The Brezis-Nirenberg problem for fractional elliptic operators
| Autor: | Marcos Montenegro, Ko-Shin Chen, Xiaodong Yan |
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| Rok vydání: | 2016 |
| Předmět: |
Pure mathematics
General Mathematics Operator (physics) 010102 general mathematics Zero (complex analysis) Boundary (topology) 01 natural sciences Dirichlet distribution 010101 applied mathematics Elliptic operator symbols.namesake Dirichlet eigenvalue Bounded function symbols 0101 mathematics Critical exponent Mathematics |
| Zdroj: | Mathematische Nachrichten. 290:1491-1511 |
| ISSN: | 0025-584X |
| DOI: | 10.1002/mana.201600072 |
| Popis: | Let L= div (A(x)∇) be a uniformly elliptic operator in divergence form in a bounded open subset Ω of Rn. We study the effect of the operator L on the existence and nonexistence of positive solutions of the nonlocal Brezis–Nirenberg problem (−L)su=un+2sn−2s+λuinΩ,u=0on∂Ω where (−L)s denotes the fractional power of −L with zero Dirichlet boundary values on ∂Ω, 0 2s and λ is a real parameter. By assuming A(x)≥A(x0) for all x∈Ω¯ and A(x)≤A(x0)+|x−x0|σIn near some point x0∈Ω¯, we prove existence theorems for any λ∈(0,λ1,s(−L)), where λ1,s(−L) denotes the first Dirichlet eigenvalue of (−L)s. Our existence result holds true for σ>2s and n≥4s in the interior case (x0∈Ω) and for σ>2s(n−2s)n−4s and n>4s in the boundary case (x0∈∂Ω). Nonexistence for star-shaped domains is obtained for any λ≤0. |
| Databáze: | OpenAIRE |
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