On a p-Laplace equation with multiple critical nonlinearities
By Roberta Filippucci, Patrizia Pucci, F. Robert. 2008
155 citations recorded by the source corpus.
Abstract Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that − Δ p u − μ | x | − p u p − 1 = | x | − s u p ⋆ ( s ) − 1 + u p ⋆ − 1 admits a positive weak solution in R n of class D 1 p ( R n ) ∩ C 1 ( R n ∖ { 0 } ) , whenever μ μ 1 , and μ 1 = [ ( 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 ∈ D 1 p ( R n ) is a weak solution in R n of − Δ p u − μ | x | − p | u | p − 2 u = | x | − s | u | p ⋆ ( s ) − 2 u + | u | q − 2 u , then u ≡ 0 when either 1 q p ⋆ , or q > p ⋆ and u is also of class L loc ∞ ( R n ∖ { 0 } ) .
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/dbfd936ec7ed2d45bc0d706482cf4dd038b437c1
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In Applied Mathematics
In Computational Theory and Mathematics
In Mathematical Physics
Citations in the corpus
- Borderline Variational Problems Involving Fractional Laplacians and Critical Singularities (2015, methodology): “1) above is the non-local counterpart of the one studied by Filippucci-Pucci-Robert in [17], who treated the case of the p-Laplacian in an equation involving both the Sobolev and the Hardy-Sobolev critical exponents.”