On a p--Laplace equation with multiple critical nonlinearities

2008/07/06 by Roberta Filippucci, Filippucci, Roberta, Patrizia Pucci +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.0807.0913

26 pages

openalex publication_date 2008/07/06 · arxiv created 2008/09/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the Mountain--Pass Theorem of Ambrosetti and Rabinowitz we prove that -Δ_p u-μ|x|^-pu^p-1=|x|^-su^\crits-1+u^\crit-1 admits a positive weak solution in \rn of class \dunp∩ C¹(\rn∖\0\), whenever μ<μ₁, and μ₁=[(n-p)/p]^p. The technique is based on the existence of extremals of some Hardy--Sobolev type embeddings of independent interest. We also show that if u∈\dunp is a weak solution in \rn of -Δ_p u-μ|x|^-p|u|^p-2u=|x|^-s|u|^\crits-2u+|u|^q-2u, then u≡0 when either 1<q<\crit, or q>\crit and u is also of class L^∞_\text\scriptsizeloc(\rn∖\0\).

Citations

• On the elliptic problems involving multi-singular inverse square potentials and multi-critical Sobolev–Hardy exponents

Cited by

• Global Compactness Result for a Brézis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator

Related

• On the critical p-Laplace equation
• On the classification of entire solutions to the critical p-Laplace equation
• A note on the classification of positive solutions to the critical p-Laplace equation in ℝⁿ
• Classification results for bounded positive solutions to the critical p-Laplace equation
• A new minimax principle and application to the p-Laplace equation