequality for intrinsic nonlinear potentials and quasilinear\n elliptic equations

2018/12/08 by Igor E. Verbitsky, Verbitsky, Igor E. · 1 citation
Computer Science · Mathematics · #42B37 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Primary 35J92 #Secondary 35J20

paper · pdf · doi:10.48550/arxiv.1812.03418

openalex publication_date 2018/12/08 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

We prove an analogue of Wolff's inequality for the so-called intrinsic\nnonlinear potentials associated with the quasilinear elliptic equation n-
Delta_p u =
sigma u^q
quad
textin
;
;
mathbbRⁿ, in the\nsub-natural growth case 0<q< p-1, where \Δ_pu = ÷( |\∇ν|^p-2 \∇ u ) is the p-Laplacian, and \σ is a nonnegative\nmeasurable function (or measure) on \ℝⁿ.\n As an application, we give a necessary and sufficient condition for the\nexistence of a positive solution u \∈ L^r(\ℝⁿ) (0<r<\∞)\nto this problem, which was open even in the case p=2.\n Our version of Wolff's inequality for intrinsic nonlinear potentials relies\non a new characterization of discrete Littlewood-Paley spaces f^p,\nq(\σ) defined in terms of characteristic functions of dyadic cubes in\n\ℝⁿ.\n

Cited by

• Some capacitary strong type inequalities and related function spaces

Related

• Wolff potential estimates for supersolutions of equations with generalized Orlicz growth
• Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms
• Positive solutions of quasilinear elliptic equations with Fuchsian potentials in Wolff class
• Nonlinear potential estimates for sublinear problems with applications to elliptic semilinear and quasilinear equations
• Semilinear elliptic problems in ℝ^N: the interplay between the potential and the nonlinear term