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On the influence of second order uniformly elliptic operators in nonlinear problems
Author(s) -
Montenegro Marcos,
Moura Renato J.
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201300249
Subject(s) - mathematics , bounded function , domain (mathematical analysis) , order (exchange) , exponent , mathematical analysis , nonlinear system , elliptic curve , identity (music) , identity matrix , function (biology) , range (aeronautics) , critical exponent , type (biology) , pure mathematics , eigenvalues and eigenvectors , geometry , scaling , linguistics , philosophy , physics , materials science , finance , quantum mechanics , evolutionary biology , acoustics , economics , composite material , ecology , biology
The purpose of the present paper is to discuss the role of second order elliptic operators of the type L = ∑ i , j = 1 n D i ( a i j( x ) D j )on the existence of a positive solution for the problem involving critical exponent− L u=u n + 2 n − 2+ λ uinΩ ,u = 0on ∂ Ω ,where Ω is a smooth bounded domain in R n , n ≥ 3 , and λ is a real parameter. In particular, we show that if the function det ( A ( x ) )has an interior global minimum point x 0 such that A ( x ) is comparable to A ( x 0 ) + | x − x 0 | γ I n , where A ( x ) = ( a i j( x ) ) and I n is the identity matrix of order n , then the range of values of λ for which the problem above has a positive solution can change drastically from γ ≤ 2 to γ > 2 .